An Accurate Heuristic for a Problem of Shparlinski

ABSTRACT

In this note, we apply classical results from number theory to give an affirmative, but heuristic, answer to the question of Shparlinski (Japanese J. Math., 2012) whether there exist infinitely many primes p of the form p = k² + lk + 1, with integers k, l, such that k > 0 and . Based on a heuristic argument, we provide a formula for the number of such primes, which is surprisingly accurate as computations show.

2000 AMS Subject Classification:

• 11A41
• 11N13

KEYWORDS:

• primes in arithmetic progressions
• Siegel-Walfisz theorem

Acknowledgment

I would like to thank Igor Shparlinski for his help and valuable comments, as well as the anonymous referee for various constructive suggestions.