Problems & Puzzles: Conjectures

Conjecture 31. The Fermat-Catalan & Beal's Conjectures

Here we deal with the Diophantine equation x^p + y^q = z^r, for positive x, y, z, p, q & r values. As a matter of fact this equation generates two conjectures:

The Fermat-Catalan Conjecture

x^p + y^q = z^r
has only a finite number of solutions if x, y & z are coprimes and 1/p+1/q+1/r<=1.

As a matter of fact nowadays only 10 solutions are known:

1+2^3 = 3^2 (Catalan)
2^5+7^2 = 3^4
7^3+13^2 = 2^9
2^7+17^3 = 71^2
3^5+11^4 = 122^2
17^7+76271^3 = 21063928^2
1414^3+2213459^2 = 65^7
9262^3+15312283^2 = 113^7
43^8+96222^3 = 30042907^2
33^8+1549034^2 = 15613^3

The last five solutions were found by F. Beukers & D. Zagier (*)

The Beal's Conjecture

For any solution to x^p + y^q = z^r, if (p, q & r)>2 then (x, y & z) are not coprimes

Two small examples are:

2^3+2^3 = 2^4
3^3+6^3 = 3^5

BTW, solving this conjecture - or finding a counterxample - has a cash prize of \$100,000 USD.

A good starting point to search for the 11th solution and/or the counterexample to Beal's conjecture, is this web page by Peter Norvig, who is Chief of the Computational Sciences Division at the NASA Ames Research Center.

Questions:

1) Can you find an eleventh solution to x^p + y^q = z^r or demonstrate that there are no more solutions?
2) Can you find a
counterexample to the Beal's conjecture?

______________
(*) According to p. 383, "Prime Numbers, a computational perspective", R. Crandall & C. Pomerance, Springer-Verlag, N.Y., 2001, 2nd printing.

Other web-references are:
http://mathworld.wolfram.com/Fermat-CatalanConjecture.html
http://mathworld.wolfram.com/BealsConjecture.html

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