Abstract

The Bernoulli numbers are among the most interesting and important number sequences in mathematics. They first appeared in the posthumous work “Ars Conjectandi" (1713) by Jakob Bernoulli (1654-1705) in connection with sums of powers of consecutive integers. Bernoulli numbers are particularly important in number theory, especially in connection with Fermat's last theorem. They also appear in the calculus of finite diferences (Nörlund (1924)), in combinatorics, and inother fields.LetusdefineSk(x)=1k +2k +...+xk wherexandkare nonnegative integers. Equations of the shape Sk(x) = yn have been considered by several authors (Bennett, Györy, Hajdu, Pintér, Schäffer, Tijdeman, ... ). In 1985, J. W.S. Cassels solved the Diophantine equation (x-1)3 +x3 +(x+1)3 = y2 in integers x and y, showing that the only solutions satisfy x = 0; 1; 2 and 24. In this talk, we first give old/new results about the Diophantine equation (x + 1)k + (x + 2)k + ... + (x+r)k = yn (*). We prove that (*) has finitely many integer solutions with x, y > 0, n > 1, k≠3. Then, extending some old results about the Diophantine equation (x - d)2 + x2 + (x + d)2 = yn (**), we give an explicit formula for all positive integer solutions of the equation (**) when n is na odd prime and d = pr, p > 3 a prime. These works are joint with Daniele Bartoli and Maohua Le.

Local e Data

Tema: SOME RECENT RESULTS ON THE DIOPHANTINE EQUATIONS WITH POWER SUMS
Palestrante: Gökhan Soydan Bursa Uludag University (Turquia)
Data: 15/09/2021