Pseudoprime Statistics to $10^{19}$

Abstract

A base-$b$ pseudoprime (psp) is a composite $N$ satisfying $b^{N-1}\equiv 1\pmod N$. We use computer searches to count odd base-3 psp near $10^n$ for $n$ up to 19. The counts indicate that a good approximation to the probability of a random odd number near $z$ being a psp is $P(z)=z^{-0.59}$. Integrating $P$ yields a psp-counting function, $Q(x)=(x^{0.41})/0.82$, which gives estimated counts close to known actual counts up to 10$^{\mbox{\ASF 19}}$, although these estimates are probably not valid for all $x$.

A table comparing pseudoprime counts up to 10$^{\mbox{\ASF 11}}$ for bases 2, 3, 5, 7, 11, 13, 17, is included.