Partitions: Exact Approximations

… there is something strange going on with Primes
Paul Erdös


Never mind the mock theta, Ramanujan’s gap, Namagiri dreams.


When Srinivasa Ramanujan wrote to G. H. Hardy in the 16th of January 1913, he had some remarkable formulas in that letter.  So remarkable are some of his formulas that mathematicians have been studying Ramanujan’s notebooks of formulas for new mathematical insights to this day, more than a hundred years later.
I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras… I have no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling”. 
Hardy invited him to England because some of the formulas “had to be true, because no one could have the imagination to make them up”.   But there were no proofs.  However, when this poor vegetarian Indian Hindu came to England, eventually Hardy showed Ramanujan (thru Littlewood) that his formula on Primes was not EXACTLY correct. So Ramanujan had to bend to Hardy and work on his proofs of some of his formulas, so when they tackled the function of Partitions P(n), Ramanujan with the help of Hardy got to point where they “cracked” Partitions (and could prove it). They developed a direct formula that computed the number of partitions pretty accurately, and at the limit (infinity) it was “perfect” — and, could by truncating the number for high partition number to an integer could guarantee to be exact: since the number of partitions of integers is an whole number (i.e., the real number series “formula” converges with an deceasing error rate). Together they “cracked” the problem where neither man could do it alone. Ramanujan supplied the “intuition” (the finding of the hidden pattern) and Hardy provided the rigor to explain why the pattern is true.  The method they created, in this instance, was called the “circle method” — and it has been used ever since by numerous mathematicians for various other results.

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A Master of Us All

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.Albert Einstein

This garden universe vibrates complete,
Some, we get a sound so sweet.
Vibrations, reach on up to become light,
And then through gamma, out of sight.
Between the eyes and ears there lie,
The sounds of color and the light of a sigh.
And to hear the sun, what a thing to believe,
But it’s all around if we could but perceive.
To know ultra-violet, infra-red, and x-rays,
Beauty to find in so many ways.

Two notes of the chord, that’s our poor scope,
And to reach the chord is our life’s hope.
And to name the chord is important to some,
So they give it a word, and the word is OM.
Graeme Edge, Moody Blues


There have been many times I have heard an individual say something to the effect: I hate (don’t like, don’t do) “math.”  Shame on our ignorant “math” educational system, but now with the Khan Academy there is no excuse for such a lament from our kids.

I always enjoyed “math” — hey, I am a nerd from the sixties.  But I “hit a wall” in mathematics in the second-year of college. And even modern “mathematicians” have to “specialize” because the “difficulties” of “mathematics”. Practically all believe that mathematics, even for the “mathematicians,” is too vast and large to now to “see” the whole elephant (so to speak).

In mathematics you don’t understand things. You just get used to them.
— John Von Neumann

Someday information science [in some form of Formatics] will encompass all of math and science, but until then:

The religion of Mathematics is a Master of Us All.

Except when in the 1700’s,
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The Number


I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.

No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways”

— G. H. Hardy

Euler Function on the Complex Plane

Yep, it is ALSO the first absolute Euler pseudoprime.  Unique.  Sui Generis.


  Just like The Man Who Knew Infinity:
 Srinivasa Ramanujan:   Sui Generis

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