Let a prime number generated by Euler's prime-generating polynomial  be known as an Euler prime. (Note that such primes are distinct from prime Euler numbers, which are known here as Euler number primes). Then the first few Euler primes occur for , 2, ..., 39, 42, 43, 45, ... (OEIS A056561), corresponding to the primes 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, ... (OEIS A005846).

As of Feb. 2013, the largest known Euler prime is , which has 398204 decimal digits and was found by D. Broadhurst (http://primes.utm.edu/primes/page.php?id=111195).

REFERENCES:

DeBenedetto, J. and Rouse, J. "A 60,000 Digit Prime Number of the Form ." 31 Jul 2012. http://arxiv.org/abs/1207.7291.

Sloane, N. J. A. Sequences A005846/M5273 and A056561 in "The On-Line Encyclopedia of Integer Sequences."