I am not sure there is an ecstasy of discovery, but there was a time of

self satisfaction knowing you have solved a problem that others do

not even look at or are willing to acknowledge as being correct.

Without the acknowledgement of the correctness of the concept discovered,

and without the associated acceptance by others, it is an empty

achievement.

I am content that I know it is correct, knowing it can never overtake the

power and momentum of prior and generally accepted simpler concepts

that the general populace can understand without question.

I am sure your accomplishment was preceded by many years of study

and hard work, questioning and solving the questions by perseverance,

combined with a sequence of good luck.

A raised glass to your revelations. May you have more.

Thanks, Michael, for your welcome toast. Your contribution, bringing contentment, is valuable because it broke new ground.

To answer your remark about needing the associated acceptance of others, I want to say yes, but at the immediate time of discovery, it does not seem to matter what others think: you perceive the evidence with such dazzling clarity, that you feel transported. That others subsequently confirm what you have discovered is reassuring, but not as intense as the dazzle of discovery.

Surprisingly the first discovery I made of this kind happened at the beginning of my academic career and inevitably led me to others along the same trail. I found that the philosopher Ortega y Gasset had sources to his thinking although this had been denied. After the discovery of the first source, confirmed by similarity in terminology and syntax, I discovered other sources, then found new sources in a related philosopher Unamuno, and yet others in a third author, a fourth, and so on to the present, always trying to clarify the thinkers through their intertexts, their dialogues with uncited others.

Nelson,

I think that I was about 9 years old and probably that I had read somewhere what an atom was. Sometime later, I remember that I made the thought experiment of cutting a substance into smaller and smaller pieces and that it was necessary for this process to have an end in order for a substance for a certain type to have a reality. It was a kind of sudden discovery that was so clear that I remember trying wakening my mother out of bed and telling her about (my) discovery. Later on in my life, I did had some ah ah experience but nothing was has beaten the day I personally discovered the atoms.

Louis, what a beautiful experience, and what a fine way you have of narrating it! It proves to me that to undergo the famous "Eureka!" experience, the researcher, even at the apex of his intellectual powers, must retain something of a child in his character.

Yes, I had a revelation related to the prime numbers.

In one of my novels where the main actors were searching the ways to open a communication-channel to an alien civilization, the distribution of primes had to be examined. So I started with a light and humorous introduction about Goldbach (who studied initially medicine in Konigsberg, the city of I. Kant and where Euler could not go round on the seven bridges) to reach to the famous conjecture: the even integers >2 can be expressed as a sum of two primes. The dummy character of the novel (an aesthete) was perplexed that such a simple statement is not proved yet, so the smart character started to explain about numerical partitions. I designed my aesthete as a person who is illiterate in maths, so we started constructing tables with the minimums and maximums of prime partitions which are adding to even numbers. As I have filled these tables, in a moment of wandering, instead of adding the primes ordered in these partitions, I multiplied them, obtaining an ordered list of semiprimes.

Here a weird thing happened. I observed that these semiprimes could be expressed as a difference between two squares, where the length of the greater square’s side is related to the sum of primes.

I became very excited as this thing could mean a new relation between additivity and multiplication.

I took a little exercise book and filled it with primes, minimax partitions and semiprimes.

My wife found this booklet in the drawer. She started to discuss with me nicely about my childhood, hobbies and priorities, but I realized that in fact she builds up in her head a checklist for psychiatric disorders (both of us are physicians, but she has more responsabilities, being my family-doctor).

As I reached the number 200, the exercise-book was filled and I was convinced that I made a discovery concerning the first 100 of even numbers.

Than catastrophe struck.

I wrote some of the numbers “on the go” and I missed a prime. Actually in a two-minutes brake I filled the number 153 as prime. Without a calculator at hand I did not realized that 153=17x9. To my amazement this error did not ruined the relation between additivity and multiplication.

So I took a new exercise book and filled it with non-prime partitions. The relation was still valid – but in higher dimensions. So, the primes delivered the simplest dimensional relation between additivity and multiplication. This dicovery was really breathtaking and seeing the numbers snapping in place caused a great joy. But I had noone to share with (mathemathical life in our city fills an empty set). Although my wife was very happy when I told her that I’m done “with the numbers”, no more exercise books.

Later I discussed with my father this issue (a retired academician on maths, nowadays working mostly in diophantinian problems) and I was happy again seeing how excited he became learning about the abovementioned relation.

So, this epiphany will serve as a basis to compare our understanding of number theory with the aliens in my to-come- novel.

András, so you are disguising your discovery (with its moment of epiphany) in the form of a novel? Do you have a publisher?

Yes, Nelson, this epiphany should be the capo d'opera of the Adventurous Mathematics.

But to find a publisher it's a more infrequent revelation. Especially during the economical crisis which will last more in this part of Europe, where the centuries of ottoman occupation were followed by 70 years of crude socialism (the light version of communism) and two world wars complicated further the interethnic relations. So, publishers for the general public are precautious and they are aiming for low-risk publications, like vampire- or detective stories and similars.

Primes could be difficult to sell.

I have contacted scientific publishers also, but the minority which answered to my request found an excuse in the lack of my formal mathemathical training.

Reading Umberto Eco's "Foucault-Pendulum" helped me to understand the world of publishers...

Primes : 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41..........

The triangles where the Hypotenuse is one greater than the length of the

longer side, and the base is a prime:

3,4,5

5, 12, 13

7, 24, 25

11, 60, 61

13, 84, 85

17, 144,145

19, 180,181

For two primes of the form n+1 when added together gives 2n+2,

which can be divided by 2 to give 2 (n+1). When you multiply

2 times any number it becomes even.

If I remember right, 2, is the only even prime number.

Can you give some examples of semi-primes.

Michael,

1 is not prime. Primes are greater than 1.

Having primes of the form n and n+1, adding them you'll get 2n+1. You can't divide that to 2 to obtain an integer. The only case is 2+3=5, undivisable to 2.

Right, 2 is the only even prime: is greater than 1 and can be divided only by 1 and itself.

The first semiprimes are the products of primes

4=2*2

6=2*3

9=3*3

10=2*5

14=2*7

15=3*5

21=3*7

22=2*11

25=5*5

and so on to the infinite

A prime that is the base of the triangle can be expressed as an even number

plus 1.

3 is thus 2 +1

5 is 4 +1

7 is 6 +1

9 is 8 +1

11 is 10 +1

etc.

Thus the 3, 4 , 5 triangle is ( n+1 ) , 2n, and 2n+1

The 5, 12, 13 triangle is ( n+1 ) , 3n, and 3n+1

The 7, 24, 25 triangle is (n+1 ), 4n, and 4n+1

The 9, 40, 41 triangle is (n+1 ), 5n, and 5n+1 ( base not prime )

The 11, 60, 61 triangle is (n+1 ), 6n, and 6n+1

The base increases by 2 at each step, but the longer

side multiplier for n increases by 1 at each step.

n itself is also increasing by 2 at each step. ( 2,4,6,8,10,.........)

When you square the numbers for the 3, 4 , 5 triangle you get:

3^2 = ( n+1 ) ( n+1) = n^2 + 2n + 1

4^ 2 = 2n x 2n = 4 n^2

5^2 = ( 2n+1 ) ( 2n+1 ) = 4n^2 + 4n +1

adding the first two together, and making it equal to the square of the

Hypotenuse:

5n^2 + 2n +1 = 4n^2 + 4n +1

which reduces to :

n^2 = 2n which is valid only when n=2.

but we already knew this because we began the first triangle

with n=2:

enough brain damage.

See Michael Stifel.

A.D. 1544.

Not related to primes, but interesting...

Michael,

when did you found this relation?

Maybe this family of Pythagorean triples should be named Stifel-Clark serie?

Prof. Abdulla, your answer makes me want to abandon my field and take up biology. You make cellular signaling sound so easy, but I imagine it is quite difficult to achieve. What do you think about its difficulty or ease?

Dear Nelson

Sure. In one of my searches (substitution of the calcaneal tendon by amniotic membrane - experimental surgery in rabbits ) the result showed that there was regeneration of the resected tendon and replaced by the membrane with similar histological characteristics of the recipient tendon. There is a possibility of amniotic membrane can be harnessed as a potential source of stem cells and this made ​​me feel happy, proud and as another contributor to improving the quality of life of many people

Dear Dr. Abdalla:

Thank you so much for reviving this string. You have seen evidence of my admiration of your research. I would collaborate with you in any project.

Dear Nelson,

My brother Jay S. Orringer, who is a plastic reconstructive surgeon in Los Angeles, California, has experimented with regenerating facial skin by using dorsal skin with similar histological characteristics. In your case, experimentation with possibly generating stem cells puts your research right on the cutting edge of medical biology.

Both you and Dr. Abdalla are worthy of everyone´s admiration for the potential healing you promote, prefaced by the joy of revelation.

Dear Nelson

Thanks for the words of encouragement

Regards

My ambition on this string was to get researchers to share their original findings. What you have done, all of you, is nothing short of amazing. I want to get Lijo Francis on here to share his discoveries in desalinization. Yes, Dr. Abdalla, this RG forum is a blessing because it is stretching our minds, broadening us to new fields of study.

A big thanks for this thread thanks for the contributers.

Regarding my research, I and my team is believing in alternative and attractive technologies for the desalination of seawater and/or waste water treatment. The result will be more economic production of drinking water and the reuse and reduction of wastewater. The two technologies are namely Forward Osmosis (FO) and Membrane Distillation (MD). Both are membrane based technologies. You know in Saudi Arabia (where I am working) majority of drinking water is produced by a conventional technology called Multi Stage Flash (MSF), in which seawater is boiled by burning the fuel (at the moment Saudi has plenty of fuel eventually the resources are diminishing) and allow to condense the resultant steam and can be used as potable water after some post treatments. So imagine the cost associated with this technology. Other important conventional desalination technology is known as Reverse Osmosis (RO), which requires very high pressures (>60 bars) to produce drinking water from seawater. So this is also expensive but chepaer than MSF.

We have a patended technology for FO and its improved designs are also under patent pending status. FO can desalinate seawater and treat wastewater simultaneously without using high pressure and at ambient temperature.

We have another patent (on membrane development) and one more patent pending technology namely MD, with which it is possible to desalinate seawater or anykind of wastewater using renewable energies.It requires ambient pressure and low temperature 40-60 degree celcius to operate and produce high quality water. We can get this temperature from solar or from various industries (e.g. thermal plants). This is stand alone process and possible to construct very small to large scale. Since these technologies are not yet commercialized. Many research groups are doing lot of efforts to make it into the market and I am proud to be a member of this family.

Regards

Lijo is very modest when describing his discoveries in public, but in private he is very excited about these accomplishments. Thanks for your fine contribution to this thread, my friend.

Andra's asked when I wound this relationship.

My first answer about 3,4,5 and 5,12,13 etc. was a couple of years ago.

I could look up the date that I recorded the concept in the Orange Survey

Books that I use to record Ideas, but ..... I have put them away for

a while in the Basement as my interests have evolved now that I

have found Research Gate.

The second one is one I explored a while back, but this forum got me to

thinking about it, so I just typed it here as I thought the process.

eg .... (n+1), 2n, 2n+1 for 3, 4, 5 etc.

My Orange Survey Books have served me well for several years and I am in

Book number 6 at page 583 A and 583 B. I sign and date each page

as I think of things, pose a question, and make observations, and follow

up with calculations, then pose another equation, and do it again.

I have been working on Growth and Expansion of Planets since

March 31, 2000, so I am a month shy of 14 years, with over 6 feet of

2 inch ring binders full of emails and expanding earth calculations,

and comments, then the discussion group gradually faded away,

and now the replacement group is down to about 3 to five people left,

and I do not participate much any more. It is a typical bell curve

of interest in a subject. it eventually fades away.

When the Olympics are over, maybe I will get them out and see if I

can come up with any new ideas again.

There must be some relationship to the Pythagorean Pyramid:

1

1,1

1,2,1

1,3,3,1

1,4,6,4,1

1,5,10,10,5,1

1,6,15,20,15,6, 1

When I do this stuff, I try to draw pictures, or graphical representations

of what I am thinking about, and If I make a mistake, I just say so in

the survey book, and start again with a new drawing, and a new calculation.

When You run out of ideas, just put it away for a while, and do something else.

I, too, am interested in the awe of innovation in both sciences and humanities. I try to treat this subject at the end of an article on Charles Darwin's prose style which is uploaded in my publications. And I have tried to describe it in an unpublished poem that I uploaded recently.. Each of my publications has been motivated by the aesthetic joy I have felt at discovering something I had not previously known in either the humanities or in science. It is a very elusive subject so that it is difficult to discuss adequately without pointing to my own writing as I felt it. Sorry I can't be of more help.  Kay (Charles Kay Smuth)