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 at Paderborn University.[1][2] The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. That is to say, that

Catalan's conjecture  the only solution in the natural numbers of

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.[3]

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 for the bound,[4] resolving Catalan's conjecture for all but a finite number of cases.

Catalan's conjecture was proven by Preda Mihăilescu in April 2002. 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.[5] In 2005, Mihăilescu published a simplified proof.[6]

Pillai's conjecture

Unsolved problem in mathematics:

Does each positive integer occur only finitely many times as a difference of perfect powers?

Pillai's conjecture concerns a general difference of perfect powers (sequence A001597 in the 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.[7]

The general conjecture would follow from the ABC conjecture.[7][8]

Paul Erdős conjectured that the ascending sequence of perfect powers satisfies for some positive constant c and all sufficiently large n.

Pillai's conjecture means that for every natural number n, there are only finitely many pairs of perfect powers with difference n. The list below shows, for n  64, all solutions for perfect powers less than 1018, as OEIS: A076427. See also OEIS: A103953 for the smallest solution (> 0).

n solution
count
numbers k such that k and k + n
are both perfect powers
n solution
count
numbers k such that k and k + n
are both perfect powers
118 33216, 256
2125 340none
321, 125 3531, 289, 1296
434, 32, 121 36264, 1728
524, 27 37327, 324, 14348907
60none 3811331
751, 9, 25, 121, 32761 39425, 361, 961, 10609
831, 8, 97336 4049, 81, 216, 2704
9416, 27, 216, 64000 4138, 128, 400
1012187 420none
11416, 25, 3125, 3364 431441
1224, 2197 44381, 100, 125
13336, 243, 4900 4544, 36, 484, 9216
140none 461243
1531, 49, 1295029 47681, 169, 196, 529, 1681, 250000
1639, 16, 128 4841, 16, 121, 21904
1778, 32, 64, 512, 79507, 140608, 143384152904 49332, 576, 274576
1839, 225, 343 500none
1958, 81, 125, 324, 503284356 51249, 625
20216, 196 521144
2124, 100 532676, 24336
22227, 2187 54227, 289
2344, 9, 121, 2025 5539, 729, 175561
2451, 8, 25, 1000, 542939080312 5648, 25, 169, 5776
252100, 144 57364, 343, 784
2631, 42849, 6436343 580none
2739, 169, 216 591841
2874, 8, 36, 100, 484, 50625, 131044 6044, 196, 2515396, 2535525316
291196 61264, 900
3016859 620none
3121, 225 6341, 81, 961, 183250369
3244, 32, 49, 7744 64436, 64, 225, 512

See also

Notes

  1. Weisstein, Eric W., Catalan's conjecture, MathWorld
  2. Mihăilescu 2004
  3. Victor-Amédée Lebesgue (1850), "Sur l'impossibilité, en nombres entiers, de l'équation xm=y2+1", Nouvelles annales de mathématiques, 1re série, 9: 178–181
  4. Ribenboim, Paulo (1979), 13 Lectures on Fermat's Last Theorem, Springer-Verlag, p. 236, ISBN 0-387-90432-8, Zbl 0456.10006
  5. Bilu, Yuri (2004), "Catalan's conjecture", Séminaire Bourbaki vol. 2003/04 Exposés 909-923, Astérisque, vol. 294, pp. 1–26
  6. Mihăilescu 2005
  7. 1 2 Narkiewicz, Wladyslaw (2011), Rational Number Theory in the 20th Century: From PNT to FLT, Springer Monographs in Mathematics, Springer-Verlag, pp. 253–254, ISBN 978-0-857-29531-6
  8. Schmidt, Wolfgang M. (1996), Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467 (2nd ed.), Springer-Verlag, p. 207, ISBN 3-540-54058-X, Zbl 0754.11020

References

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