Beal conjecture

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

Ax+By=Cz (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(am+bm)]m+[b(am+bm)]m=(am+bm)m+1

for m3, and a,b such that the are nonzero. But computerized searching brings forth quite a few additional solutions, such as:

33+63 =35
39+543 =311
36+183 =38
76+77 =983
274+1623 =97
2113+31653 =4224
3863+48253 =5794
3073+6144 =52193
54003+904 =6304
2173+56423 =6514
2713+8134 =75883
6023+9034 =87293
6243+143523 =3125
18623+577223 =37244
22463+44924 =741183
18383+974143 =55144

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