Catalan's conjecture
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu.
23 and 32 are two powers of natural numbers, whose values 8 and 9 respectively are consecutive. The theorem states that this is the only case of two consecutive powers. That is to say, that the only solution in the natural numbers of
- xa − yb = 1
for a, b > 1, x, y > 0 is x = 3, a = 2, y = 2, b = 3.
History
The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (x, y) was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case b = 2.[1]
In 1976, Robert Tijdeman applied Baker's method in transcendence theory to establish a bound on a,b and used existing results bounding x,y in terms of a, b to give an effective upper bound for x,y,a,b. Michel Langevin computed a value of exp exp exp exp 730 for the bound.[2] This resolved Catalan's conjecture for all but a finite number of cases. However, the finite calculation required to complete the proof of the theorem was nonetheless too time-consuming to perform.
Catalan's conjecture was proven by Preda Mihăilescu in April 2002, so it is now sometimes called Mihăilescu's theorem. The proof was published in the Journal für die reine und angewandte Mathematik, 2004. It makes extensive use of the theory of cyclotomic fields and Galois modules. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki.
Generalization
It is a conjecture that for every natural number n, there are only finitely many pairs of perfect powers with difference n; see the list.
(For the smallest number (>0), see 
A103953, and see A076427 for number of solutions (except 0))
| n | numbers k such that k and k + n are both perfect powers | n | numbers k such that k and k + n are both perfect powers |
| 1 | 0, 8 | 33 | 16, 256 |
| 2 | 25 | 34 | None |
| 3 | 1, 125 | 35 | 1, 289, 1296 |
| 4 | 0, 4, 32, 121 | 36 | 0, 64, 1728 |
| 5 | 4, 27 | 37 | 27, 324, 14348907 |
| 6 | None | 38 | 1331 |
| 7 | 1, 9, 25, 121, 32761 | 39 | 25, 361, 961, 10609 |
| 8 | 0, 1, 8, 97336 | 40 | 9, 81, 216, 2704 |
| 9 | 0, 16, 27, 216, 64000 | 41 | 8, 128, 400 |
| 10 | 2187 | 42 | None |
| 11 | 16, 25, 3125, 3364 | 43 | 441 |
| 12 | 4, 2197 | 44 | 81, 100, 125 |
| 13 | 36, 243, 4900 | 45 | 4, 36, 484, 9216 |
| 14 | None | 46 | 243 |
| 15 | 1, 49, 1295029 | 47 | 81, 169, 196, 529, 1681, 250000 |
| 16 | 0, 9, 16, 128 | 48 | 1, 16, 121, 21904 |
| 17 | 8, 32, 64, 512, 79507, 140608, 143384152904 | 49 | 0, 32, 576, 274576 |
| 18 | 9, 225, 343 | 50 | None |
| 19 | 8, 81, 125, 324, 503284356 | 51 | 49, 625 |
| 20 | 16, 196 | 52 | 144 |
| 21 | 4, 100 | 53 | 676, 24336 |
| 22 | 27, 2187 | 54 | 27, 289 |
| 23 | 4, 9, 121, 2025 | 55 | 9, 729, 175561 |
| 24 | 1, 8, 25, 1000, 542939080312 | 56 | 8, 25, 169, 5776 |
| 25 | 0, 100, 144 | 57 | 64, 343, 784 |
| 26 | 1, 42849, 6436343 | 58 | None |
| 27 | 0, 9, 169, 216 | 59 | 841 |
| 28 | 4, 8, 36, 100, 484, 50625, 131044 | 60 | 4, 196, 2515396, 2535525316 |
| 29 | 196 | 61 | 64, 900 |
| 30 | 6859 | 62 | None |
| 31 | 1, 225 | 63 | 1, 81, 961, 183250369 |
| 32 | 0, 4, 32, 49, 7744 | 64 | 0, 36, 64, 225, 512 |
Pillai's conjecture
Pillai's conjecture concerns a general difference of perfect powers (sequence A001597 in OEIS): it is an open problem initially proposed by S. S. Pillai, who conjectured that the gaps in the sequence of perfect powers tend to infinity. This is equivalent to saying that each positive integer occurs only finitely many times as a difference of perfect powers: more generally, in 1931 Pillai conjectured that for fixed positive integers A, B, C the equation 
has only finitely many solutions (x,y,m,n) with (m,n) ≠ (2,2). Pillai proved that the difference 
for any λ less than 1, uniformly in m and n.[3]
The general conjecture would follow from the ABC conjecture.[3][4]
Paul Erdős conjectured that there is some positive constant c such that if d is the difference of a perfect power n,[clarification needed] then d>nc for sufficiently large n.
See also
References
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