Powers in arithmetic progressions

The question that at most how many squares one can find among $N$ consecutive terms of an arithmetic progression, has attracted a lot of attention. An old conjecture of Erd\H{o}s predicted that this number $P_N(2)$ is at most $o(N)$; it was proved by Szemer\'edi. Later, using various deep tools, Bombieri, Granville and Pintz showed that $P_N(2) < O(N^{2/3+o(1)})$, which bound was refined to $O(N^{3/5+o(1)})$ by Bombieri and Zannier. There is a conjecture due to Rudin which predicts a much stronger behavior of $P_N(2)$, namely, that $P_N(2)=O(\sqrt{N})$ should be valid. An even stronger form of this conjecture says that we have $$ P_2(N)=P_{24,1;N}(2)=\sqrt{\frac{8}{3}N}+O(1) $$ for $N\geq 6$, where $P_{24,1;N}(2)$ denotes the number of squares in the arithmetic progression $24n+1$ for $0 \leq n < N$. This stronger form has been recently proved for $N \leq 52$ by Gonz\'alez-Jim\'enez and Xarles. In the talk we take up the problem for arbitrary $\ell$-th powers. First we characterize those arithmetic progressions which contain the most $\ell$-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 $\ell$. Then we formulate analogues of Rudin's conjecture for general powers $\ell$, and we prove these conjectures for $\ell=3$ and $4$ up to $N=19$ and $5$, respectively. The new results presented are joint with Sz. Tengely.Non UBCUnreviewedAuthor affiliation: university of DebrecenFacult