A197632 Lerch primes: odd primes that divide their Lerch quotients A197630.

3, 103, 839, 2237

Offset: 1

Author

Jonathan Sondow, Oct 16 2011

Keywords

• nonn
• more

Comments

Odd primes p such that Sum_{a=1..p-1} a^(p-1) - (p-1)! == p (mod p^3). (The congruence holds mod p^2 for any odd prime p; see Lerch (1905).)

Marek Wolf has computed that if a 5th Lerch prime p exists, then 4496113 < p < 18816869 or 18977773 < p < 32452867 or p > 32602373.

Can a number be simultaneously a Lerch prime and a Wilson prime A007540?

René Gy (see links) has shown that a number is simultaneously a Lerch prime and a Wilson prime if and only if it satisfies the congruence (p - 1)! + 1 == 0 (mod p^3). - John Blythe Dobson, Feb 23 2018

Named after the Czech mathematician Mathias Lerch (1860-1922). - Amiram Eldar, Jun 23 2021

Examples

			The 27th prime is 103, and A197631(27) = 0, so 103 is a member.
		

Links

• John Blythe Dobson, A note on Lerch primes, arXiv:1311.2242 [math.NT], 2014.
• John Blythe Dobson, A Characterization of Wilson-Lerch Primes, Integers, Vol. 16 (2016), A51.
• René Gy, Generalized Lerch Primes, Integers, Vol. 18 (2018), A10.
• M. Lerch, Zur Theorie des Fermatschen Quotienten (a^(p-1)-1)/p = q(a), Mathematische Annalen, Vol. 60, No. 4 (1905), pp. 471-490.
• Jonathan Sondow, Lerch quotients, Lerch primes, Fermat-Wilson quotients, and the Wieferich-non-Wilson primes 2, 3, 14771, In: M. Nathanson (ed.), Combinatorial and Additive Number Theory. Springer Proceedings in Mathematics & Statistics, Vol. 101, Springer, New York, NY, 2014, pp. 243-255, preprint, arXiv:1110.3113 [math.NT], 2011-2012.

Crossrefs

Cf. A000720, A007540, A197630, A197631, A308963.

Programs

Mathematica

Cases[Prime[Range[2, 500]], p_ /; Divisible[(Sum[(k^(p-1)-1)/p, {k, 1, p-1}] - ((p-1)! + 1)/p)/p, p]] (* Jean-François Alcover, Nov 21 2018 *)

PARI

is(p)=my(m=p-1,P=p^3); !sum(k=1, m, Mod(k,P)^m,-p-m!) && isprime(p) \\ Charles R Greathouse IV, Jun 18 2012

Formula

A197630(A000720(a(n))) == 0 (mod a(n)).

A197631(A000720(a(n))) = 0.