A189117 -id:A189117 - cp's OEIS frontend

A023055 (Apparently) differences between adjacent perfect powers (integers of form a^b, a >= 1, b >= 2).

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 37, 38, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 60, 61, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 85, 87, 89, 92, 93, 94, 95, 97, 99, 100

Offset: 1

David W. Wilson

Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference, between 8 and 9.

Alf van der Poorten, Remarks on the sequence of 'perfect' powers

Cf. A001597, A023057.

Cf. A189117 (conjectured number of pairs of consecutive perfect powers differing by n).

A023056 a(n) is least k such that k and k+n are adjacent nontrivial powers of positive integers, or 0 if no such k apparently exists.

Original entry on oeis.org

8, 25, 1, 4, 27, 0, 9, 97336, 16, 2187, 3125, 2197, 36, 0, 49, 128, 64, 225, 81, 196, 100, 0, 2025, 1000, 144, 42849, 169, 484, 0, 6859, 0, 7744, 256, 0, 289, 1728, 14348907, 1331, 361, 2704, 400, 0, 441, 0, 9216, 0, 529, 21904, 576, 0, 625, 0, 676, 0, 729, 5776, 784, 0

Offset: 1

David W. Wilson

nonn

Searching up to 10^22, the largest term for n <= 1000 is a(618) = 421351^3 = 74805251419106551. - T. D. Noe, Apr 21 2011

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A189117 (conjectured number of pairs of consecutive perfect powers differing by n).

Cf. A103954. (the powerful (A001694) analogous sequence).

nextPerfectPowers[n_] := Block[{k = n + 1}, While[GCD @@ Last /@ FactorInteger@ k == 1, k++ ]; k]; t = Table[0, {100}]; t[[3]] = 1; m = 0; While[m < 14400000, n = nextPerfectPowers@ m; d = n - m; If[d < 100 && t[[d]] == 0, t[[d]] = m; Print[{d, m}]]; m = n]; t (* Robert G. Wilson v, May 29 2009 *)
(* checked against *) mx = 14400000; pp = Union[ Join[{1}, Flatten[ Table[n^i, {n, 2, Sqrt@mx}, {i, 2, Log[n, mx]}]]]]; d = Rest@ pp - Most@ pp; pp[[ # ]] & /@ Flatten[ Table[ Position[d, n, 1, 1], {n, 56}] /. {{} -> {0}}] /. {List -> 0} (* Robert G. Wilson v, May 29 2009 *)

A076427 Number of solutions to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1.

Original entry on oeis.org

1, 1, 2, 3, 2, 0, 5, 3, 4, 1, 4, 2, 3, 0, 3, 3, 7, 3, 5, 2, 2, 2, 4, 5, 2, 3, 3, 7, 1, 1, 2, 4, 2, 0, 3, 2, 3, 1, 4, 4, 3, 0, 1, 3, 4, 1, 6, 4, 3, 0, 2, 1, 2, 2, 3, 4, 3, 0, 1, 4, 2, 0, 4, 4, 4, 0, 2, 5, 2, 0, 4, 4, 6, 2, 3, 3, 2, 0, 4, 4, 4, 0, 2, 2, 2, 0, 3, 3, 6, 0, 3, 4, 4, 2, 4, 5, 3, 2, 4, 10

Offset: 1

T. D. Noe, Oct 11 2002

This is the classic Diophantine equation of S. S. Pillai, who conjectured that there are only a finite number of solutions for each n. A generalization of Catalan's conjecture that a^x-b^y=1 has only one solution. For n <=100, a total of 274 solutions were found for perfect powers less than 10^12. No additional solutions were found for perfect powers < 10^18.

			a(4)=3 because there are 3 solutions: 4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2.

R. K. Guy, Unsolved Problems in Number Theory, D9.
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

T. D. Noe, Solutions to Pillai's Equation for n<=100
Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
Dorin Andrica and George C. Ţurkaş, An elliptic Diophantine equation from the study of partitions, Stud. Univ. Babeş-Bolyai Math. (2019) Vol. 64, No. 3, 349-356.
M. A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Dana Mackenzie, 2184: An absurd (and adsurd) tale, Integers (2018) 18, Article #A33.
Roswitha Rissner and Daniel Windisch, Absolute irreducibility of the binomial polynomials, arXiv:2009.02322 [math.AC], 2020.
Eric Weisstein's World of Mathematics, Pillai's Conjecture

Cf. A189117, A001597, A074981.

A264752 First differences of A106543.

Original entry on oeis.org

4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Gionata Neri, Jul 10 2016

nonn

It is conjectured that, for n > 1, a(n) < 4.

Cf. A106543, A189117.

cp's OEIS Frontend

A023055 (Apparently) differences between adjacent perfect powers (integers of form a^b, a >= 1, b >= 2).

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A023056 a(n) is least k such that k and k+n are adjacent nontrivial powers of positive integers, or 0 if no such k apparently exists.

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A076427 Number of solutions to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1.

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A264752 First differences of A106543.

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