Euler Number Prime

An Euler number prime is an Euler number  such that the absolute value  is prime (the absolute value is needed since  takes on alternating positive and negative values for even indices). Note that these numbers are distinct from a different type of prime known as Euler primes.

The first few Euler number primes  occur for , 6, 38, 454, 510, ... (OEIS A103234), corresponding to 5, , , ... (OEIS A092823).

As of August 2020, the largest known prime Euler number is , which has 1062 decimal digits and was proven to be prime by D. Broadhurst in 2002.

The following table summarizes searches for larger Euler number primes, of which there are none up to a limit of at least .

REFERENCES:

Sloane, N. J. A. Sequences A092823 and A103234 in "The On-Line Encyclopedia of Integer Sequences."