Leírás
Austrian-Hungarian Diophantine Number Theory seminar
Abstract: The question of how many squares one can find among N consecutive terms of an arithmetic progression, has attracted a lot of attention. Among many other results and questions, a sharp conjecture of Rudin predicts that for this maximum PN (2), we have PN (2)=P24,1;N (2)=(8N/3)1/2+O(1) for N ≥ 6, where P24,1;N (2) denotes the number of squares in the arithmetic progression 24n+1 for 0≤ n<N. In the talk we take up the problem for arbitrary l-th powers. First we characterize those arithmetic progressions which contain the most l-th powers asymptotically. In fact, we can give a complete description, and it turns out that basically the "best" arithmetic progression is unique for any l. Then we formulate analogues of Rudin's conjecture for general powers l, and we prove these conjectures for l=3 and 4 up to N=19 and 5, respectively. The new results presented are joint with Sz. Tengely.
link: https://uni-salzburg.webex.com/uni-salzburg/j.php?MTID=m8db92b832403ae67a12be278f6b61989