Fermat-Catalan conjecture
Contents
Statement
The equation:
with positive integers such that are relatively prime and:
has only finitely many solutions, where, in the special case that or , we count all solutions with distinct values of the corresponding exponent as equivalent.
(Note that sometimes the conjecture is written with strict inequality . This is an equivalent formulation because in all the cases of exact equality with , it is known that there are no solutions).
Status
The conjecture is open. A certain finite list of solutions has been found, but the conjecture does not claim that this is the complete list. Note that to avoid unnecessary duplication, we do not list the mirror solutions to each solution thatcan be obtained by interchanging with and with . The solutions found so far are:
Equation | Value of | Comment | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
, | 1 | 2 | 3 | - | 3 | 2 | 1 | 8 | 9 | Catalan's conjecture states that this is the only solution with . That conjecture has been proved. | |
2 | 7 | 3 | 5 | 2 | 4 | 32 | 49 | 81 | 0.95 = 19/20 | ||
13 | 7 | 2 | 2 | 3 | 9 | 169 | 343 | 512 | 17/18 | ||
2 | 17 | 71 | 7 | 3 | 2 | 128 | 4913 | 5041 | 41/42 | ||
3 | 11 | 122 | 5 | 4 | 2 | 243 | 14641 | 14884 | 0.95 = 19/20 | ||
33 | 1549034 | 15613 | 8 | 2 | 3 | 1406408618241 | 2399506333156 | 3805914951397 | 23/24 | ||
1414 | 2213459 | 65 | 3 | 2 | 7 | 2827145944 | 4899400744681 | 4902227890625 | 41/42 | ||
9262 | 15312283 | 113 | 3 | 2 | 7 | 794537372728 | 234466010672089 | 235260548044817 | 41/42 | ||
17 | 76271 | 21063928 | 7 | 3 | 2 | 410338673 | 443688652450511 | 443689062789184 | 41/42 | ||
43 | 96222 | 30042907 | 8 | 3 | 2 | 11688200277601 | 890888060733048 | 902576261010649 | 23/24 |
Relation with other facts/conjectures
Stronger facts and conjectures
Conjecture | Statement | Status | Proof of implication |
---|---|---|---|
abc conjecture | For every ,there exists such that for all , | open | abc conjecture implies Fermat-Catalan conjecture |
Conjecture/fact | Statement | Status | Nature of relationship |
---|---|---|---|
Catalan's conjecture | The only solution to for and positive is | proved | If a complete set of solutions to the Fermat-Catalan conjecture is found and proved to be complete, Catalan's conjecture would be a special case. |
Fermat's last theorem | has no solutions for and positive integers | proved | The Fermat-Catalan conjecture would imply that there are only finitely many solutions. The Fermat-Catalan conjecture along with an explicit list of solutions would prove Fermat's last theorem. |
Beal conjecture | has no solutions for relatively prime with all at least | The Fermat-Catalan conjecture would imply that there are only finitely many solutions. If none of the solutions to the Fermat-Catalan problem provides a counterexample to the Beal conjecture, this would prove the Beal conjecture. | |
Faltings' theorem | This states that certain kinds of algebraic curves have only finitely many rational points | proved | Faltings' theorem implies that for fixed choice of , there are only finitely many solutions. The Fermat-Catalan conjecture goes further: it says that for all but finitely many permissible choices of , there are no solutions. |