2026/07/21 by Hai-Liang Wu
#math.NT #math.CO
Inspired by recent works on Diophantine tuples over finite fields, in this paper we consider Diophantine tuples related to primitive elements of finite fields. Let \mathbbFq be the finite field with q elements and \mathbbFq^*=\mathbbFq∖\0\ be the multiplicative cyclic group of all non-zero elements over \mathbbFq. An element g∈\mathbbFq is called primitive if g generates the group \mathbbFq^*. A set \x1,x2,⋯,xm\⊆\mathbbFq^* of m elements is said to be a P-Diophantine m-tuple over \mathbbFq if xixj+1 is primitive for any 1≤ i≤ j≤ m. Let Nm denote the number of P-Diophantine tuples over \mathbbFq. Then we obtain the asymptotic formula m!⋅ Nm=((φ(q-1))/(q-1))m(m+1)/2qm+Om,r(qm-(1)/(2)+r), where φ(⋅) is the Euler totient function and r∈(0, 1/2) is an arbitrary real number. Moreover, we prove that there exists a P-Diophantine m-tuple over \mathbbFq whenever q≥ exp(exp(m(m+1))).