Wieferich pair

Wieferich pair

In mathematics, a Wieferich pair is a pair of prime numbers "p" and "q" that satisfy

:"p""q" − 1 ≡ 1 (mod "q"2) and "q""p" − 1 ≡ 1 (mod "p"2)

Wieferich pairs are named after German mathematician Arthur Wieferich.

The only known Wieferich pairs (sequence OEIS2C|id=A124121 and OEIS2C|id=A124122 in OEIS) are (2, 1093), (3, 1006003), (5, 1645333507), (83, 4871), (911, 318917), and (2903, 18787). [MathWorld|title=Double Wieferich Prime Pair|urlname=DoubleWieferichPrimePair]

Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof [cite journal | author=Preda Mihăilescu | authorlink=Preda Mihăilescu | title=Primary Cyclotomic Units and a Proof of Catalan's Conjecture | journal=J. Reine Angew. Math. | volume=572 | year=2004 | pages=167–195 . MathSciNet Featured Review MathSciNet | id = 2076124.] of Mihăilescu's theorem (formerly known as Catalan's conjecture). [ [http://www.math.leidenuniv.nl/~jdaems/scriptie/Catalan.pdf "A Cyclotomic Proof of Catalan's Conjecture"; Jeanine Daems] ]

ee also

*Wieferich prime

References

*
*
* cite journal | author=Ray Steiner | title= Class number bounds and Catalan's equation
journal=Math. Comp. | volume=67 | issue=213 | year=1998 | pages=1317–1322 | url=http://www.ams.org/mcom/1998-67-223/S0025-5718-98-00966-1/home.html | doi=10.1090/S0025-5718-98-00966-1


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