Problems & Puzzles: Puzzles

Puzzle 559.- Mauldin / Tijdeman-Zagier Conjecture

Perhaps you know very well that related to the Fermat-Catalan conjecture there are only 10 known solutions for this equation:

x^p + y^q = z^r; {x, y, z} = coprimes; 1/p+1/q+1/r<1

Please notice that at in each solution one power is 2.

Q1. Would you like to try for an 11th solution?

I have been told by Luis Rodríguez that there is a prize of $100,000 USD for a solution where all exponents are greater than 2.

Q2 Would you like to try for a prized solution?

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n.b.

1) Perhaps it would be nice to mention that the following powers p,q,r generate no further solutions: {2,3,7},{2,3,8},{2,3,9} and {2,4,5}, according to Crandall/Pomerance in "Prime Numbers, a computational approach". It also states: "there are many other triples of exponents for which it has been proved that there are no nontrivial solutions".There are references to numerous authors, in particular Beukers, who gives an overview in his intro of: http://www.math.uu.nl/people/beukers/Fermatlectures.pdf (Jan van Delden)

2) Please also see this link before trying some calculations by your own:http://www.norvig.com/beal.html (C. Rivera)