Beal conjecture

The Beal conjecture is the following conjecture in number theory:

Unsolved problem in mathematics:
If where A, B, C, x, y, z are positive integers and x, y, z are ≥ 3, do A, B, and C have a common prime factor?
(more unsolved problems in mathematics)
If
,
where A, B, C, x, y, and z are positive integers with x, y, z ≥ 3, then A, B, and C have a common prime factor.

Equivalently,

The equation has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z ≥ 3.

The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem.[1][2] Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample.[3] The value of the prize has increased several times and is currently $1 million.[4]

In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation,[5] the Mauldin conjecture,[6] and the Tijdeman-Zagier conjecture.[7][8][9]

  1. ^ "Beal Conjecture". American Mathematical Society. Retrieved 21 August 2016.
  2. ^ "Beal Conjecture". Bealconjecture.com. Retrieved 2014-03-06.
  3. ^ Cite error: The named reference Mauldin was invoked but never defined (see the help page).
  4. ^ Cite error: The named reference BealPrize was invoked but never defined (see the help page).
  5. ^ Bennett, Michael A.; Chen, Imin; Dahmen, Sander R.; Yazdani, Soroosh (June 2014). "Generalized Fermat Equations: A Miscellany" (PDF). Simon Fraser University. Retrieved 1 October 2016.
  6. ^ "Mauldin / Tijdeman-Zagier Conjecture". Prime Puzzles. Retrieved 1 October 2016.
  7. ^ Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review. 1 (1).
  8. ^ Michel Waldschmidt (2004). "Open Diophantine Problems". Moscow Mathematical Journal. 4: 245–305. arXiv:math/0312440. doi:10.17323/1609-4514-2004-4-1-245-305. S2CID 11845578.
  9. ^ Cite error: The named reference PrimeNumbers was invoked but never defined (see the help page).