Fermat–Catalan conjecture

The states, that the equation am + bn = ck has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) satisfying the following restrictions:


 * 1) All variables are positive integers;
 * 2) a, b and c are coprime; and
 * 3) 1/m + 1/n + 1/k < 1.

There are currently ten known solutions:


 * 1) 1 + 8 = 9;
 * 2) 32 + 49 = 81;
 * 3) 169 + 343 = 512;
 * 4) 128 + 4,913 = 5,041;
 * 5) 243 + 14,641 = 14,884;
 * 1,406,408,618,241 + 2,399,506,333,156 = 3,805,914,951,397;
 * 2,827,145,944 + 4,899,400,744,681 = 4,902,227,890,625;
 * 794,537,372,728 + 234,466,010,672,089 = 235,260,548,044,817;
 * 410,338,673 + 443,688,652,450,511 = 443,689,062,789,184; and
 * 11,688,200,277.601 + 890,888,060,733,048 = 902,576,261,010,649.