The pseudoprimes to 25⋅10⁹

Abstract

The odd composite n ⩽ 25 ⋅ 10 9 n \leqslant 25 \cdot {10^9} such that 2 n − 1 ≡ 1 ( mod n ) {2^{n - 1}} \equiv 1\;\pmod n have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Carmichael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences.