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Собрание статейСринивасы Рамануджана

Сриниваса Рамануджан · 1911 · пересказ с английского оригинала

Collected Papers of Srinivasa Ramanujan

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Уровень1 · Как в книге2 · Формулы с объяснением3 · Формулы словами4 · Идеи и примеры5 · Общая картина
ГолосМужской · Charon
РисункиГрафика тушьюБелым по чёрному

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Тот же плеер, который продукт даёт для вашей книги: голос, рисунки по мере того, как они рисуются, текст.

Первая статья, которую гений-самоучка опубликовал в 23 года в Мадрасе: странные закономерности чисел Бернулли, чисел, стоящих за суммами степеней. Большинство его утверждений верны, некоторые — догадки, одно ложно; лекции показывают, какое есть какое.

Это пробная часть: 1 из 37 (статьи).

Из лекций

The function x cot x is written two ways, compared term by term in a box, which hands out the numbers; then a bar half as long shows the work saved by pairing with its twin.
Where the Bernoulli Numbers Live

Write it twice, compare

That function is the cotangent from school trigonometry, multiplied by the angle. Written out near zero as a long sum of powers, its successive terms carry the Bernoulli numbers one after another; Ramanujan takes them all as positive and calls the zeroth one minus one so that the first term fits.

On the left, the denominator rule marked as a true theorem of Clausen and von Staudt; on the right, the prime-numerator guess broken by 174611 = 283 × 617.
Reading Laws From the Specimens

True rule, failed guess

A close cousin of the rule says that the part of each Bernoulli number after its whole part is made of one over each of those primes, added up, minus one.

A single box, whole-number series, with arrows to three results it would give: fractional parts, equal denominators, oddness; the box is marked unproved, while the three results are marked true.
From Rough Size to Exact Value, and Proofs Left Open

One unproved claim

Then he tries to prove his own patterns. His plan is to show that certain infinite sums, built from one over the squares of shifted whole numbers and from the primes above three, can be rewritten as series whose coefficients are all whole numbers.

The Bernoulli number with index 444: a fraction from the prime rule, a whole part of 631 digits starting 1751, a numerator of 639 digits starting 1588, and known last digits; part of it rests on the unproved prime guess.
How Big, and What It Ends With

The giant B444

Third, the showpiece: the Bernoulli number with index four hundred and forty-four. In 1911 writing it out by hand was out of the question, yet Ramanujan describes it in detail.

The fifth roots of one on a pentagon bring in the golden ratio ≈ 1.618 and whole numbers 1, 3, 4, 7, 11; each five-interval formula keeps Bernoulli numbers spaced ten apart; the four-interval case uses √2; one trick, many shortcuts.
The General Pattern, and What the Editors Said

Pentagon and golden ratio

To show it working, he carries out the five-interval case. The fifth roots of one bring in the golden ratio, the number about one point six one eight that appears in sunflowers, in the Fibonacci numbers and in classical architecture.

Why read it: a self-taught mind rediscovering a theorem, finding shortcuts, guessing patterns; later with Hardy he adds rigour; recap of the interval formulas, the verdict and his earliest manner.
The General Pattern, and What the Editors Said

Before the rigour

Why is this still worth reading? Because it shows a mathematical mind at the very start: self-taught, without access to the literature, rediscovering a famous theorem, finding fast computational shortcuts, and guessing patterns from data with a sure instinct for which ones matter.

Как это сделано

Примеры для: a curious adult with school maths who likes number puzzles and knows Ramanujan's story; add one modern instance per idea, such as a calculator, computer algebra or cryptography

Источник

Лекции пересказывают английского текст: the Cambridge 1927 edition (Internet Archive pli.kerala.rare.28155). Автор умер в 1920 году; произведение перешло в общественное достояние. Пересказ, рисунки и голос сделаны ИИ и сверены с книгой второй моделью; вручную ничего не правилось. Как используется ИИ.

Лекции можно свободно читать, делиться ими и цитировать со ссылкой на эту страницу.

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