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Vandiver's Criteria

Let p be an irregular prime, and let P=rp+1 be a prime with P<p^2-p. Also let t be an integer such that t^3≢1 (mod P). For an irregular pair (p,2k), form the product

Q_(2k)=t^(-rd/2)product_(b=1)^m(t^(rb)-1)^(b^(p-1-2k)),
(1)

where

m=1/2(p-1)
(2)
d=sum_(n=1)^(m)n^(p-2k).
(3)

If Q_(2k)^r≢1 (mod P) for all such irregular pairs, then Fermat's last theorem holds for exponent p.

See also

Fermat's Last Theorem, Irregular Pair, Irregular Prime

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References

Johnson, W. "Irregular Primes and Cyclotomic Invariants." Math. Comput. 29, 113-120, 1975.

Referenced on Wolfram|Alpha

Vandiver's Criteria

Cite this as:

Weisstein, Eric W. "Vandiver's Criteria." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VandiversCriteria.html

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