On the integer points on some special hyper-elliptic curves over a finite field

AbstractIf lr(p) is the least positive integral value of x for which y2 ≡ x(x + 1) ⋯ (x + r − 1)(modp) has a solution, we conjecture that lr(p) ≤ r2 − r + 1 with equality for infinitely many primes p. A proof is sketched for r = 5. A further generalization to y2 ≡ (x + a1) ⋯ (x + ar) is suggested, where the a's are fixed positive integers