1729

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

Only the self-educated are free.
DMK

Srinivasa Ramanujan, Architect Rational, (Tamil: ஸ்ரீனிவாஸ ராமானுஜன்; 22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Living in India with no access to the larger mathematical community, which was centred in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician G. H. Hardy, in the same league as mathematicians such as Euler and Gauss. He died at the age of 32. [Wikipedia, revised]

His introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and re-discovered Euler’s identity independently. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.

Nobody in British India was qualified to appreciate his work. In the early 1900’s the center of mathematics in English was in the universities in England.

In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. The first two returned Ramanujan’s papers without comment.  On 16 January 1913, Ramanujan wrote to G. H. Hardy. Hardy noticed that letter contained already know results but also some results were “seemed scarcely possible to believe.” Hardy commented that “they [theorems] defeated me completely; I had never seen anything in the least like them before”. He figured that Ramanujan’s theorems “must be true, because, if they were not true, no one would have the imagination to invent them“. Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardy concluded that the letters were “certainly the most remarkable I have received” and commented that Ramanujan was “a mathematician of the highest quality, a man of altogether exceptional originality and power”. Hardy recognizing the brilliance of his work, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge. Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs and working styles. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man and relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan’s education without interrupting his spell of inspiration.

Ramanujan died of illness, malnutrition, and possibly liver infection in 1920. (His age was 32: 32 is the first member of the 31–aliquot tree.)  An interesting number. 32 is the smallest number n with exactly 7 solutions to the equation φ(x) = n. It is also the sum of the totient function for the first 10 integers.

During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known. He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research.  The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.

For Architects, the world exists primarily to be analyzed, understood, explained – and re-designed. External reality in itself is unimportant, little more than raw material to be organized into structural models. What is important for Architects is that they grasp fundamental principles and natural laws, and that their designs are elegant, that is, efficient and coherent.  Architects show the greatest precision in thought and speech of all the types. [Please Understand Me II]

“I quickly came to recognize that my instincts had been correct; that the mathematical universe had much of value to offer me, which could not be acquired in any other way. I saw that mathematical thought, though nominally garbed in syllogistic dress, was really about patterns; you had to learn to see the patterns through the garb. That was what they called “mathematical maturity”. I learned that it was from such patterns that the insights and theorems really sprang, and I learned to focus on the former rather than the latter.” –– Robert Rosen

What is life?

In 1919 Ramanujan was deathly ill while on a long ride back to India, from February 27 to March 13 on the steamship Nagoya. All he had was a pen and pad of paper, and he wanted to write down his equations before he died.

Ramanujan believed that 17 new functions he discovered were “mock modular forms” that looked like theta functions when written out as an infinite sum (their coefficients get large in the same way), but weren’t super-symmetric. Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri.

“Sir, an equation has no meaning for me unless it expresses a thought of GOD.”

“Architects often seem difficult to know. They are inclined to be shy except with close friends, and their reserve is difficult to penetrate. Able to concentrate better than any other type, they prefer to work quietly at their computers or drafting tables, and often alone. Architects also become obsessed with analysis, and this can seem to shut others out. Once caught up in a thought process, Architects close off and persevere until they comprehend the issue in all its complexity. Architects prize intelligence, and with their grand desire to grasp the structure of the universe, they can seem arrogant and may show impatience with others who have less ability, or who are less driven.” — [David Keirsey]

Ramanujan died before he could prove his hunch. But more than 90 years later, mathematicians proved that these functions mimicked modular forms, but don’t share their defining characteristics, such as super-symmetry.  The expansion of mock modular forms helps physicists compute the entropy, or level of disorder, of black holes.  In developing mock modular forms, Ramanujan was decades ahead of his time; mathematicians only figured out which branch of math these equations belonged to in 2002.

“Authority derived from office, credential, or celebrity does not impress them. Architects are interested only in what make sense, and thus only statements that are consistent and coherent carry any weight with them.” [Please Understand Me II]

“Ramanujan’s legacy, it turns out, is much more important than anything anyone would have guessed when Ramanujan died”

The findings were presented at the Ramanujan 125 (5^3) conference at the University of Florida, ahead of the 125th anniversary of the mathematician’s birth on Dec.(12th month) 22, 2012.

Every positive integer is one of Ramanujan’s personal friends.
John Littlewood

42 (forty-two) is the natural number immediately following 41 and directly preceding 43. The number has received considerable attention in popular culture as a result of its central appearance in The Hitchhiker’s Guide to the Galaxy as the “Answer to The Ultimate Question of Life, the Universe, and Everything“.  Doug Adams wrote the Guide as satire.

Seriously: hey, Doug. You are one off.  42-1=41  Forty-one is a very very interesting number. {0,1,7,41,71}:24

Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime. It is also the sum of the first six prime numbers (2 + 3 + 5 + 7 + 11 + 13), and the sum of three primes (11 + 13 + 17). Forty-one is also the 12th supersingular prime, a Sophie Germain prime and a Newman–Shanks–Williams prime. 41 is the smallest Sophie Germain prime to start a Cunningham chain of the first kind of three terms, {41, 83, 167}. It is an Eisenstein prime, with no imaginary part and real part of the form 3n − 1. 41 is a Proth prime as it is  5 × 2³ + 1. The number figures in the polynomial f(n) = n² + n + 41, which yields primes for −40 ≤ n < 40. This is also called the lucky number of Euler Prime. It is the biggest of such primes.  Forty-one is the sum of two squares, 4² + 5². Adding up the sums of divisors for 1 through 7 yields 41.  41 is the smallest integer whose reciprocal has a 5-digit repetend. That is a consequence of the fact that 41 is a factor of 99999.  It is a centered square number.  The Square root of 5 (an irrational number) — Squared — is 5.

Re: Hollywood fantasy, Jeremy Irons is set to co-star in the production of “The Man Who Knew Infinity,” the biopic of Srinivasa Ramanujan, with Dev Patel starring as the revered Indian mathematician.  Filming is scheduled for 2014.

Any intelligent fool can make things bigger and more complex… It takes a touch of genius – and a lot of courage to move in the opposite direction.  – Albert Einstein

Other Architect Rationals:  Emmy Noether, Paul Dirac, Robert Rosen, David Keirsey, Albert Einstein, Lonnie Athens, David Bohm