Question posted on August 12, 2018
We consider the product P of the first N primes.
We split this product into two factors A and B
so that P = AB and A > B.
We consider the distance d = A – B between A and B.
What is the splitting that gives the smallest distance d?
Is there a pattern?
I have just posted my first observations about this
in the following document:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
With best regards, Jean-Claude
Dear Jean,
If P = p₁p₂p₃....pn.
Let A=∏ipi and B=(P/A) be a product decomposition of P.
Let S={(A,B);P=AB where A and B are a product decomposition}
How many decomposition we have?
The answer is 2ⁿ -1.
Now, consider the distance function d(A,B)=|A-B|.
If n is finite, then, Minimum d(A, B), A, B ∈ S
exists.
( Hint: (i) The minimum maybe 1, (ii) d has a maximum too)
One need to design a program for this purpose.
No explicit formula guaranteed. This fact is a direct
consequence of the irregular distribution of primes.
Best regards
A challenging problem!
As 2 is the unique even prime number, it is obvious that one number (A or B) is even and the other is odd. Therefore, D ≥ 1.
For example, if N = 2, then P = 2*3. We can choose A = 2, B = 3, so D = B – A = 1.
If N = 3, then P = 2*3*5. If A = 2*3 = 6 and B = 5, so D = A – B = 1.
If N = 4, then P = 2*3*5*7. If A = 2*7 = 14 and B = 3*5 = 15, so D = B – A = 1. If N = 5, then P = 2*3*5*7*11. In this case, we can consider that A = n1*n2, with n1 < n2 and B = P/A. There are 4+3+2+1 = 10 cases. The minimum D is obtained for A = 5*11 = 55 and B = 2*3*7 = 42, so D = 13.
If N = 6, then P = 2*3*5*7*11*13. In this case, we can consider that A = n1*n2*n3, with n1 < n2 < n3 and B = P/A. If we compute all the possible values of A, the B and D can be obtained easily. It is sufficient to retain the case with the minimum value of D.
For N ≥ 7, one can write a program based on a similar method. It’s the only approach I can imagine right now.
I thank Issam for his answer posted 3 hours ago.
It is a good idea to consider the total number K
of ways to split P into A and B.
We can add 1 to Issam’s number
to include the case A = P and B = 1.
Then we have to divide this number by 2,
because of my condition A > B.
For every splitting, the distance d = A – B
is either 1 or a product of primes
that are all greater than the N-th prime
possibly just one such prime,
and possibly exactly the smallest of them.
This may be a way to find at least K - 1 new primes,
at the possible high cost that when the distance
is a product of more than one prime,
we have to factor the distance to find new primes.
To find whether there is a splitting that gives 1,
we can use the quadratic formula for the computation of A
that I have posted in my document
Splitting in Euclid's proof to find the next prime
https://www.researchgate.net/publication/326971510_Splitting_in_Euclid's_proof_to_find_the_next_prime
replacing q by 1 in that formula.
It is faster to first check whether the discriminant
is a perfect square, as suggested by Issam
in another answer posted on August 11, 2018
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem#view=5b6f02e7d7141b40d6262023
that was an answer to my other related question.
To determine whether the distance is exactly the next prime,
we can use Issam’s theorem posted on July 27, 2018:
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem#view=5b5b0af7979fdc9de53c53b8
I will write some observations concerning
how we can try to choose what factors go into A
(and of course, this gives what factors go into B)
in order to obtain the minimum distance d = A – B.
I will post these observations as soon as possible
in my document
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
There is no problem with the maximum distance:
It is obtained with A = P and B = 1.
The maximum distance is P – 1.
With best regards, Jean-Claude
Answer posted on August 13, 2018
I thank Gheorghe a lot for his answer and all the computation.
His computation agrees 100% with the computation
that I posted yesterday in my document
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
I also solved and posted the cases of the first 6
and the first 7 primes yesterday.
Gheorghe’s idea that to obtain the minimum distance,
we may first try to consider the case
where about one half the first N primes
go into A is wonderful.
It is a very interesting conjecture.
I will try to solve and post the case of the first 8 primes today.
With best regards, Jean-Claude
Jean-Claude,
In fact, I suggest to use k prime numbers to compute A = ni1*ni2*...nik with ni1
I thank Gheorghe a lot for his second answer and the clarification.
It is correct that I misunderstood his first answer.
His method is easy to use and saves a lot of time.
I will keep trying to improve my first draft and first results.
If I find any improvements, I will post an updates in the following document:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
With best regards, Jean-Claude
Answer posted on August 14, 2018
I thank Gheorghe a lot for his comment posted 2 hours ago.
There is no doubt that a more theoretical approach
not only “should” but in addition “must” be considered.
It is usual in sciences to first be curious about a question,
to first make some naïve observations,
and then construct better and better tools
to study and answer the question.
I have started a list of reference to articles
with some relations to my question
at the end of the following document:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
There are good chances that there are some relations
between my question and some of these articles.
In this document, I have not yet had time to explain
the naïve method that I have been using so far
to find the minimum distance d = A – B.
I will explain my high school method as soon as possible.
I have been extremely surprised by the choice of factors
for A, and then automatically B,
in the case of the N = 9 first primes, from 2 to 23.
My current question originated from my previous question:
Is there a published improvement of Euclid’s Theorem?
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem
Both questions are strongly related to the following
high school problem:
Given the product P and the sum S of two real numbers x and y
find x and y.
This is immediately solved by the quadratic formula,
which, as it is usual with degree 2, gives
“two” solutions: (x, y) and (y, x),
that is, the same {x, y}.
The next question is when P and S are integers
find in what cases x and y are also integers.
As observed by Issam,
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem#view=5b6f02e7d7141b40d6262023
we must first check when the discriminant
of the quadratic equation is a perfect square.
I have seen this irritating high school problem
a lot of times in Number Theory.
Of course, the authors of research articles
in Number Theory do not want to look ridiculous
with a high school problem.
It would be interesting to check all the different ways
the authors of articles containing this irritating
high school question have dealt with it.
I am very slow to improve my document
about the current question.
My first priority is to finish my book of logic:
Construction of my book of logic
https://www.researchgate.net/publication/325861861_Construction_of_my_book_of_logic
I also have to dramatically improve this document
that describes what book I am constructing:
I want my book of logic to be rigorous AND gentle
AND restricted to what is used the most by users,
learners, and teachers of mathematics.
The main difficulty of the writing of this book
is in the first AND.
To make clear that there is no book like mine
at the present time, I have started to write a list
of all the existing books of logic, set theory,
class theory and foundations of mathematics:
List of books on the foundations of mathematics
https://www.researchgate.net/publication/326356380_List_of_books_on_the_foundations_of_mathematics
Everyone is very welcome to check whether
there is a book on this list that is rigorous AND gentle
AND time saving for everyone interested.
I do pretend that there is none.
Thanks to the hard work of many mathematicians
the foundations of mathematics have become
not only very beautiful, but in addition very useful,
but nobody has a fast access to this wonderful
part of mathematics until my book
will finally be finished and published.
Until then, only a few mathematicians
can see this wonderful mathematical paradise.
With best regards, Jean-Claude
Answer posted on August 14, 2018
I have tried for one hour to upload my new updated version of
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
When I click on “Upload” it stays on “Upload”.
I have clicked on “Upload” for one hour.
The screen stays on “Upload”.
It is very disgusting to have waste time like this.
Before this, I have uploaded new versions
of my documents without any problem.
It is the first time that this happens.
So, at the present time, the space for this document
is still here, but it is empty, because before uploading
we must first remove the old version.
So, I have removed the old version
but it is impossible to upload the new version.
I have solved the case N = 10
corresponding to the first 10 primes from 2 to 29.
The minimum distance for d = A – B is 1811.
I will try to upload again tomorrow.
With best regards, Jean-Claude
Answer posted on August 15, 2018
I thank Peter a lot for his answer posted 12 hours ago.
I will remember Évariste’s birth date forever.
Perhaps the birth dates of mathematicians
who worked around that time
will give some answers to the other cases of my question:
1736—1813: Joseph-Louis Lagrange
1746—1818: Gaspard Monge
1749—1827: Pierre-Simon Laplace
1749—1822: Jean Baptiste Joseph Delambre
1752—1833: Adrien-Marie Legendre
1755—1836: Marc-Antoine Parseval
1765—1825: Johann Friedrich Pfaff
1765—1822: Paolo Ruffini
1766—1826: John Farey Sr.
1768—1830: Joseph Fourier
1768—1822: Jean-Robert Argand
1776—1831: Sophie Germain
1777—1859: Louis Poinsot
1777—1855: Carl Friedrich Gauss
1778—1853: Józef Maria Hoene-Wroński
1781—1840: Siméon Denis Poisson
1781—1848: Bernard Bolzano
1784—1846: Friedrich Bessel
1784—1873: Charles Dupin
1785—1836: Claude-Louis Navier
1786—1856: Jacques Philippe Marie Binet
1788—1827: Augustin-Jean Fresnel
1788—1867: Jean-Victor Poncelet
1789—1857: Augustin-Louis Cauchy
1790—1868: August Ferdinand Möbius
1792—1856: Nikolai Lobachevsky
1793—1880: Michel Chasles
1793—1841: George Green
1796—1878: Irénée-Jules Bienaymé
1796—1863: Jakob Steiner
1798—1867: Karl Georg Christian von Staudt
1801—1883: Joseph Plateau
1802—1829: Niels Henrik Abel
1802—1860: János Bolyai
1803—1882: Christian Heinrich von Nagel
1803—1855: Jacques Charles François Sturm
1804—1851: Carl Gustav Jacob Jacobi
1805—1859: Peter Gustav Lejeune Dirichlet
1805—1865: William Rowan Hamilton
1806—1843: Robert Murphy
1806—1871: Augustus De Morgan
1809—1877: Hermann Grassmann
1809—1882: Joseph Liouville
1809—1880: Benjamin Peirce
1810—1893: Ernst Kummer
1811—1832: Évariste Galois
1811—1874: Otto Hesse
1814—1894: Eugène Charles Catalan
1814—1897: James Joseph Sylvester
1815—1897: Karl Weierstrass
1819—1903: Sir George Stokes
1821—1895: Arthur Cayley
1821—1894: Pafnuty Chebyshev
1823—1892: Enrico Betti
1823—1852: Gotthold Eisenstein
1823—1891: Leopold Kronecker
1826—1866: Bernhard Riemann
1830—1903: Luigi Cremona
1831—1916: Richard Dedekind
1831—1889: Paul du Bois-Reymond
1831—1901: Peter Tait
1832—1903: Rudolf Lipschitz
1832—1910: Eugène Rouché
1832—1918: Peter Ludwig Mejdell Sylow
https://en.wikipedia.org/wiki/Niels_Henrik_Abel
https://en.wikipedia.org/wiki/Jean-Robert_Argand
https://en.wikipedia.org/wiki/Friedrich_Bessel
https://en.wikipedia.org/wiki/Enrico_Betti
https://en.wikipedia.org/wiki/Jacques_Philippe_Marie_Binet
https://en.wikipedia.org/wiki/Paul_du_Bois-Reymond
https://en.wikipedia.org/wiki/János_Bolyai
https://en.wikipedia.org/wiki/Bernard_Bolzano
https://en.wikipedia.org/wiki/Eugène_Charles_Catalan
https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy
https://en.wikipedia.org/wiki/Arthur_Cayley
https://en.wikipedia.org/wiki/Michel_Chasles
https://en.wikipedia.org/wiki/Pafnuty_Chebyshev
https://en.wikipedia.org/wiki/Luigi_Cremona
https://en.wikipedia.org/wiki/Richard_Dedekind
https://en.wikipedia.org/wiki/Paul_du_Bois-Reymond
https://en.wikipedia.org/wiki/Jean_Baptiste_Joseph_Delambre
https://en.wikipedia.org/wiki/Augustus_De_Morgan
https://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet
https://en.wikipedia.org/wiki/Charles_Dupin
https://en.wikipedia.org/wiki/Gotthold_Eisenstein
https://en.wikipedia.org/wiki/John_Farey_Sr.
https://en.wikipedia.org/wiki/Joseph_Fourier
https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel
https://en.wikipedia.org/wiki/Évariste_Galois
https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
https://en.wikipedia.org/wiki/Sophie_Germain
https://en.wikipedia.org/wiki/Hermann_Grassmann
https://en.wikipedia.org/wiki/George_Green_(mathematician)
https://en.wikipedia.org/wiki/William_Rowan_Hamilton
https://en.wikipedia.org/wiki/Otto_Hesse
https://en.wikipedia.org/wiki/Carl_Gustav_Jacob_Jacobi
https://en.wikipedia.org/wiki/Leopold_Kronecker
https://en.wikipedia.org/wiki/Ernst_Kummer
https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange
https://en.wikipedia.org/wiki/Pierre-Simon_Laplace
https://en.wikipedia.org/wiki/Adrien-Marie_Legendre
https://en.wikipedia.org/wiki/Joseph_Liouville
https://en.wikipedia.org/wiki/Rudolf_Lipschitz
https://en.wikipedia.org/wiki/Nikolai_Lobachevsky
https://en.wikipedia.org/wiki/August_Ferdinand_Möbius
https://en.wikipedia.org/wiki/Gaspard_Monge
https://en.wikipedia.org/wiki/Augustus_De_Morgan
https://en.wikipedia.org/wiki/Robert_Murphy_(mathematician)
https://en.wikipedia.org/wiki/Claude-Louis_Navier
https://en.wikipedia.org/wiki/Christian_Heinrich_von_Nagel
https://en.wikipedia.org/wiki/Marc-Antoine_Parseval
https://en.wikipedia.org/wiki/Benjamin_Peirce
https://en.wikipedia.org/wiki/Johann_Friedrich_Pfaff
https://en.wikipedia.org/wiki/Joseph_Plateau
https://en.wikipedia.org/wiki/Louis_Poinsot
https://en.wikipedia.org/wiki/Siméon_Denis_Poisson
https://en.wikipedia.org/wiki/Jean-Victor_Poncelet
https://en.wikipedia.org/wiki/Bernhard_Riemann
https://en.wikipedia.org/wiki/Paolo_Ruffini
https://en.wikipedia.org/wiki/Jakob_Steiner
https://en.wikipedia.org/wiki/Karl_Georg_Christian_von_Staudt
https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet
https://en.wikipedia.org/wiki/Jacques_Charles_François_Sturm
https://en.wikipedia.org/wiki/Peter_Tait_(physicist)
https://en.wikipedia.org/wiki/Christian_Heinrich_von_Nagel
https://en.wikipedia.org/wiki/Karl_Georg_Christian_von_Staudt
https://en.wikipedia.org/wiki/Peter_Ludwig_Mejdell_Sylow
https://en.wikipedia.org/wiki/James_Joseph_Sylvester
https://en.wikipedia.org/wiki/Karl_Weierstrass
https://en.wikipedia.org/wiki/Józef_Maria_Hoene-Wroński
With best regards, Jean-Claude
Dear Jean,
I think Peter's answer was to show a nice observation related to
your statement
[ The minimum distance for d = A – B is 1811. ]
He connected the number 1811 with Galois' year of birth .-:).
Best regards
Answer posted on August 16, 2018
I thank Issam for his answer posted 5 hours ago.
It is crystal clear that Peter’s comment posted 18 hours ago
is about my result d = 1811 when N = 10
posted just above his comment.
Peter’s comment is wonderful, because we never have
too many opportunities to learn the history of mathematics.
Since the birth date of Évariste Galois gives the answer
to the case N = 10, I thought that we should look at
the birth dates of mathematicians who were contemporaries
of Évariste to find the answers for the other values of N.
Before looking at these birth dates, I was wrong
on several points. For example, I thought that
Carl Friedrich Gauss lived before Évariste.
This was wrong: The lifespan of Carl covers 100%
of the short lifespan of Évariste.
The life of Évariste was doubly horribly dramatic:
First, the publication of his work was blocked
by the most powerful mathematician of that time
in France, namely, Augustin-Louis Cauchy.
It turned out that contrary to what this blockage
means, namely, “Galois Theory is garbage”,
it turned out that not only this was wrong,
but horribly wrong. The work of Évariste
was not only good, not only very good,
not only excellent, not only outstanding:
It was a major historical event in the history
of mathematics. And even much more than this:
https://en.wikipedia.org/wiki/Galois_theory#Aftermath
https://en.wikipedia.org/wiki/List_of_things_named_after_Évariste_Galois
With best regards, Jean-Claude
I confirm D=1811 obtained for N=10, A=2*7*13*19*23, B=P/A, D=|A-B|.
With the same program, for N=11 I obtained Dmin=1579 with A=3*11*19* 23*31, B=P/A, D=|A-B|.
For N=12, D=18859 obtained for A=2*5*11*23*29*37, while for N=13, D=95533 obtained for A=3*13*17*23*31*37.
Answer posted on August 16, 2018
I thank Gheorghe a lot for his answer and all his computation
for the cases N = 10 and N = 11 posted 12 minutes ago.
Yesterday, I posted the same results in the document:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
As always in sciences, it is very important
to have same results obtained independently.
I have not yet been able to solve the case
of the N = 12 first primes from 2 to 37.
I am trying to improve my method.
With best regards, Jean-Claude
Answer posted on August 16, 2018
I have just finished to solve the case N = 12,
that is, the case of all the 12 primes from 2 to 37.
I have posted my result in the document
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
I have found a faster method.
I will first test it for N = 13,
that is, all the primes from 2 to 41,
and if my new method is OK,
I will explain it in the above document,
I hope tomorrow.
With best regards, Jean-Claude
Answer posted on August 16, 2018
I have just finished to solve the case N = 13,
that is, the case of all the 13 primes from 2 to 41.
I have posted my result in the document
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
I have found a faster method.
I will write it in the above document as soon as possible.
This writing will take at least one day.
With best regards, Jean-Claude
Observations:
Based on the sequence of values of the difference D
as mentioned by Jean and Gheorghe.
1,1,13,17,1,41,
157= 157
1811= 1811
18859= 18859
95533= 83×1151
17659= 17659
1995293= 1995293
We observe that
(1) All available D's are primes with one exception 95533= 83×1151.
(2) The values of the sequence D are not montone.
One can raise the following questions:
(1) What is the probability that D is a prime number?
(2) Is there any relation between the distinct values for D?
Conjecture 1. p(D is prime)>1/2
Conjecture 2. gcd(Di,Dj)=1, for i ≠ j.
More values of D may reveal some new observations.
Best
Answer posted on August 17, 2018
I am very grateful to Gheorghe, Issam, and Peter
for their answers, computation, and comments.
I am very impressed that Gheorghe solved
the cases N = 14 and N = 15, especially the
case N = 15, which is so much more difficult
than the previous cases.
It would be very helpful for me if
Gheorghe, Issam, Peter, and Wulf Rehder
would agree to write a joint publication with me containing:
What is the splitting P = AB of the product P of the first N primes
that gives the smallest distance between A and B?
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B
Is there a published improvement of Euclid’s Theorem?
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
Splitting in Euclid's proof to find the next prime
https://www.researchgate.net/publication/326971510_Splitting_in_Euclid's_proof_to_find_the_next_prime
I could work only part time on this possible joint publication,
because I really must finish my book of logic as soon as possible.
It is really a fact that this book will save a lot time and will be
very useful to all users, learners, and teachers of mathematics.
At the present time, all of them are under torture, because
they all need a book of logic that is rigorous, AND gentle,
AND restricted to what is used the most.
On the other hand, it is obvious that a publication on P = AB
would be published a lot faster and will be much better
if we are 5 to work part time on it, rather than if do it alone.
I am late to update my document
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
I will do my best to improve it as soon as possible.
With best regards, Jean-Claude
For N=16, A=2*7*13*23*29*31*37*41, D=208303.
For N=17, A=3*11*13*29*31*41*47*59, D=2396687.
For N=18, A=2*7*11*23*29*31*41*43*61, D=58 513 111.
For N=19, A=5*7*19*23*29*43*47*53*59, D=299 808 329.
For N=8, A=2*7*13*17=3094, B=3*5*11*19=3135, D=B-A=41.
For N=9, A=2*17*19*23=14858, B=3*5*7*11*13=15015, D=B-A=157.
Answer 27 posted on August 18, 2018
I thank Gheorghe a lot for his last 3 answers.
I am very impressed that he solved the cases
from N = 16 to N =19 so fast.
All his new results will help a lot to check several conjectures.
So far, I have solved only the cases from N = 1 to N = 13,
that I posted on August 16 in the document
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
My results for N = 8 and 9 are the same as Gheorghe’s
results. It is a lot safer to have this double independent
checking.
Peter’s idea to look at the ratio A/B is very important.
It gave me several new ideas.
I have to check whether any of my new ideas is useful.
I will keep improving the above-mentioned document
in several directions, including ideas related to Peter’s idea
and all Gheorghe’s new results.
I hope that Gheorghe, Issam, Peter, and Wulf will agree
to write a joint publication with me on all this.
With best regards, Jean-Claude
For N=20, A=11*13*23*29*31*43*47*59*67=23622001517543, B = 23619540863730 and D=2460653813.
For N=21, A=11*19*37*41*47*53*59*61*71=201811443968167, B=201820470625410 and D=9026657243.
Thank you, Jean-Claude, for your proposal. I agree to write a joint publication on the splitting P = AB of the product P of the first N primes that gives the smallest distance between A and B. I imagine we have different approaches for this problem.
Answer 32 posted on August 19, 2018
I thank Peter a lot for the answer 28 posted on August 18, 2018,
and for all the explanations.
Peter’s additional way to see my question is very original
and may lead to computer programs for my question
that are already available.
Here are some links related to Peter’s additional way:
https://en.wikipedia.org/wiki/Packing_problems
http://www.packomania.com/
https://www2.stetson.edu/~efriedma/packing.html
http://mathworld.wolfram.com/SquarePacking.html
https://en.wikipedia.org/wiki/Optimization_problem
https://en.wikipedia.org/wiki/NP_(complexity)
https://en.wikipedia.org/wiki/NP-completeness
https://en.wikipedia.org/wiki/List_of_NP-complete_problems
One-Dimensional Bin-Packing Problems with Branch and Bound Algorithm
Niluka P. Rodrigo, Wasantha B. Daundasekera, and Athula I. Perera
International Journal of Discrete Mathematics
Volume 3, issue 2, pages 36--40
DOI: 10.11648/j.dmath.20180302.12
June 2, 2018
http://www.sciencepublishinggroup.com/j/dmath
http://www.sciencepublishinggroup.com/journal/paperinfo?journalid=605&doi=10.11648/j.dmath.20180302.12
http://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20180302.12.pdf
Concerning Peter’s remark about large primes,
the theorem is the following:
The distance between two consecutive primes
divided by the value of any of these two primes
can be made as small as we want, that is,
given any epsilon > 0, we can find a positive integer N
such for any pair of consecutive primes after the N-th prime,
this ratio is smaller than epsilon:
https://en.wikipedia.org/wiki/Bertrand%27s_postulate#Better_results
With best regards, Jean-Claude
Answer 33 posted on August 19, 2018
I thank Gheorghe a lot for the answers 29, 30, and 31
and for the huge amount of computation.
=============================
I also thank him for accepting to work with me
on a joint publication about the question of this discussion.
=============================
I hope that Issam, Peter, and Wulf will also agree
to work on this joint publication, so that we could
finish it without a long delay.
=============================
I have the following to add to my proposal
posted in answer 23 of August 17, 2018:
=============================
A very interesting direction is the case of repeated factors.
=============================
I have posted the following two results:
=============================
By using twice the factor 2, we can obtain the distance 19
by using the primes from 2 to 17.
=============================
By using twice the factor 3 and twice the factor 17,
we can obtain the distance 23
using the primes from 2 to 19.
=============================
These two results with repeated factors are posted
in my following document:
Splitting in Euclid's proof to find the next prime
https://www.researchgate.net/publication/326971510_Splitting_in_Euclid's_proof_to_find_the_next_prime
=============================
I am writing all my methods in my following document:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
=============================
I am typing the above document with Microsoft Word,
not yet with LaTeX, because it is a draft or ideas,
it is not at all the document to submit for publication.
With Microsoft Word, I can use colors,
while we are not allowed to submit documents
with colors for publication.
=============================
I have not started to type in LaTeX.
I must first learn how to install LaTeX on my computer.
=============================
In my document, I feel free to write a lot of details
that would not be accepted for publication.
I have not started the work to sort what will be
good for publication, and what looks ridiculous
for publication.
=============================
I have only started to look for any kind of
related publications, and I have posted
the 3 references that I have found so far
in the above-mentioned document.
=============================
With best regards, Jean-Claude
Answer 34 posted on August 19, 2018
I have just posted a first use
of the following wonderful suggestion from Peter:
Answer 22 posted by Peter Breuer on August 17, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b773167d7141b1f581dabf3
to consider the ratio A/B.
I have posted this first use in my following document:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
It is under the title “Ratio A/B”
followed by “Example: N = 6”.
Peter’s method is really spectacular.
With best regards, Jean-Claude
For N=6, A=2*7*13=182, B=3*5*11=165, so D=A-B=17. This confirms the results given by Jean-Claude.
For N=7, A=5*11*13=715, B=2*3*7*13=714, so D=1. This confirms the results given by Jean-Claude.
For N=22, A=7*11*13*19*23*29*31*37*43*47*61=1.793796113720611e+015,
B=2*3*5*17*41*53*59*67*71*73*79=1.793762815477830e+015, D=3.329824278100000e+010.
Answer 39 posted on August 21, 2018
I am very grateful to Peter for the answer 35,
posted on August 19, 2018
with all the comments and relations, notably:
Packing problem.
Two-computer job scheduling.
Long Processing Time.
The ratio A/B.
Partition problem.
It seems to me that there are good chances that several articles
may be published related to the current article under preparation.
This would give more value to the current article.
So, everyone, including co-author(s) of the current article,
is very welcome to construct related articles as soon as possible.
I will always be very happy to receive references to such other articles.
The current article is to save time to anyone interested in related problems.
My idea is to present different approaches on small examples,
as clearly as possible, so that everyone interested
can see what is going on in a few minutes.
My current two documents
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
Splitting in Euclid's proof to find the next prime
https://www.researchgate.net/publication/326971510_Splitting_in_Euclid's_proof_to_find_the_next_prime
are not the publishable version of the results.
They contain many trivial comments that are not publishable.
They also contain some primitive methods
that are not publishable.
With best regards, Jean-Claude
Answer 40 posted on August 21, 2018
I am very grateful to Gheorghe for the answers 36, 37, and 38,
posted on August 19, 2018.
It is very helpful that Gheorghe has checked all my results
in addition to finding the minimum distances far beyond
what I have been able to find.
As always, I am far behind schedule with updating the documents:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
Splitting in Euclid's proof to find the next prime
https://www.researchgate.net/publication/326971510_Splitting_in_Euclid's_proof_to_find_the_next_prime
Notably, all the new Gheorghe’s results are missing.
At the present time, I am working on a new method
that is very different from my previous “New” methods.
As Peter has been talking several times about the ratio A/B,
I am following my own ideas about this.
I do not yet know whether my own ideas will work.
With best regards, Jean-Claude
Hello Peter ! I don't have haskell/hugs but I am very interested to see your program. Is it possible to give some mathematical explanations concerning your "naive program" ?
Answer 44 posted on August 25, 2018
I thank a lot Peter and Gheorghe for the answers 41, 42, and 43.
Concerning Haskell/hugs, I have found the following:
https://en.wikipedia.org/wiki/Haskell_(programming_language)
https://www.haskell.org/hugs/
I have not had time to read Peter’s program carefully,
and I have not had time to try to download Haskell/hugs
and see whether I am able to install it and use it.
Concerning advanced computer methods related
to my joint article under preparation, everyone,
including co-author(s) of my joint article, is welcome
to construct and publish articles related to my joint
article. Such other articles would increase the value
of my joint article.
I am not qualified in computer science, consequently,
I would not be able to contribute to articles in computer
science, and in addition, I am overflowed by the construction
of my book of logic, the construction of the present joint article,
and I also need to publish an article on an alternative Rolle’s
theorem for space curves, and an article on longest sequences
of consecutive integers that are all composites containing
only the first N primes.
The subject of my joint article is a gentle introduction to:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
Splitting in Euclid's proof to find the next prime
https://www.researchgate.net/publication/326971510_Splitting_in_Euclid's_proof_to_find_the_next_prime
What is the splitting P = AB of the product P of the first N primes
that gives the smallest distance between A and B?
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B
Is there a published improvement of Euclid’s Theorem?
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem
What are the pairs of consecutive positive integers whose product is a primorial?
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial
What are the products n(n + 1)(n + 2) equal to a primorial?
https://www.researchgate.net/post/What_are_the_products_nn_1n_2_equal_to_a_primorial
I am very grateful to Gheorghe for having accepted to be co-author.
I hope that Issam and Peter will also accept to be co-authors
of this introductory article. Anyone who could also contribute
to my joint article and who could make it better and faster
would be welcome to be co-author.
With best regards, Jean-Claude
Answer 45 posted on August 25, 2018
I have improved the A/B method to find
the minimum distance between A and B.
I have posted it in Parts 26, 27, and 28 of my document:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
In part 28, I apply the method to the case N = 3.
I have also a draft for the cases N = 4 and 5.
The A/B method works in a wonderful way.
I must improve this draft
before I can add it to my document,
I hope tomorrow.
I keep improving all this every day.
The idea to use A/B came from Peter’s comments.
Answer 22 posted by Peter Breuer on August 17, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b773167d7141b1f581dabf3
Answer 28 posted by Peter Breuer on August 18, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b78808afdda4a6f91379db4
Answer 35 posted by Peter Breuer on August 19, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b7a318c36d23533985091f6
I feel that there is a good chance that the A/B method
will give the fastest method.
With best regards, Jean-Claude
Answer 47 posted on August 25, 2018
I thank Peter a lot for the answer 46 and all the explanations.
Peter’s idea to consider the logarithm of the products,
which transforms products into sums, and to distribute
the logarithms of the factors of P into two boxes A and B
in such a way that the sums of logarithms in each of these
two boxes be as equal as possible is a wonderful idea.
To help the readers, all this is in the following answers:
Answer 28 posted by Peter Breuer on August 18, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b78808afdda4a6f91379db4
Answer 35 posted by Peter Breuer on August 19, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b7a318c36d23533985091f6
Answer 43 posted by Peter Breuer on August 25, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b81156e36d23517ef0675d2
Answer 46 posted by Peter Breuer on August 25, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b81dc1836d235babb1efb03
I am also very grateful to Peter for having checked
the enormous amount of computation achieved by Gheorghe.
This is extremely helpful for the construction of my current
joint article on the extension of Eucllid’s proof
from Euclid’s q = P + 1
to q = A – B, with P = AB, and A > B.
This is one more reason why Peter should be one
of the co-authors of my joint article on all this.
In my answer 44, I said any other articles using
advanced computer science constructed by anyone
including co-author(s) of my current joint article
will be very welcome. I need to add that any other
articles, with or without computer science will be
very welcome. It is always better for articles
to have more citations and references.
Concerning downloading, installing, and learning
computer programs, it has never happened to me
so far that I could do it without a huge waste of time
with stupidities. I will post any stupid problem
that I may have with Haskell.
At the present time, I have all my mind
on the A/B method. This method works
in a wonderful way. I think that it will be
the fastest method. I will try to post
any progress I am making on this
every day in the parts 27, 28, … of my document:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
I apologize to Gheorghe for having misspelled
his first name in my answer 44.
RG allowed me to correct this in answer 44
without shutting down my RG file.
With best regards, Jean-Claude
Answer 50 posted on August 26, 2018
=============================
I am very grateful to Peter for the answers 48 and 49
and for all the information, explanations and clarifications.
=============================
I am extremely impressed that Peter’s program
solved the case N = 23 in a few seconds.
Already the case N = 22 first solved by Gheorghe
and checked by Peter, was and still is
of astronomic magnitude.
The case N = 23 is above the roof of the universe.
=============================
Concerning A < B or A > B, I have specified
in all my questions, answers, and documents
that for me, A > B.
=============================
Concerning repeated factors,
I have posted the following two results:
=============================
In the case of the first 7 primes from 2 to 17,
we can obtain the distance 19,
that is, we can obtain the next prime after 17,
by using twice the factor 2.
=============================
In the case of the first 8 primes from 2 to 19,
we can obtain the distance 23,
that is, we can obtain the next prime after 19,
by using twice the factor 3 and twice the factor 17.
=============================
These two results with repeated factors are posted
in my following document:
Splitting in Euclid's proof to find the next prime
https://www.researchgate.net/publication/326971510_Splitting_in_Euclid's_proof_to_find_the_next_prime
=============================
Concerning recommendations on RG, it would be very
helpful to be given a choice of options, instead of just
“Recommend”. Some of such options could be:
The answer was helpful.
The answer was very helpful.
The answer was very very helpful.
The answer gives a nice minor new result.
The answer gives a nice new result.
The answer gives an important new result.
The answer gives a very important new result.
The answer contains a new major result.
The answer contains a new major result
that is a historical event in mathematics.
=============================
I am very busy with the A/B method.
I have posted the beginning of the beginning
in Parts 26, 27, and 28 of my following
documents:
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
In this beginning of the beginning,
it is impossible to see how wonderful
the A/B method is.
But in my drafts with N = 5, N = 6,
not N = 7, because N = 7 is trivial,
N = 8, and N = 9, I see that the method
is super-fantastic.
I have no doubt that it is the best method.
I have not posted my drafts, because they
are so badly written that I have difficulties
to understand what I wrote. But it is obvious:
The A/B method is the most best
of the best methods.
=============================
The idea to use A/B came from Peter’s comments:
Answer 22 posted by Peter Breuer on August 17, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b773167d7141b1f581dabf3
Answer 28 posted by Peter Breuer on August 18, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b78808afdda4a6f91379db4
Answer 35 posted by Peter Breuer on August 19, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b7a318c36d23533985091f6
=============================
With best regards, Jean-Claude
Answer 51 posted on August 27, 2018
I have dramatically improved the A/B method.
Since I believe that it is the best method,
I have moved it to the top of my document
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
Part 1 contains a short introduction
to P = AB, A > B, d = A – B.
Part 2 contains the best method
to check whether d = 1.
Parts 3, 4, 5, 6, and 7, contain applications
of the method of Part 2
to the cases N = 1, 2, 3, 4, and 5.
Part 8 explains the A/B method.
Part 9 gives the A/B algorithm.
Part 10 Applies the A/B algorithm to the case N = 6.
Parts 11—29 contain mostly unpublishable remarks.
The writing is still extremely disgusting.
I will keep un-disgusting it every day.
I also have horrible drafts
that I have not included in my document.
These horrible drafts convince me
that the A/B method is the most marvelous
method of history of mathematics.
With best regards, Jean-Claude
Based on Peter's answer 28 posted on August 18, 2018, I considered the splitting P = AB of the product P of the first N primes that gives :
Rmin=min[|A/B-1| |B/A-1|].
For N=2, Rmin=|2/3-1|=0.3333.
For N=3, Rmin=|5/6-1|=0.1667.
With an extension of the previous Matlab program, I obtained:
For N=6, Rmin=|165/182-1|=0.0934
For N=7, Rmin=|714/715-1|=0.0013986.
There are the same numbers A and B giving the smallest D=|A-B|. If B < A, then Rmin=|B/A-1|=1-B/A.
Answer 55 posted on August 27, 2018
I thank Gheorghe a lot for the answers 52, 53, and 54.
It is interesting to see how fast the ratio A/B
goes to 1 from above when N increases,
and how fast the ratio B/A goes to 1 from below
when N increases.
We see that the distance from 1 decreases fast.
We may also note that A/B – B/A =(A^2 – B^2)/P.
I keep working on the construction of my version
of the A/B method, which originated from comments
from Peter. I try to decrease the number of divisions.
At the present time, it is impossible to test my version,
because it is far to be finished.
I am posting every day what I have constructed
so far to show where I am in the construction.
With best regards, Jean-Claude
For N=8, Rmin=|3094/3135-1|=0.0131
For N=9, Rmin=|14858/15015-1|=0.0131.
There are the same numbers A and B giving the smallest D=|A-B|. If B < A, then Rmin=|B/A-1|=1-B/A.
As for the smallest D=|A-B|, increasing N do not give a monotone variation of Rmin.
For N=10, Rmin=|79534/81345-1|= 0.022263,
For N=81, Rmin=|447051/448630-1|=0.0035196.
Same remarks.
For N=11, Rmin=|447051/448630-1|=0.0035196 (it was a mistake for N in my previous message).
For N=12, Rmin=|2714690/2733549-1|=0.006899.
For N=13, Rmin=|17395070/17490603-1|=0.00546196.
Same remarks.
Dear Jean,
< It would be very helpful for me if Gheorghe, Issam, Peter, and Wulf Rehder
would agree to write a joint publication with me containing...>
Well, computations can be done up to (finite) nth digits for a given P=AB to calculate the minimum difference D=min(A-B) or the maximum sum S = max(A+B). No general results obtained yet.
The main issue is
(1) What is the mathematical significance of D and S?
(2) Are there any precise consequences for calculating D and S that provide new information about gaps between primes?
I proved that:
If D =1, then the primorial P can be written
as a product of two consecutive integers.
(The proof is available on another thread, by Jean, concerning primorials)
Are there more results can be drawn?
The available results are not enough to write an article. Agree?
Best regards
Dear Peter,
Your numerical method to solve the (difference problems) is perfect.
No doubt, your program is helpful to tackle similar problems.
Best regards
Answer 62 posted on August 28, 2018
I thank Gheorghe, Issam, and Peter a lot for the answers 56, 57, 58, 59, 60, and 61. I cannot see Peter’s new program, because I do not have C on my computer. I am very impressed that Peter has solved the case N = 24. I am always extremely happy to know the very different points of view from everyone. I think that we will never find two people on this planet who agree on everything. One thinks that only Option O-4739-3972 for problem P-4739 is acceptable, another one thinks that Option O-68257-771 for problem P-68257 is totally unacceptable. I always expect that the points of view will be very different, some very negative, some very positive. What is the most important for me, is that everyone feels free to tell me what they think, even if what they feel is extremely negative on every point. I will always try to explain my point of view as clearly as possible and answer every criticism. So, for example, there are many very different opinions about what an article should be or not be. I respect all these opinions and I will always try to explain my own opinion as clearly as possible. I have full respect of the opinions of Peter and Issam that there is nothing in my 2 documents and 4 questions for an article. I am very happy that they are telling me what they really think. My opinion is the opposite of their opinion, I will try my best to explain mine, and will try my best to understand theirs. The verb “Understand” has two totally different meaning. One is when someone tells someone else “I do not understand you”, it means “I disagree with you”. When I say, “I will try to understand theirs”, I am using a totally different meaning. I am using the meaning that a certain word can have different meanings, and if I cannot determine from the context which of these different meanings they are using, then I will ask about this. At the present time, I am convinced that there is already enough in my 2 documents and 4 questions for an article/note. Because I have all my mind in the construction of my version of the A/B method, I cannot give a good list showing that there is enough, so, below, I give a first draft of list, and it is likely that I forget the most important points, because my mind is totally in the construction of my version of the A/B method. The reasons that am able to see right now why I am convinced that there is enough for an article/note are the following:
1. An article does not have to contain the solution of a problem.
2. It is always immensely more useful to submit new ideas to the mathematical community rather than keeping these ideas in a box.
3. The idea to extend Euclid’s proof
from Euclid’s q = P + 1
to the extension q = A – B, P = AB, A > B,
this idea must be published because it is a wonderful source of new ideas.
(Note: When I am interested to find the next prime, I denote it by q,
and I when I am interested to find the minimum distance, I denote it by d.
So, q = d).
4. The idea that in some cases, we can find the next prime must be published. The article does not have to solve the problem of what are all such cases. It is better to submit this question to the mathematical community.
5. The idea that in some cases, we can find the next prime by using repeated factors must be published.
6. Proposing our best program to find the minimum d saves time to everyone interested in related subjects
7. The idea that some primorials are products of two consecutive integers must be published. It raises the question “What are all such primorials”. An article does not have to solve problems. It is very good for an article to raise good questions. It seems that Issam and me found this idea independently. I can search for the first time each of us mentioned it in all the questions and answers.
8. The idea that some primorials are the products of 3 consecutive integers must be published.
I stop here, I really want to work on my version of the A/B method.
It is very likely that I am forgetting the most important. If so, I will post an updated list.
With best regards, Jean-Claude
For N=14, Rmin=|114371070/114388729-1|=1.54377E-4
A=7*13*23*31*41*43=114388729
B=2*3*5*11*17*19*29*37=114371070.
For N=15, Rmin=|783152070/785147363-1|=0.00254129746.
A=7*11*13*17*29*37*43=785147363
B=2*3*5*19*23*31*41*47.
Same remarks:
There are the same numbers A and B giving the smallest D=|A-B|. If B < A, then Rmin=|B/A-1|=1-B/A.
As for the smallest D=|A-B|, increasing N do not give a monotone variation of Rmin.
Dear Jean,
We found your questions are interesting and we shared you almost all. My previous response was a motivation to explore and investigate more results rather than evaluation.
Many of the raised questions are still conjectures, no final answers confirmed.
There are few values, D=1, were discovered. Are these values the only ones or more cases can be determined?
It is equivalent to say: how many primorials exist as a product of two consecutive integers? It is equivalent to ask: How many primorials P exist such that 4P +1 is a complete square?
Similar questions can be raised for the general case P =m(m+1)...(m+k).
I assumed that the proposed article should give answers for many of the previous claims. Besides that, you already gathered the responses in a section at your page, which may be in due time, deserve to submit the work for reviewing.
Best wishes
Answer 66 posted on August 27, 2018
I thank Gheorghe, Peter, and Issam a lot for the answers 63, 64, and 65. I cannot answer more right now, because I have several emergencies. I do not have one molecule of food left in my apartment. It may be better that I do something about it before I collapse.
With best regards, Jean-Claude
For N=18, Rmin=|342444658094/342503171205-1|=1.708396E-4
A=3*5*13*17*19*37*47*53*59=342503171205
B=2*7*11*23*29*31*41*43*61=342444658094 < A
For N=19, Rmin=|2803119896185/280349704514-1|+1.0694379E-4
A=2*3*11*13*17*31*37*41*61*67=2803419704514
B=5*7*19*23*29*43*47*53*59=2803119896185 < A.
For N=20, Rmin=| 23619540863730/23622001517543-1|=1.041678797E-4
A=11*13*23*29*31*43*47*59*67=23622001517543
B=2*3*5*7*17*19*37*41*53*61*71=23619540863730.
For N=21, Rmin=| 201813981102615/20181733409378-1|=1.9583526E-5
A=2*11*13*29*37*43*53*59*67*73=20181733409378
B=3*5*7*17*19*23*31*41*47*61*71=201813981102615.
Answer 70 posted on August 29, 2018
=======================
I have posted my most recent version
of the A/B algorithm in the following document:
Algorithm about extensions of Euclid's proof
https://www.researchgate.net/publication/327287317_Algorithm_about_extensions_of_Euclid's_proof
=======================
For the first time, it is easy to read
and understand.
=======================
I thank Peter and Gheorghe a lot
for the answers 67, 68, and 69.
=======================
I am very impressed by Peter’s algorithm
and by the enormous amount of work
achieved by Peter and Gheorghe
to make this algorithm successfully work
on their computers.
I am very impressed by the enormous amount
of checking achieved by Peter and Gheorghe
on the cases from N = 1 to N = 24.
I am very impressed that Gheorghe could understand
this algorithm so fast and successfully apply it
with total success.
=======================
All the checking achieved by Gheorghe, Issam,
Peter, Stanley, and Wulf (in alphabetic order)
is wonderful, because it makes all the results
obtained several times by different methods
very trustable.
=======================
Unlike Gheorghe, I have not been able
to understand Peter’s algorithm yet.
=======================
My top priority is to improve my version
of the A/B algorithm, so that anyone interested
can determine whether it is slower or faster
than Peter’s algorithm, or whether a combination
of the two algorithms would be better.
=======================
With best regards, Jean-Claude
Thank you, Jean-Claude, for your last message. In fact, I used my own methods to write some A-B and A/B Matlab programs. For both A-B and A/B criteria I obtained the same splitting P = AB of the product P of the first N primes. Is this always true ? For any value of N ? If YES, which is the proof ?
Answer 72 posted on August 29, 2018
=================
I thank Gheorghe a lot for the answer 71.
=================
It is impossible for me to answer his question,
because I do not have access to his definitions
of A, B, and P in his 2 programs in MatLab.
=================
In the 3 related documents that I have posted on Research Gate,
in the 4 related questions that I have posted on Research Gate,
and in all the answers that I have posted on Research Gate,
I have always defined A, B, N, P, and d in exactly the same
following way:
=================
I consider a positive integer N.
I consider the product P of the first N primes.
I consider a splitting of P
into two positive integer factors A and B
such that P = AB, A > B, and d = A – B.
=================
For example, when N = 2, the first two primes are 2 and 3.
The product P of these first two primes
is P = (2 times 3) = 6.
I can factor P with the first factor greater
than the second in the following two ways:
P = 6 = 3 times 2
and P = 6 = 6 times 1.
There are two possible splits of P such that
A is a positive integer,
B is a positive integer,
P = AB
and A > B.
The first one is A = 3 and B = 2.
The second one is A = 6 and B = 1.
In both cases, we have AB = P
by the choice that I have made of A and B.
=================
The 3 related documents that I have posted
on Research Gate are the following:
=================
Minimum distance from splitting the product of the first N primes
https://www.researchgate.net/publication/326985440_Minimum_distance_from_splitting_the_product_of_the_first_N_primes
Splitting in Euclid's proof to find the next prime
https://www.researchgate.net/publication/326971510_Splitting_in_Euclid's_proof_to_find_the_next_prime
Algorithm about extensions of Euclid's proof
https://www.researchgate.net/publication/327287317_Algorithm_about_extensions_of_Euclid's_proof
=================
The 4 related questions that I have posted
on Research Gate are the following:
=================
What is the splitting P = AB of the product P of the first N primes
that gives the smallest distance between A and B?
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B
Is there a published improvement of Euclid’s Theorem?
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem
What are the pairs of consecutive positive integers whose product is a primorial?
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial
What are the products n(n + 1)(n + 2) equal to a primorial?
https://www.researchgate.net/post/What_are_the_products_nn_1n_2_equal_to_a_primorial
=================
With best regards, Jean-Claude
Q15, A75, August 31, 2018
==================
I thank Peter a lot for his answers 73 and 74,
for all the very clear explanations,
and for having looked at my version
of A/B method to find (A, B)
that gives the minimum distance.
==================
My program considers all divisors of A
and all divisors of B, including all divisors
that are NOT primes.
==================
So, the number of primes on the numerator
and the denominator of A/B changes
every time this gives a better choice of (A, B).
==================
We can start from the worst choice
A = P and B = 1,
and still obtain the best choice of (A, B).
==================
For every new better choice of (A, B),
my program first computes
the square root S of A/B,
and then for every divisor b of B
from the smallest to the largest,
it computes the product bS = (b times S),
and then it looks at every divisor a of A
from the smallest to the largest
and it checks whether this divisor a
satisfies the condition b < a < bS.
==================
Checking this condition takes almost
no computer time: There are no divisions,
no square roots just check
the two inequalities b < a and a < bS
==================
Very soon, the difference (S – 1)
becomes very small,
the interval (b, bS) becomes very small
and it becomes exceptional that the
divisor a of A is in this very small
interval, so that from this point,
the program runs very fast
==================
After a small number of exchanges
of divisors a and b between numerator
and denominator of A/B, the program
arrives at the best choice of (A, B).
==================
Then it makes a last run through
the divisors b of B, and if it does not
find any better choice of (A, B)
it means that (A, B) gives the minimum
distance, and the program stops
at this point.
==================
I know the concept of recursion very well.
I am writing a book of logic starting from
the very logical beginning of mathematics,
that is, starting from almost nothing,
just an elementary knowledge of English,
and no knowledge at all of 1, 2, 3, ….
I am restricting this book to what is needed
to arrive rigorously as soon as possible
to the wonderful construction of the set N
of natural numbers found by John von Neumann
https://en.wikipedia.org/wiki/John_von_Neumann
https://en.wikipedia.org/wiki/Von_Neumann_ordinal
From this, the axioms of Giuseppe Peano
https://en.wikipedia.org/wiki/Giuseppe_Peano
https://en.wikipedia.org/wiki/Peano_axioms
become theorems.
==================
I have used recursion in computer programs,
but I have not practiced this for a long time.
==================
I will improve my program on the A/B method,
create a subroutine for the first choice of (A, B)
and a subroutine for next better choices of (A, B).
==================
Later, I will try to download Python-Sage
https://en.wikipedia.org/wiki/Python_(programming_language)#Development
https://en.wikipedia.org/wiki/SageMath
try to learn how to use it,
and try to test my program.
but at the present time, I have terrible problems
with my computer, and this is an understatement.
==================
Meanwhile, I think that it is much better
that I write and post a description of my project
of joint minor article on an introduction
to the miracles created by the extensions
from Euclid’s proof of the infinitude of primes.
There were no reasons to think that this would
be interesting, but it turns out that it is
a wonderful source of miracles.
So, the objective of this article is to tell
to the mathematical community
that it is indeed a source of miracles.
The objective is not at all to solve any problem.
It is much better to offer these problems
to the mathematical community
instead of eating all these marvels
and leaving nothing to eat to other mathematicians.
==================
I hope that this description that I still have
to write will convince Issam, Peter, Stanley,
and Wulf to co-authoring this first minor
joint article in order to make it better and faster.
==================
With best regards, Jean-Claude
Q15, A77, August 31, 2018
==================
I am very grateful to Peter for the answer 76
and for not ignoring my A/B program.
==================
Every time my program finds a divisor a of A
in the interval (b, bS)
it constructs the better choice (New A, New B)
from the old choice (Old A, Old B)
by defining [New A] and [New B]
in terms of [Old A] and [Old B]
in the following way:
[New A] = (b/a) times [Old A],
[New B] = (a/b) times [Old B].
==================
This moves the factor a from [Old A] into [New B]
and
it moves the factor b from [Old B] into [New A].
==================
Equivalently, we can say that
it moves the factor a from the numerator of [Old A/B]
into the denominator of [New A/B]
and
it moves the factor b from the denominator of [Old A/B]
into the numerator of [New A/B]
==================
With best regards, Jean-Claude
Q15, A79, August 31, 2018
==================
I thank Peter a lot for the answer 78
and the explanation of what is not clear in my program.
==================
As I was in a hurry when I wrote the first drafts
of my program, I wrote a “For i from 1 to i_max”
instead of a “While [Not_found] = True”.
Because of this, what I want to do in my program
is different from what it looked like, and from
it still looks like, because I have not had time
to make the corrections.
==================
For the readers who try to understand what
I am talking about, I recall that my program
is posted in the following document:
Algorithm about extensions of Euclid's proof
https://www.researchgate.net/publication/327287317_Algorithm_about_extensions_of_Euclid's_proof
Currently, it is wrong with “For” instead of “While”.
==================
The objective of my program is the following:
Given a positive integer N,
I denote by P the product of the first N primes.
The objective it to find two positive integers
A and B such that P = AB, A > B,
and the distance d between A and B
is minimum, that is, d = A – B is the minimum
over all possible choices of A and B.
==================
In the above-mentioned document, I explain
very carefully why I use the square root S of A/B.
==================
Here are the 4 steps of my program:
==================
1. I make a first easy (bad) choice of A and B:
I travel the list of the first N primes ordered
in increasing order, and take the first prime into A,
the second into B, the third into A, the fourth into B, …
If A < B, I interchange A and B to get A > B.
==================
2. I construct the list of all divisors of A
in increasing order, from 1 to A
and the list of all divisors of B from 1 to B.
==================
3. I compute the square root S of A/B.
==================
4. “While” I have not found a better
choice of A and B, I travel the list of
divisors of B in increasing order, and
for every divisor b of B, I compute
the product bS = [b times S],
and I search for
the first divisor a of A such that b < a < bS.
==================
If by chance I find such a divisor a of A,
then I define
[New A] = (b/a) times [Old A],
[New B] = (a/b) times [Old B].
==================
This moves the factor a from [Old A] into [New B]
and
it moves the factor b from [Old B] into [New A].
==================
Equivalently, we can say that
it moves the factor a from the numerator of [Old A/B]
into the denominator of [New A/B]
and
it moves the factor b from the denominator of [Old A/B]
into the numerator of [New A/B].
==================
In the above-mentioned document,
I prove that the condition b < a < bS
guarantees that the new distance between
the new values of A and B is smaller
that the old distance, and that the new
value of A is greater than the new value of B,
and we still have P = [New A][New B].
=========================
If in Step 4 the program finds a divisor of b of B
and a divisor a of A such that b < a < bS,
then as soon as is has computed
[New A] and [New B],
it stops Step 4,
and it goes back to Step 2,
and it repeats the Steps 2, 3, and 4,
with a new S, so that S – 1 > 0 is much smaller,
and the new intervals (b, bS) are much smaller
than before
until the program can restart from Step 2,
perform Step 3, and travel all Step 4 without
ever finding a and b such that b < a < bS.
=================
If in Step 4, for every divisor b of B
the program never finds a divisor
a of A such that b < a < bS,
this means that there are no pairs
of factors (a, b) that we can interchange
between A and B to make d = A – B smaller,
and therefore, d is the minimum distance.
===================
I will be very happy to clarify any
point that is not clear in my program.
===================
I have not had time to change “For”
with “While”. I will do it as soon as
I can. I agree that “For” is wrong.
===================
I must also write a subroutine
“Find a first bad choice of A and B”
and subroutine “Find a better choice”.
=======================
With best regards, Jean-Claude
It's always good to see people digesting their ideas on a particular problem. It remain for you to choose the best contributions.
For N=22:
A=2*3*5*7*13*31*37*41*47*53*71*79=1 793 779 635 410 490
B=11*17*19*23*29*43*59*61*67*73=1 793 779 293 633 437
Rmin=|B/A-1|=0.190534582111823E-6
Q15, A85, September 1, 2018
==================
I thank a lot Peter, Jamilu, and Gheorghe
for the answers 80—84.
==================
I am immensely impressed by the enormous amount
of very successful work achieved by Peter
in a very short time, and all the huge difficulties
that he has solved in this short time.
My posted draft of program still has many mistakes
and unsolved difficulties.
I cannot be more impressed that Peter has solved
all this so fast.
==================
Peter’s results are beyond the limits of the best computers.
==================
For example, in my answer
Answer 5 to Question 16, posted on August 23, 2018
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial#view=5b7f6657fdda4a179d1bc716
I explained that with my current system,
I could not go beyond 14 non-zero digits,
and my ceiling was the case N = 12.
As the magnitude of the difficulties dramatically
increases from one value of N to the next,
the case N =31 reached by Peter
is very far above the roof of the universe.
I will need a very strong parachute to come back
from the dream of N = 31 down to planet Earth.
==================
I still hope that Issam, Peter, Stanley, and Wulf
will accept to co-authoring a first minor
introductory joint article in order to help me
to make this introductory article better and faster.
==========================
Anyone, including co-authors, is welcome
to publish other articles in parallel
to my joint article.
==============================
For example, Wulf has already a very strong
article ready on his wonderful method
of transforming primorials into factorials,
and also looking
at near misses of well-known results,
all these with outstanding references.
Wulf has posted his fantastic discoveries
in answers 12, 14, 15, 19, and 21 to the following question:
What are the pairs of consecutive positive integers whose product is a primorial?
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial
============
I also recall that Issam starting to look
for interesting conjectures in his following answer:
Answer 21 posted by Issam on August 17, 2018
https://www.researchgate.net/post/What_is_the_splitting_P_AB_of_the_product_P_of_the_first_N_primes_that_gives_the_smallest_distance_between_A_and_B#view=5b7683c34921ee119406bcbf
============
I will try to improve my description
of the first introductory joint article,
whose draft of a first paragraph
is posted in the following document:
Description of my project of extensions from Euclid's proof
https://www.researchgate.net/publication/327350053_Description_of_my_project_of_extensions_from_Euclid's_proof
============
With best regards, Jean-Claude
For N=24:
A=5*17*29*31*37*41*43*53*79*83*89=154 171 363 634 898 185
B= 2*3*7*11*13*19*23*47*59*61*67*71*73=154 170 926 013 430 326
For N=25:
A=2*3*11*19*29*31*37*41*43*47*67*79*83=1 518 410 187 442 699 518
B=5*7*13*17*23*53*59*61*71*73*89*97 =1 518 409 177 581 024 365
Peter, my program uses only integers. For a first number n1 I consider all products of k primes, vith k taking values from 2 up to M=floor(N/2). The other number n2 is computed as P/n1. Hence, A=max(n1,n2) and B=min(n1,n2). For each couple (n1,n2) I compute r=min(|n1/n2-1|,|n2/n1-1|). If r
Q15, A92, September 2, 2018
==================
I thank a lot Peter and Gheorghe for the answers 86—91.
==================
I have written a first part of a summary
of results related to the extension
Q = A – B, P = AB, A > B,
from Euclid’s Q = P + 1, in the document:
Description of my project of extensions from Euclid's proof
https://www.researchgate.net/publication/327350053_Description_of_my_project_of_extensions_from_Euclid's_proof
=================
I have started to correct my program.
I am horrified by the huge number of mistakes
that I have made in my program.
I am amazed that Peter was able to survive
all my mistakes and write a corrected program,
while it will take me several more days
to correct my program. It is still horrible.
It is posted in the following document:
Algorithm about extensions of Euclid's proof
https://www.researchgate.net/publication/327287317_Algorithm_about_extensions_of_Euclid's_proof
=================
With best regards, Jean-Claude
Peter, you're right: sumk m!/k!(m-k)! is 2^m IF k takes all possible values from 0 to m. In my case, k takes values from 2 to the integer part of N/2. So, the result is smaller than 2^m.
Q15, A96, September 3, 2018
==================
I thank a lot Peter and Gheorghe for the answers 93, 94, and 95.
==================
The following possible minor improvement
would not change the complexity
but may save some time:
The case N = 7 is the last one that gives
a minimum distance d = 1, with
A = 715, B = 714, and, d = 715 – 714 = 1.
The minor improvement for N > 7 would
be to start the search with
either A = 715X and B = 714Y
or A = 714X and B = 715Y.
==================
I have started the beginning of the beginning
of a list of results. It is posted in the following
document:
List of results on extensions from Euclid's proof
https://www.researchgate.net/publication/327392627_List_of_results_on_extensions_from_Euclid's_proof
=======================
I have noticed that N = 9 is the first time
where A contains 5 consecutive primes.
=======================
With best regards, Jean-Claude
Dear Peter,
The value of A - B is based on the original product AB.
Observe that if P = AB and A - B =1, then
P = B(B+1).
Therefore D=1 is equivalent to say:
The Primorial can be expressed as a product of two consecutive integers
and this also equivalent to say 4AB +1 is a complete square.
Hence D = A - B=1 if and only if 4AB+1 is a complete square and then
one can search for complete squares.
We discover only a few values where 4AB +1 is a complete square.
AB=2, AB=2x3, AB=2x3x5, AB=2x3x5x7,2x3x5x7x11x13x17
AB=2 is equivalent to A=2 B=1 A-B=1 and 4(2) +1 = 3^2
AB=2 x3 is equivalent to A=3 B=2, A-B=1 and 4(2x3)+1=5^2
AB=2 x3x5 is equivalent to A=2x3 B=5, A-B=1 and 4(2x3x5)+1=11^2
AB=2 x3x5x7 is equivalent to A=3x5 , B=2x7, A-B=1
and 4(2x3x5x7) +1=(29)^2
AB=2x3x5x7x11x13x17 is equivalent to A=5x11x13, B=2x3x7x17,
A - B =1 and 4(2x3x5x7x11x13x17) +1=(1429)^2.
Are there more?
You can see the similar problem as Brocard's Problem that observed by Wulf.
https://en.wikipedia.org/wiki/Brocard%27s_problem
Best wishes
Q15, A99, September 3, 2018
==================
I thank a lot Peter and Issam for the answers 97 and 98.
==================
Concerning the search for P = n(n + 1)
other than the 5 that I have posted
in the description of my question 16:
What are the pairs of consecutive positive integers whose product is a primorial?
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial
I have the following pieces of information:
=========================
On August 28, 2018, Stanley Rabinowitz checked
that from N = 1 to N = 10,000:
there are no other solutions
than the five in my question 16:
Answer 9 to my question 16 posted Stanley Rabinowitz on August 28, 2018;
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial#view=5b8578a7d7141bbf7467dab6
====================
The current best method for computer
search for P = B(B + 1) is posted
in the description of my question 16
What are the pairs of consecutive positive integers whose product is a primorial?
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial
with an improvement in my answer 1:
Answer 1 to my question 16 posted on August 22, 2018
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial#view=5b7d91b9a5a2e252907ab79f
====================
In spite of the fact that it is a high school result
obtained from the quadratic formula,
with x = B as the variable, and from the
discriminant (4P + 1) of the quadratic formula,
the equivalence [ P = B(B + 1)]
equivalent to [(4P + 1) is a perfect square]
is not only the best for finding theoretical results,
but in addition, it has been a source
of outstanding results obtained by Wulf.
==================
This high school method is the first that
I used before finding better methods:
Answer 13 to my question 14 posted on August 10, 2018
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem#view=5b6e58d8d7141b3de56cf732
Issam tested this method the next day
and posted the results in the next answer:
Answer 14 to my question 14 posted on August 11, 2018
https://www.researchgate.net/post/Is_there_a_published_improvement_of_Euclids_Theorem#view=5b6f02e7d7141b40d6262023
==================
I did not post this high school method
in the description of my question 16,
because my above method is much better
for computers, and I never suspected
that a high school formula could be
a source of miracles.
==================
It was a huge surprise for me when
Wulf started finding more and more
fantastic results from this high school
formula for the discriminant
of a quadratic equation.
==================
It is a wonderful miracle that Issam posted it:
Answer 2 to my question 16 posted
by Issam Kaddoura on August 22, 2018
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial#view=5b7d9a9a2a9e7a2c2f42de3e
========================
Thanks to this miracle, Wulf Rehder
has obtained outstanding results:
What are the pairs of consecutive positive integers whose product is a primorial?
https://www.researchgate.net/post/What_are_the_pairs_of_consecutive_positive_integers_whose_product_is_a_primorial
========================
With best regards, Jean-Claude
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