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Double Wieferich Prime Pair

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A pair of prime numbers (p,q) such that

p^(q-1)=1 (mod q^2) and q^(p-1)=1 (mod p^2).

The only known examples are (2, 1093), (3, 1006003), (5 , 1645333507), (83, 4871), (911, 318917), and (2903, 18787).

If the equation of Catalan's Diophantine problem

x^p-y^q=+/-1

has a nontrivial solution in integers x,y and primes p,q greater than 3, then (p,q) must be a double Wieferich pair, as proved in 2000 by Mihailescu (Steiner 1998, Peterson 2000).

See also

Catalan's Conjecture, Wieferich Prime

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References

Peterson, I. "MathTrek: Zeroing In on Catalan's Conjecture." Dec. 4, 2000. https://web.archive.org/web/20040205230110/http://www.sciencenews.org/20001202/mathtrek.asp.Steiner, R. "Class Number Bounds and Catalan's Equation." Math. Comput. 67, 1317-1322, 1998.

Referenced on Wolfram|Alpha

Double Wieferich Prime Pair

Cite this as:

Weisstein, Eric W. "Double Wieferich Prime Pair." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DoubleWieferichPrimePair.html

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