# 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 BealConjecture 2013-03-22 13:16:53 2013-03-22 13:16:53 mathcam (2727) mathcam (2727) 16 mathcam (2727) Conjecture msc 11D41 Beal’s conjecture