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 April 2024, 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 others up to a limit of at least
.