Generalized Diophantine 𝑚-tuples

Abstract

For non-zero integers n n and k ≥ 2 k\geq 2 , a generalized Diophantine m m -tuple with property D k ( n ) D_k(n) is a set of m m positive integers { a 1 , a 2 , … , a m } \{a_1,a_2,\ldots , a_m\} such that a i a j + n a_ia_j + n is a k k -th power for 1 ≤ i > j ≤ m 1\leq i> j\leq m . Define M k ( n ) ≔ sup { | S | : S M_k(n)≔\sup \{|S| : S has property D k ( n ) } D_k(n)\} . In this paper, we study upper bounds on M k ( n ) M_k(n) , as we vary n n over positive integers. In particular, we show that for k ≥ 3 k\geq 3 , M k ( n ) M_k(n) is O ( log ⁡ n ) O(\log n) and further assuming the Paley graph conjecture, M k ( n ) M_k(n) is O ( ( log ⁡ n ) ϵ ) O((\log n)^{\epsilon }) . The problem for k = 2 k=2 was studied by a long list of authors that goes back to Diophantus.