In number theory, a Pierpont prime is a prime number of the form
2 u ⋅ 3 v + 1 {\displaystyle 2^{u}\cdot 3^{v}+1\,}
for some nonnegative integers u and v. That is, they are the prime numbers p for which p − 1 is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding. Except for 2 and the Fermat primes, every Pierpont prime must be 1 modulo 6. The first few Pierpont primes are:
It has been conjectured that there are infinitely many Pierpont primes, but this remains unproven.
Distribution
A Pierpont prime with v = 0 is of the form 2 u + 1 {\displaystyle 2^{u}+1} , and is therefore a Fermat prime (unless u = 0). If v is positive then u must also be positive (because 3 v + 1 {\displaystyle 3^{v}+1} would be an even number greater than 2 and therefore not prime), and therefore the non-Fermat Pierpont primes all have the form 6k + 1, when k is a positive integer (except for 2, when u = v = 0).
Empirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed; there are 42 Pierpont primes less than 106, 65 less than 109, 157 less than 1020, and 795 less than 10100. There are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. Thus, it is expected that among n-digit numbers of the correct form 2 u ⋅ 3 v + 1 {\displaystyle 2^{u}\cdot 3^{v}+1} , the fraction of these that are prime should be proportional to 1/n, a similar proportion as the proportion of prime numbers among all n-digit numbers. As there are Θ ( n 2 ) {\displaystyle \Theta (n^{2})} numbers of the correct form in this range, there should be Θ ( n ) {\displaystyle \Theta (n)} Pierpont primes. Andrew M. Gleason made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, and more specifically that there should be approximately 9n Pierpont primes up to 10n. According to Gleason's conjecture there are Θ ( log N ) {\displaystyle \Theta (\log N)} Pierpont primes smaller than N, as opposed to the smaller conjectural number O ( log log N ) {\displaystyle O(\log \log N)} of Mersenne primes in that range.
Primality testing When 2 u > 3 v {\displaystyle 2^{u}>3^{v}} , 2 u ⋅ 3 v + 1 {\displaystyle 2^{u}\cdot 3^{v}+1} is a Proth number and thus its primality can be tested by Proth's theorem. On the other hand, when 2 u < 3 v {\displaystyle 2^{u}<3^{v}} alternative primality tests for M = 2 u ⋅ 3 v + 1 {\displaystyle M=2^{u}\cdot 3^{v}+1} are possible based on the factorization of M − 1 {\displaystyle M-1} as a small even number multiplied by a large power of 3.
Pierpont primes found as factors of Fermat numbers As part of the ongoing worldwide search for factors of Fermat numbers, some Pierpont primes have been announced as factors. The following table gives values of m, k, and n such that
The left-hand side is a Fermat number; the right-hand side is a Pierpont prime.
As of 2023, the largest known Pierpont prime is 81 × 220498148 + 1 (6,170,560 decimal digits), whose primality was discovered in June 2023.
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