Beal's conjecture

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Beal's conjecture is a conjecture in number theory proposed by the Texas billionaire and amateur mathematician Andrew Beal.

While investigating generalizations of Fermat's last theorem in 1993, Beal formulated the following conjecture:

If

,

where A, B, C, x, y and z are positive integers with x,y,z > 2 then A, B, and C must have a common prime factor.

By computerized searching, greatly accelerated by aid of modular arithmetic, this conjecture has been verified for all values of all six variables up to 1000^[1]So in any counterexample, at least one of the variables must be greater than 1000.

To illustrate, the solution 3³ + 6³ = 3⁵ has bases with a common factor of 3, and the solution 7⁶ + 7⁷ = 98³ has bases with a common factor of 7. Indeed the equation has infinitely many solutions, including for example

for any a, b, m > 3. But no such solution of the equation is a counterexample to the conjecture, since the bases all have the factor a^m + b^m in common.

It can happen that the exponents are pairwise coprime, as for example in 27⁴ + 162³ = 9⁷.

Beal's conjecture is a generalization of Fermat's last theorem, which corresponds to the case x = y = z. If a^x + b^x = c^x with , then either the bases are coprime or share a common factor. If they share a common factor, it can be divided out of each to yield an equation with smaller, coprime bases. In either case, a counterexample to Fermat's Last Theorem would yield a counterexample to Beal's conjecture.

The conjecture is not valid over the larger domain of Gaussian integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided (−2 + i)³ + (−2 − i)³ = (1 + i)⁴.^[2] In this respect Beal's statement differs from Fermat's; given a specific small integer exponent, Fermat's Last Theorem tends to be true, and indeed easier to prove, in a suitable extension of the ring of integers than in the ring of integers alone.

Beal has offered a prize of US$100,000 for a proof of his conjecture or a counterexample^[3].

[edit] References

1. ^ Beal's Conjecture: A Search for Counterexamples
2. ^ Neglected Gaussians
3. ^ The Beal Conjecture

[edit] External links

• http://www.bealconjecture.com/
• http://www.math.unt.edu/~mauldin/beal.html
• http://www.ams.org/notices/199711/beal.pdf
• A search for counterexamples - http://www.norvig.com/beal.html
• http://planetmath.org/encyclopedia/BealsConjecture.html