The Fermat–Catalan conjecture 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 (a, b, c, m, n, k):

  1. (1, 2, 3, m, 3, 2) for any m>6: 1 + 8 = 9;
  2. (2, 7, 3, 5, 2, 4): 32 + 49 = 81;
  3. (13, 7, 2, 2, 3, 9): 169 + 343 = 512;
  4. (2, 17, 71, 7, 3, 2): 128 + 4,913 = 5,041;
  5. (3, 11, 122, 5, 4, 2): 243 + 14,641 = 14,884;
  6. (33, 1549034, 15613, 8, 2, 3): 1,406,408,618,241 + 2,399,506,333,156 = 3,805,914,951,397;
  7. (1414, 2213459, 65, 3, 2, 7): 2,827,145,944 + 4,899,400,744,681 = 4,902,227,890,625;
  8. (9262, 15312283, 113, 3, 2, 7): 794,537,372,728 + 234,466,010,672,089 = 235,260,548,044,817;
  9. (17, 76271, 21063928, 7, 3, 2): 410,338,673 + 443,688,652,450,511 = 443,689,062,789,184; and
  10. (43, 96222, 30042907, 8, 3, 2): 11,688,200,277,601 + 890,888,060,733,048 = 902,576,261,010,649.

Numbers with additional facts

  • The number 343 is also the largest known undulating perfect power with an exponent larger than 2.
  • The number 14,641 is also the first palindromic fourth power with more than one digit. When treated as consecutive single digits, it is also the fourth line of Pascal's triangle.
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