A053289 First differences of consecutive perfect powers (A001597).

3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, 35, 19, 18, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 23, 68, 71, 10, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 100

Offset

1,1

Comments

Michel Waldschmidt writes: Conjecture 1.3 (Pillai). Let k be a positive integer. The equation x^p - y^q = k where the unknowns x, y, p and q take integer values, all >= 2, has only finitely many solutions (x,y,p,q). This means that in the increasing sequence of perfect powers [A001597] the difference between two consecutive terms [the present sequence] tends to infinity. It is not even known whether for, say, k=2, Pillai's equation has only finitely many solutions. A related open question is whether the number 6 occurs as a difference between two perfect powers. See Sierpiński [1970], problem 238a, p. 116. - Jonathan Vos Post, Feb 18 2008

References

Wacław Sierpiński, 250 problems in elementary number theory, Modern Analytic and Computational Methods in Science and Mathematics, No. 26, American Elsevier, Warsaw, 1970, pp. 21, 115-116.

S. S. Pillai, On the equation 2^x - 3^y = 2^X - 3^Y, Bull, Calcutta Math. Soc. 37 (1945) 15-20.

Links

Daniel Forgues and T. D. Noe, Table of n, a(n) for n = 1..10000

Rafael Jakimczuk, Gaps between consecutive perfect powers, International Mathematical Forum, Vol. 11, No. 9 (2016), pp. 429-437.

Holly Krieger and Brady Haran, Catalan's Conjecture, Numberphile video (2018).

Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.

Formula

a(n) = A001597(n+1) - A001597(n). - Jonathan Vos Post, Feb 18 2008

From Amiram Eldar, Jun 30 2023: (Start)

Formulas from Jakimczuk (2016):

Lim sup_{n->oo} a(n)/(2*n) = 1.

Lim inf_{n->oo} a(n)/(2*n)^(2/3 + eps) = 0. (End)

Example

Consecutive perfect powers are A001597(14) = 121, A001597(13) = 100, so a(13) = 121 - 100 = 21.

Mathematica

Differences@ Select[Range@ 3200, # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &] (* Michael De Vlieger, Jun 30 2016, after Ant King at A001597 *)

Crossrefs

Cf. A053707 (first differences of consecutive perfect prime powers).

Cf. A001597, A025475, A053707, A069623, A219551.

Sequence in context: A016607 A262216 A076446 * A076412 A053707 A075052

Adjacent sequences: A053286 A053287 A053288 * A053290 A053291 A053292

Keyword

nonn

Author

Labos Elemer, Mar 03 2000

Status

approved