The definition of a prime number leads to the first answer, and is a very old barrier, dating to c. 300 BC, but the concept of PFEFS leads to the second answer, and hope. This is explored in my blog in RG, at https://www.researchgate.net/post/Introduction_RG-blog
What is your qualified opinion?
NOTE: The results shown by me in this thread, include the contribution of several individuals-- some of which I do not know personally, and some of which may come spontaneously online. Therefore, the results may change over time, as it is being refined by intersubjectivity. To mark this temporary but transient nature, the term WORK shall be used. Repeated updated is suggested, so as not to use any outdated terms. Any suggestions are always welcome! It is a new model for an online class, where anyone may join.
Why not?
If you work with Polynomial Ideals
then the interesting expansion is of the form:
p1(x,y,...)*f1(x,y,...)+p2(x,y,...)*f2(x,y,...)+...+pm(x,y,...)*fm(x,y,...)
Groebner basis is a great example here...
The problem of partioning a large number into its prime numbers turns out to be a hard, laborious hacking, and an important problem in mathematics and in code. But is it?
Well, cybersecurity itself depends on its difficulty for very large numbers. So, the answer is important.
2x3 is a product, 2+2+2 is a sum. By definition, factorization of an integer is an expression of that integer as a product of other integers; prime factorization of a positive integer >1 is the expression of that integer as a product of primes and it is unique by the fundamental theorem of arithmetic.
1 satisfies the defining property of a prime, since it has no positive divisors other than 1 and itself (which in case of 1 is also 1), but if it was counted as a prime, then the prime factorization of any integer would not be unique, sine 1 is the multiplicative identity and thus it can be added to any factorization as many times as one wishes.
To the best of my knowledge, the reason that 1 is not counted as a prime is for the prime factorization to be unique, which is fundamental to number theory.
SK: Thanks. No one is contradicting the fundamental theory of arithmetic. But, when one looks at the partition of positive integers, an even more fundamental truth may be at play.
In simple arithmetic of positive integers, multiplication is repeated addition. The same is obtained if signed integers or so-called "real" numbers are used. We suggest that the addition partition allows one to calculate primes more effectively versus the multiplication partition, which leads nowhere but brute force.
For example, take 2047. An oscillating chord may be split in 23 oscillations of a single 89 block, or 89 oscillations of a 23 block. A chord would not carry 89 oscillations AND 23 oscillations. This can be verified experimentally. The model must use sum, not multiplication. This must have influence in music, also.
SK: BTW, I accept you reason that 1 is not counted as a prime, because of the prime factorization. It has to be unique, which is fundamental to number theory.
But, there is an even higher reason when looking at partitions of different orders. 1 is not a prime number because we would have to exclude it in many theorems valid for prime numbers, and include in none. Thus, a fundamental set argument emerges: 1 is not of the same class as the set of prime numbers. This is evident when considering the addition model of the formation of prime numbers, using waves.
EG: I will read about PNEFS on your blog; It seems very interesting.
SK: the PNEFS proposes a wave model for vibrations, linking periodicity with factoring of integers, to find prime numbers.
From physics to mathematics!
Here, interference makes it easier to measure periodicity, and this should not come as a big surprise. Physicists routinely use scattering of electromagnetic waves and interference measurements to determine periodicity of physical objects such as crystal lattices.
In an important work, Bennett (the physicist Charles Bennett, 1973) showed that any classical computation can be transformed into a reversible form, which allows quantum mechanics to be used in computation! Thus, we can use localized oscillations in a quantum model. This allows one to use an efficient method for computing the greatest common divisor (GCD) of two integers. The GCD is the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).
SK: It may be useful to note, in considering the PNEFS method, that the Euclidean algorithm itself is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers.
Generally, the GCD never requires more steps than five times the number of digits (base 10) of the smaller integer (Gabriel Lamé ,1844).
EG: The property you describe is sometimes referred to as GCD "switching".
I have noticed that when the dividend and the divisor are relatively prime and >1, then the Euclidean algorithm yields the two Bezout coefficients that have the maximum product (or the minimum product in absolute value, since they have opposite signs); I do not know if there exists a general proof for this, but if the algorithm terminates in one or two steps, it is not difficult to prove.
SK: Another way to see it, is that the reality of wire oscillations, when held between two fixed points, can be fragmented into two different views.
In the first, there are infinite frequencies separated by infinitesimal differences. This is the Fourier view, where the model frequencies are independent of the wire modes, so they do not depend on the wire.
The second view is determined mathematically by applying the first, even though fictional. To every element of the wire, no matter how physically small, one can assign laws of motion, differentially. This second view is calculated mathematically step-by-step in the Appendix of
https://www.researchgate.net/publication/333677283/
with the free book New Physics with the Euler-Lagrange Equation: Going Beyond Newton.
Then, the second view emerges as the physical reality we are supposed to see, out of a fictional reality. This second reality still is of two infinite types, which are obviously incorrect: infinite frequencies going to infinity. We can correct that by a physical realization into PNEFS. Only physical frequencies exist. We see that in a continued fraction expansion, or in an harmonic composition shown above.
These physical frequencies are seen, also experimentally, following a sum partition (as a linear combination of independent oscillators) of the figure above, not as a product.
In the case of 6, one sees oscillations in 3 sets of 2, just with different phases.
.
Check https://www.cuemath.com/numbers/factors-of-6/
https://www.omnicalculator.com/math/factor
Factor is about multiplication so 6 = 1 x 6, 6 = 2 x3 so factors of 6 are 1, 2,3 and 6.
Nice play of operator . + operator is equivalent to x with respect to an integer . except 0,1 .
There exists an important difference when using sums instead of products to "factorize" (or represent) integers.
Since 1 is the identity element under multiplication, it does not change any product when added as a factor to that; i.e. multiplication is insensitive to 1.
On the other hand, 1 is the generator of the group of integers under addition, since every integer n is a multiple of 1; i.e. n=1+1+…+1 (n terms) and in this sense, this is the only irreducible “additive” factorization of n.
In the example of 6, 2+2+2 is reduced to 1+1+1+1+1+1, while 2*3 cannot be further reduced; it is irreducible.
Hence, using additive factorizations, only 1 can be considered as a “prime”, all other integers are “composites”.
SK: I disagree with the concept.
Using addition, the element "0" is the identity element -- as it does not change any addition. The number "1" is not a prime, whatever model is used. The addition model is also physically intuitive, directly connecting to localized waves, and periodicity, visually.
Multiplication is just revealed to be a kind of "shorthand" notation of repeated addition.
WORK: In using the model of addition for partioning, the terms "factorization", "composite number", and "prime number" have to defined accordingly. To wit, one defines:
"composite number": the equivalence class of the sums of positive integers that are not 1, not a prime number, and represent the same value.
Example: the number 6 can be represented by 3+3 and by 2+2+2, or 6 is the equivalence class (2+2+2, 3+3). . Hence, 6 admits the prime numbers 2 and 3.
The conventional definition agrees with this result for s prime , although a different method is used to calculate it. This is assured by the laws of arithmetic, as multiplication is repeated addition.
This also makes factorization unchanged, as a result -- although using different algorithms. This allows one to connect to all existing theorems on factorization.
NOTE: The results shown by me in this thread, include the contribution of several individuals-- some of which I do not know personally, and some of which may come spontaneously online. Therefore, the results may change over time, as it is being refined by intersubjectivity. To mark this temporary but transient nature, the term WORK shall be used. Repeated update is suggested, so as not to use any outdated terms. Any suggestions are always welcome! It is a new model for an online class, where anyone may join, reflecting the nature of online fora.
EG: Where could one read the details of these results? Are they accessible?
By the way, any partition of a set defines an equivalence relation on that set. Here, what is the induced equivalence relation? Is this a new congruence relation?
WORK: Division, as prime numbers are usually defined by, is a difficult integer operation, not in procedure but in results. The division of two integer numbers is not always an integer number.
However, the sum of two integer numbers is always an integer number. It is a complete field, that of finite integer numbers.
Basing factorization on integer addition, thus, not only makes it trivial but also makes it complete. No longer, one sees fractional results.
WORK: In a physical line, we expect the oscillation mode with higher losses to fade first. This means that the mode we see more easily are the lower modes, with less losses. Thus, one expects to see 2+2+2 more easily than 3+3. This seems not important if a physical line is not considered.
WORK: In finding prime numbers by exploring periodicity, one is tempted to use the modular exponential function.
This works as follows. Assume that N is positive and has only two distinct prime factors:
N = p1p2
First, pick any integer a between 2 and N-1, and compute the greatest common divisor (gcd) (N,a). This can be done very efficiently using Euclid’s algorithm, as we saw before.
From now on, let us assume that gcd (N,a)=1 , that is, a and N are co-prime. Let r be the even period of a modulo N, such that a^r = 1 (mod N).
At this point, given N, we have found some pair a,r such that a and N are co-prime, and r is even, and r is the smallest integer that (a^r -1) is a multiple of N, that is a^r = 1 (mod N). Use the identity
a^r -1 = [a^(r/2)-1] [a^(r/2) +1]
to split a^r -1.
Test that a^(r/2) +1 is not a multiple of N, change a or r if needed. Then, neither is a^(r/2)-1 a multiple of N, although their product is.
This is possible only if p1 is a prime factor of N and a^(r/2)-1, or p2 is a prime factor of N and a^(r/2)+1. We have split a^r - 1, non-trivially.
It is worth noting that when the modulus N is the product of two distinct primes p1,p2, there exist exactly four different solutions for the pair of integers (ar\2-1,ar\2+1), as it can be seen by setting x= ar\2 and solving the congruence x2=1 (modN), where N=p1p2.
The existence of the previous four solutions is ensured by a theorem called the Chinese Remainder Theorem, which is a basic result in elementary number theory.
Two of the four solutions are the trivial ones x=1,N-1, which are present in any case N>1.
These two solutions are excluded in EG’s analysis, since then x-1=0, which is a trivial multiple of the product of two primes, or x+1=N, which is again a multiple of the product of two primes.
The other two solutions are non-trivial and include pairs (ar\2-1,ar\2+1) of integers where one integer is a multiple of one of the primes and the other integer is a multiple of the other prime.
In the special case where the modulus N is the product of two twin primes, i.e. of two primes that differ by 2, for instance, 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on, one of the two non-trivial solutions is exactly the pair of the two twin primes, since then the two primes differ by 2, which is equal to the difference of the two integers ar\2+1,ar\2-1 and for this reason such a solution is possible. So, in this case, one can calculate, by the previous analysis, the two twin primes themselves, not just multiples of them.
SK: good observations. I left out the trivial solutions, mainly because they do not satisfy the conditions I stated. That is also why 1 is not a prime number.
The pair (a^(r\2) -1, a^(r\2)+1 ) as an algebraic solution to a^r - 1, can efficiently split even 0, and that gives us the FFT, as I showed earlier above, then using all four solutions! This creates the imaginary plane, and allows us to visualize √-1 as a simple 180 degree rotation, not as an irrational, and use it effectively in mathematics and quantum mechanics. We show that in RG, at
Preprint Overview of Complex Analysis and Applications
WORK: The above led us to consider the famous phrase "a negative times a negative equals a positive" as an easily accessible truth.
We did this by considering a negative number as a 180 rotation, so their product is a 360 degree rotation, back to positive. One cannot achieve the same ease of visualization by considering a negative as a "loss", as done in the movie "Stand and Deliver" at
https://youtu.be/2k_jS1zVLWw
WORK: In simple arithmetic of positive integers, multiplication is repeated addition.
We are then suggesting in this question that the addition partition allows one to calculate primes more effectively, versus the traditional partition in the Fundamental Algebra Theorem using multiplication.
This question is taking us from nowhere to no place.
Why do people like to create problems where none exists?
JRS: Today is not April 1st. But, what you call "nowhere" is the factorization of integers. And what you call "no place" is its finite solution. Both are very valuable, and only the former is known to humans. Nature; however, has had the latter for 13.8 billion years. Time for us to catch up?
As above, to summarize and expand.
Given N, as the product of two primes, as used also in cybersecurity, how can one find them?
Suppose we have found some pair (a,r) such that a and N are co-prime, r is even, and r is the smallest integer that (a^r -1) is a multiple of N, that is a^r = 1 (mod N). There are two non-trivial solutions to (a,r) -- as guaranteed by the Chinese Remainder Theorem.
To split a^r -1, we use the identity
a^r -1 = [a^(r/2)-1] [a^(r/2) +1].
We can use an algebra calculator, also, with Hyper Calc, as
(x^(y / 2) - 1) (x^(y / 2) + 1) which the calculator evaluates as
x^y - 1.
It is then an easier matter to find the solution (x,y) that factors N, by periodicity. Here, x and N are co-prime, y is even, and y is the smallest integer that (x^y -1) is a multiple of N, that is x^y = 1 (mod N).
As noted by SK, if N is formed by twin primes, where the two primes differ by 2, one can calculate, by the previous analysis, the two twin primes themselves, not just multiples of them.
Thus, the use of twin primes is to be avoided in cryptography. This has been exploited recently, to break RSA, and is explained in
https://arstechnica.com/information-technology/2022/03/researcher-uses-600-year-old-algorithm-to-crack-crypto-keys-found-in-the-wild/
Therefore, one should sufficiently randomize the two prime numbers, when used to generate RSA keys.
WORK: the pair (a^(r\2) -1, a^(r\2)+1 ) is an algebraic solution to (a^r - 1), and can efficiently split even 0, as shown above.
This gave us the FFT, using all four solutions. This created the imaginary plane, and allowed us to visualize √-1 as a simple 180 degree rotation, not as an irrational, or somehow an imaginary number, and use it effectively in mathematics and quantum mechanics in the real world.
Now, we can use the same algebra expression to split an irrational -- a number that cannot be counted. Consider the identity:
(√2 - 1)(√2 + 1) = 1
Then, although we cannot calculate √2, we can split √2 into two parts. The same, simpler, for
√2. √2 =2.
We are able to calculate with irrationals.The same goes for the Euler identity, e^(i.pi)=-1.
This makes irrationals part of the real world, answering a question in RG.
WORK: On Procedures and proofs
The use of examples to point to abstract concepts and to general procedures is commonplace pedagogical practice (see e.g. Mason).
In the field of number theory, a case in point might be an explication of the Euclidean algorithm for the greatest common divisor of two natural numbers.
Beginning with, say, 194 and 40 the demonstration proceeds:
194=4 x 40+ 34
40=1 x 34+6
34=5 x 6+4
6=1 x 4+2
4=2 x 2+0
In order to apply the procedure to another pair of natural numbers, the student needs to become aware of the status of each number in each row of the procedure, and how each row relates to the next. That is, not only to agree that each line is a true statement, but to appreciate how it has been initiated and structured.
Read more online, at
Author: Tim Rowland
Title: Proofs in Number Theory: History and Heresy
PUB DATE: 2002-07-00
In: Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education (26th, Norwich, England, July 21-26, 2002)
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