A152155 Minimal residues of Pepin's Test for Fermat Numbers using the base 3.

0, -1, -1, -1, -1, 10324303, -6586524273069171148, 110780954395540516579111562860048860420, 5864545399742183862578018016183410025465491904722516203269973267547486512819

Offset: 0

Author

Dennis Martin (dennis.martin(AT)dptechnology.com), Nov 27 2008

Keywords

• sign

Comments

For n>=1 the Fermat Number F(n) is prime if and only if 3^((F(n) - 1)/2) is congruent to -1 (mod F(n)).

Any positive integer k for which the Jacobi symbol (k|F(n)) is -1 can be used as the base instead.

Examples

			a(4) = 3^(32768) (mod 65537) = 65536 = -1 (mod F(4)), therefore F(4) is prime.
a(5) = 3^(2147483648) (mod 4294967297) = 10324303 (mod F(5)), therefore F(5) is composite.
		

References

• M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001, pp. 42-43.

Links

• Dennis Martin, Table of n, a(n) for n = 0..11
• Chris Caldwell, The Prime Pages: Pepin's Test.

Crossrefs

A000215, A019434, A152153, A152154, A152156.

Programs

Maple

f:= proc(n) local F;
   F:= 2^(2^n) + 1;
   `mods`(3 &^ ((F-1)/2), F)
end proc:
seq(f(n), n=0..10); # Robert Israel, Dec 19 2016

PARI

a(n)=centerlift(Mod(3,2^(2^n)+1)^(2^(2^n-1))) \\ Jeppe Stig Nielsen, Dec 19 2016

Formula

a(n) = 3^((F(n) - 1)/2) (mod F(n)), where F(n) is the n-th Fermat Number, using the symmetry mod (so (-F(n)-1)/2 < a(n) < (F(n)-1)/2).