Beal conjecture

Let $A, B, C, x, y, z$ be nonzero integers such that $x$ , $y$ , and $z$ are all $\geq 3$ , and

A^{x}+B^{y}=C^{z}

(1)

Then $A$ , $B$ , and $C$ (or any two of them) are not relatively prime.

It is clear that the famous statement known as Fermat’s Last Theorem would follow from this stronger claim.

Solutions of equation (1) are not very scarce. One parametric solution is

[a(a^{m}+b^{m})]^{m}+[b(a^{m}+b^{m})]^{m}=(a^{m}+b^{m})^{m+1}

for $m\geq 3$ , and $a, b$ such that the are nonzero. But computerized searching brings forth quite a few additional solutions, such as:

	$\displaystyle 3^{3}+6^{3}$	$\displaystyle=3^{5}$
	$\displaystyle 3^{9}+54^{3}$	$\displaystyle=3^{11}$
	$\displaystyle 3^{6}+18^{3}$	$\displaystyle=3^{8}$
	$\displaystyle 7^{6}+7^{7}$	$\displaystyle=98^{3}$
	$\displaystyle 27^{4}+162^{3}$	$\displaystyle=9^{7}$
	$\displaystyle 211^{3}+3165^{3}$	$\displaystyle=422^{4}$
	$\displaystyle 386^{3}+4825^{3}$	$\displaystyle=579^{4}$
	$\displaystyle 307^{3}+614^{4}$	$\displaystyle=5219^{3}$
	$\displaystyle 5400^{3}+90^{4}$	$\displaystyle=630^{4}$
	$\displaystyle 217^{3}+5642^{3}$	$\displaystyle=651^{4}$
	$\displaystyle 271^{3}+813^{4}$	$\displaystyle=7588^{3}$
	$\displaystyle 602^{3}+903^{4}$	$\displaystyle=8729^{3}$
	$\displaystyle 624^{3}+14352^{3}$	$\displaystyle=312^{5}$
	$\displaystyle 1862^{3}+57722^{3}$	$\displaystyle=3724^{4}$
	$\displaystyle 2246^{3}+4492^{4}$	$\displaystyle=74118^{3}$
	$\displaystyle 1838^{3}+97414^{3}$	$\displaystyle=5514^{4}$

Mysteriously, the summands have a common factor $>1$ in each instance.

Dan Vanderkam has verified the Beal conjecture for all values of all six variables up to 1000, and he provides source code for anyone who wants to repeat the verification for himself. A 64-bit machine is required. See http://www.owlnet.rice.edu/ danvk/beal.html

This conjecture is “wanted in Texas, dead or alive”. For the details, plus some additional , see http://www.math.unt.edu/ mauldin/beal.htmlMauldin.

Title	Beal conjecture
Canonical name	BealConjecture
Date of creation	2013-03-22 13:16:53
Last modified on	2013-03-22 13:16:53
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	16
Author	mathcam (2727)
Entry type	Conjecture
Classification	msc 11D41
Synonym	Beal’s conjecture