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# Consecutive integers which are perfect powers

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## Mihăilescu's theorem

In 1844, Eugène Charles Catalan proposed the conjecture (proved in 2002 by Preda Mihăilescu, hence **Mihăilescu's theorem**)

Conjecture (Catalan's conjecture, 1844).(Catalan)

8 and 9 are the only consecutive integers which are perfect powers,

Interestingly, 2 and 3 are the only consecutive primes.

## Consecutive perfect powers with difference equal to *k*

A001597 Perfect powers: where is an integer and

- {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, ...}

Consecutive perfect powers with difference equal to

- 1: (8, 9).
- 2: (25, 27), ?
- 3: (1, 4), (125, 128), ?
- 4: (4, 8), (32, 36), (121, 125), ?
- 5: (27, 32), ?

It seems that all those lists are finite. (CONJECTURE? or PROOF?)