Computational discovery of the fifth Lerch prime: 42447347

[Veljko Vranić]

Dear SeqFan members,

I completed an exhaustive search of all primes through 50,000,000 and found the new Lerch prime

p = 42,447,347.

It satisfies

sum(a^(p-1), a=1..p-1) - (p-1)! - p == 0 (mod p^3).

The search reproduced Marek Wolf’s previously reported negative intervals and found exactly:

3, 103, 839, 2237, 42447347.

The new value was verified with separate Rust and CPython implementations. Source code, manifests, results, and verification transcripts are available here:

https://github.com/veljkovranic/lerch

Would anyone on the list be willing to independently verify p=42447347, preferably with a separate implementation? Even confirmation of the defining congruence would be extremely helpful before I propose updating OEIS A197632: https://oeis.org/A197632.

Best,
Veljko Vranic

[Geoffrey Caveney]

Veljko, this is a significant accomplishment. Well done!

Your discovery raises the question, what is the value of the Bernoulli number B_42,447,346 ? (See equation (6) on p. 5 of Sondow's preprint in the links of A197632.)

Geoffrey

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[Hugo Pfoertner]

PARI:  is(42447347) gives 1, so 42447347 is confirmed as a term. But it is by orders of magnitude too slow to search significantly beyond limit 100000.

On Thu, Aug 20, 2026 at 9:36 AM Veljko Vranić <bybl...@gmail.com> wrote:

Dear SeqFan members,

I completed an exhaustive search of all primes through 50,000,000 and found the new Lerch prime

p = 42,447,347.

It satisfies

sum(a^(p-1), a=1..p-1) - (p-1)! - p == 0 (mod p^3).

Would anyone on the list be willing to independently verify p=42447347, preferably with a separate implementation? Even confirmation of the defining congruence would be extremely helpful ...

[Veljko Vranić]

Thank you both!

Hugo, thank you for the independent PARI confirmation.

Geoffrey, I computed the Bernoulli number exactly:

B_42,447,346 = N / 254,684,082,

where N has 271,466,759 digits. Also,

42,447,347 * B_42,447,346 is congruent to
49,628,251,800,410,944,737,487 modulo 42,447,347^3.

The full numerator is available here:

https://github.com/veljkovranic/lerch/releases/tag/bernoulli-42447346

I'm also running a search in the 50 million to 200 million range. The simple 1/p heuristic suggests only about a 7% chance of another hit there, but I guess we will see.

Best,
Veljko

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