Folha: 'A surprising mathematical proof'

Reproduction of Marcelo Viana's column in Folha de S. Paulo. In 1650, Pietro Mengoli (1626–1686) defended a thesis at the University of Bologna entitled "New Mathematical Quadratures or Addition of Fractions" (in Latin). In it, he proved that the sum 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + … of the inverses of the positive integers is infinite. His argument is still taught in calculus classes today. Mengoli also proved that the sum 1/ ^1² + 1/ ^2² + 1/3² + ¹ /4² ⁺ 1/ ^5² + … of the inverses of the squares of the integers is finite, and asked what its value would be. For more than a century, this question challenged the best mathematicians in Europe, including the brothers Jacob and Johann Bernoulli. It eventually became known as the Basel Problem, in reference to the Swiss city where the Bernoulli family lived. The problem was only solved in 1735 by Leonhard Euler (1707–1783), who was also from Basel and had studied with Johann Bernoulli, but had moved to Saint Petersburg, the capital of the Russian Empire. In a work presented to the Russian Academy of Sciences, Euler proved that the sum 1/ ^1² + ^1/2² + ¹ /3² + ¹ /4² + 1/ ^5² + … is equal to ^π² /6. This success led him to study the expression ς(s) = 1/ ^1s² + 1 ^/ 2s² + 1 ^/ 3s² + 1 ^/ 4s² + 1 ^/5s² + …, which is now known as the Euler–Riemann zeta function, for other values of the exponent s. Euler realized that the function ς is related to prime numbers, and even used it to prove that the sum 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … of the reciprocals of primes is infinite. In particular, he concluded that the quantity of prime numbers must be infinite (Euclid already knew this, but Euler's proof is my favorite!). Read more: Aloisio Araújo is interviewed by Valor Econômico IMPA inaugurates showcase of awards and honors IMPA has an open position for IT Specialist 3 Euler also proved that ς(s) is an irrational number, that is, it is not a fraction of integers, whenever s is an even integer. Surprisingly, the odd case is much more difficult, and remained unsolved for over two hundred years. Thus, it was with great skepticism that, in 1978, the international mathematical community received the announcement that the Frenchman Roger Apéry (1916–1994) had proven that ς(3)=1/1 ³ +1/2 ³ +1/3 ³ +1/4 ³ +1/5 ³ +… is an irrational number. The (healthy!) skepticism of the community was not only related to the fact that it was an old and very difficult problem, it was also linked to Apéry's proof itself. Whenever we seek to prove a mathematical fact, it is crucial to choose a strategy, a logical path formed by intermediate steps that have a good chance of being true and that, in the correct sequence, lead to the intended result. Now, in the case of Apéry's strategy, many of these intermediate steps were totally surprising formulas, they seemed to have fallen from the sky: to this day we do not know how and why he intuited that they would be true. It would only take one to be false for the proof to collapse! But it turns out that all these formulas were correct, and the skeptics were convinced by the proof. On Apéry's tomb in the Père Lachaise cemetery in Paris, a mathematical inscription marks the feat: 1+1/8+1/27+1/64… ≠ p/q. To read the full article, visit the newspaper's website. Also read: 'IMPA Summer' brings together more than 400 participants; IMPA Tech releases list of those qualified for the 2nd phase.