Euler Number Prime

An Euler number prime is an Euler number E_n such that the absolute value |E_n| is prime (the absolute value is needed since E_n 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 E_n occur for n=4, 6, 38, 454, 510, ... (OEIS A103234), corresponding to 5, -61, -23489580527043108252017828576198947741, ... (OEIS A092823).

As of April 2024, the largest known prime Euler number is |E_(510)|, 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 n=100000.

search limitdatesearcher
28688Mar. 21, 2009E. Weisstein
69574Aug. 21, 2020S. Plouffe (pers. comm.)
100000Feb. 20, 2022S. Plouffe (pers. comm.)
112130Apr. 15, 2024S. Plouffe (pers. comm.)

See also

Euler Number, Euler Prime, Integer Sequence Primes, Prime Number

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

Cite this as:

Weisstein, Eric W. "Euler Number Prime." From MathWorld--A Wolfram Web Resource.

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