A076438 -id:A076438 - cp's OEIS frontend

A074981 Conjectured list of positive numbers which are not of the form r^i - s^j, where r,s,i,j are integers with r>0, s>0, i>1, j>1.

6, 14, 34, 42, 50, 58, 62, 66, 70, 78, 82, 86, 90, 102, 110, 114, 130, 134, 158, 178, 182, 202, 206, 210, 226, 230, 238, 246, 254, 258, 266, 274, 278, 290, 302, 306, 310, 314, 322, 326, 330, 358, 374, 378, 390, 394, 398, 402, 410, 418, 422, 426

Offset: 1

Zak Seidov, Oct 07 2002

This is a famous hard problem and the terms shown are only conjectured values.
The terms shown are not the difference of two powers below 10^19. - Don Reble
One can immediately represent all odd numbers and multiples of 4 as differences of two squares. - Don Reble
_Ed Pegg Jr_ remarks (Oct 07 2002) that the techniques of Preda Mihailescu (see MathWorld link) might make it possible to prove that 6, 14, ... are indeed members of this sequence.
Numbers n such that there is no solution to Pillai's equation. - T. D. Noe, Oct 12 2002
The terms shown are not the difference of two powers below 10^27. - Mauro Fiorentini, Jan 03 2020

			Examples showing that certain numbers are not in the sequence: 10 = 13^3 - 3^7, 22 = 7^2 - 3^3, 29 = 15^2 - 14^2, 31 = 2^5 - 1, 52 = 14^2 - 12^2, 54 = 3^4 - 3^3, 60 = 2^6 - 2^2, 68 = 10^2 - 2^5, 72 = 3^4 - 3^2, 76 = 5^3 - 7^2, 84 = 10^2 - 2^4, ... 342 = 7^3 - 1^2, ...

R. K. Guy, Unsolved Problems in Number Theory, Sections D9 and B19.
P. Ribenboim, Catalan's Conjecture, Academic Press NY 1994.
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

Mauro Fiorentini, Table of n, a(n) for n = 1..138
A. Baker, Review of "Catalan's conjecture" by P. Ribenboim, Bull. Amer. Math. Soc. 32 (1995), 110-112.
M. E. Bennett, On Some Exponential Equations Of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Yu. F. Bilu, Catalan's Conjecture (after Mihailescu)
J. Boéchat and M. Mischler, La conjecture de Catalan racontée a un ami qui a le temps, arXiv:math/0502350 [math.NT], 2005-2006.
C. K. Caldwell, The Prime Glossary, Catalan's Problem
T. Metsankyla, Catalan's Conjecture: Another old Diophantine problem solved, Bull. Amer. Math. Soc. 41 (2004), 43-57.
Alf van der Poorten, Remarks on the sequence of 'perfect' powers
P. Ribenboim, Catalan's Conjecture, Séminaire de Philosophie et Mathématiques, 6 (1994), pp. 1-11.
P. Ribenboim, Catalan's Conjecture, Amer. Math. Monthly, Vol. 103(7) Aug-Sept 1996, pp. 529-538.
Gérard Villemin, Conjecture de Catalan (French)
Eric Weisstein's World of Mathematics, Draft Proof of Catalan's Conjecture Circulated
Eric Weisstein's World of Mathematics, Pillai's Conjecture
Wikipedia, Catalan's conjecture
Wikipedia, Hall's conjecture

Subsequence of A016825 (see second comment of Don Reble).
n such that A076427(n) = 0. [Corrected by Jonathan Sondow, Apr 14 2014]
For a count of the representations of a number as the difference of two perfect powers, see A076427. The numbers that appear to have unique representations are listed in A076438.
For sequence with similar definition, but allowing negative powers, see A066510.
Cf. A001597, A074980, A069586, A023057, A075788-A075791, A053289, A076438, A207079, A219551.

Corrected by Don Reble and Jud McCranie, Oct 08 2002. Corrections were also sent in by Neil Fernandez, David W. Wilson, and Reinhard Zumkeller.

A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 15, 18, 19, 20, 23, 25, 26, 27, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 64, 67, 71, 72, 74, 76, 79, 81, 83, 87, 89, 95, 100, 104, 106, 107, 109, 112, 116, 118, 121, 124, 125, 126, 127, 128, 135, 139, 143, 146, 147, 148, 150, 151, 152, 153

Offset: 1

Zak Seidov, Apr 28 2005

nonn

A given a(n) can have multiple solutions with distinct (b,c), e.g., a=4 with b=2, c=2 (4 + 2^2 = 2^3) or with b=11, c=5 (4 + 11^2 = 5^3). (See also A181138.) Sequences A106266 and A106267 list the minimal values. - M. F. Hasler, Oct 04 2013
The cubes A000578 = (1, 8, 27, 64, ...) form a subsequence of this sequence, corresponding to b=0, a=c^3. If b=0 is excluded, these terms are not present, except for a few exceptions, a = 216, 343, 12167, ... (6^3 + 28^2 = 10^3, 7^3 + 13^2 = 8^3, 23^3 + 588^2 = 71^3, ...), cf. A038597 for the possible b-values. - M. F. Hasler, Oct 05 2013
This is the complement of A081121. The values do indeed correspond to solutions listed in Gebel's file. - M. F. Hasler, Oct 05 2013
B-file corrected following a remark by Alois P. Heinz, May 24 2019. A double-check would be appreciated in view of two values that were missing, for unknown reasons, in the earlier version of the b-file. - M. F. Hasler, Aug 10 2024

			a = 1,2,4,7,8,11,13,15,18,19,20,23,25,26,27,28,35,39,40,44,45,47,48,49,53, ...
b = 0,5,2,1,0, 4,70, 7, 3,18,14, 2,10, 1, 0, 6,36, 5,52, 9,96,13,4,524,26, ...
c = 1,3,2,2,2, 3,17, 4, 3, 7, 6, 3, 5, 3, 3, 4,11, 4,14, 5,21, 6, 4,65, 9, ...
Here are the values grouped together:
{{1, 0, 1}, {2, 5, 3}, {4, 2, 2}, {7, 1, 2}, {8, 0, 2}, {11, 4, 3}, {13, 70, 17}, {15, 7, 4}, {18, 3, 3}, {19, 18, 7}, {20, 14, 6}, {23, 2, 3}, {25, 10, 5}, {26, 1, 3}, {27, 0, 3}, {28, 6, 4}, {35, 36, 11}, {39, 5, 4}, {40, 52, 14}, {44, 9, 5}, {45, 96, 21}, {47, 13, 6}, {48, 4, 4}, {49, 524, 65}, {53, 26, 9}, {54, 17, 7}, {55, 3, 4}, {56, 76, 18}, {60, 2, 4}, {61, 8, 5}, {63, 1, 4}, {64, 0, 4}, {67, 110, 23}, {71, 21, 8}, ... }
a(2243) = 10000 = 25^3 - 75^2. - _M. F. Hasler_, Oct 05 2013, index corrected Aug 10 2024
a(136) = 366 = 11815^3 - 1284253^2 (has c/a(n) ~ 32.3); a(939) = 3607 = 244772^3 - 121099571^2 (has c/a(n) ~ 67.9); a(1090) = 4265 = 84521^3 - 24572364^2 (has c/a(n) ~ 19.8). - _M. F. Hasler_, Aug 10 2024

M. F. Hasler, Table of n, a(n) for n = 1..2500 (corrected and extended Aug 10 2024)
J. Gebel, Integer points on Mordell curves, negative k values [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A106266, A106267 for respective minimal values of b and c.
Cf. A023055: (Apparent) differences between adjacent perfect powers (integers of form a^b, a >= 1, b >= 2); A076438: n which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1; A076440: n which appear to have a unique representation as the difference of two perfect powers and one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and that solution has odd x or odd y (or both odd); A075772: Difference between n-th perfect power and the closest perfect power, etc.
Cf. A023055, A075772, A076438, A076440.
Cf. A054504, A081121, A081120; A179386 - A179388.

f[n_] := Block[{k = Floor[n^(1/3) + 1]}, While[k < 10^6 && !IntegerQ[ Sqrt[k^3 - n]], k++ ]; If[k == 10^6, 0, k]]; Select[ Range[ 154], f[ # ] != 0 &] (* Robert G. Wilson v, Apr 28 2005 *)

select( {is_A106265(a, L=99)=for(c=sqrtnint(a, 3), (a+9)*L, issquare(c^3-a, &b) && return(c))}, [1..199]) \\ The function is_A106265 returns 0 if n isn't a term, or else the c-value (A106267) which can't be zero if n is a term. The L-value can be used to increase the search limit but so far no instance is known that requires L>68. - M. F. Hasler, Aug 10 2024

a(n) = A106267(n)^3 - A106266(n)^2.

More terms from Robert G. Wilson v, Apr 28 2005

Definition corrected, solutions with b=0 added by M. F. Hasler, Sep 30 2013

A207079 The only nonunique differences between powers of 3 and 2.

Original entry on oeis.org

1, 5, 7, 13, 23

Offset: 1

Gottfried Helms, Feb 15 2012

The sequence is finite, this fact is a theorem in [Bennet2004].
= 3-2 = 3^2-2^3 = 2^2-3.
= 3^2-2^2 = 2^3-3 = 2^5 - 3^3.
= 2^4-3^2 = 3^2 - 2.
= 2^4-3 = 2^8 - 3^5.
= 3^3 - 2^2 = 2^5 - 3^2.

M. A. Bennett, Pillai's conjecture revisited, J. Number Theory 98 (2003), 228-235.
Douglas Edward Iannucci, On duplicate representations as 2^x+3^y for nonnegative integers x and y, arXiv:1907.03347 [math.NT], 2019. Mentions this sequence.

Cf. A053289, A074981, A076438, A219551.

A219551(a(n)) > 1. - Jonathan Sondow, Dec 10 2012

A219551 Number of positive integer solutions to the equation |2^x - 3^y| = n.

Original entry on oeis.org

0, 3, 0, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0

Offset: 0

Jonathan Sondow, Dec 09 2012

Pillai (1931) proved that a(n) is finite for all n.
Hershfeld (1936) computed a(n) for n <= 10 and proved that a(n) <= 2 for all large n.
Stroeker and Tijdeman (1982) proved that a(n) <= 2 for all n > 13.
For additional comments, references, and links, see the crossrefs.
a(n) <= 1 except for n=1, 5, 7, 13, 23:  see e,g, Bennett (2003). - Robert Israel, Mar 06 2017

			1 = 2^2 - 3 = 3 - 2 = 3^2 - 2^3.
5 = 2^3 - 3 = 2^5 - 3^3 = 3^2 - 2^2.
7 = 2^4 - 3^2 = 3^2 - 2.
23 = 2^5 - 3^2 = 3^3 - 2^2 and a(n) <= 2 for n > 13, so a(23) = 2.

S. Pillai, On the inequality 0 < a^x - b^y <= n, Journal Indian Math. Soc., 19 (1931), 1-11.
R. J. Stroeker and R. Tijdeman, Diophantine equations, Computational methods in number theory, Part 2, Math. Cent. Tracts, 155 (1982), 321-369.

M. A. Bennett, On Some Exponential Equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
M. A. Bennett, Pillai’s conjecture revisited, J. Number Theory, 98 (2003) 228-235.
A. Herschfeld, The equation 2^x - 3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231-234.
M. Waldschmidt, Perfect Powers: Pillai's works and their developments, arXiv:0908.4031 [math.NT], 2009.

Cf. A053289, A074981, A076438, A207079.

Clear[seq]; seq[m_] := seq[m] = (Clear[a]; a[A219551%20=%20seq%5Bm%5D%20(*%20_Jean-Fran%C3%A7ois%20Alcover">] = 0; Do[n = Abs[2^x - 3^y]; a[n] = a[n] + 1, {x, 1, m}, {y, 1, m}]; Table[a[n], {n, 0, 10}]); seq[m = 1]; While[seq[m] != seq[m - 1], m = 2*m]; A219551 = seq[m] (* _Jean-François Alcover, Dec 13 2012 *)

a(2n) = a(3n) = 0.

a(n) <= 2 for n > 13.

a(11) - a(30) from Robert Israel, Mar 06 2017

A076440 Numbers k which appear to have a unique representation as the difference of two perfect powers where one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = k, with a > 0, b > 0, x > 1, y > 1 and that solution has odd x or odd y (or both odd).

Original entry on oeis.org

1, 2, 10, 30, 38, 46, 122, 126, 138, 142, 146, 150, 154, 166, 170, 190, 194, 214, 222, 234, 270, 282, 298, 318, 338, 342, 354, 370, 382, 386, 406, 486, 490, 498, 502, 518, 546, 550, 566, 574, 582, 586, 594, 638, 666, 678, 686, 694, 710, 726, 730, 734, 746

Offset: 1

T. D. Noe, Oct 12 2002

There are two types of unique solutions. See A076438 for the general case. This sequence was found by examining all perfect powers (A001597) less than 2^63-1. By examining a larger set of perfect powers, we may discover that some of these numbers do not have a unique representation.

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

M. E. Bennett, On Some Exponential Equations Of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
T. D. Noe, Unique solutions to Pillai's Equation requiring an odd power for n <= 1000.
Eric Weisstein's World of Mathematics, Pillai's Conjecture.

Cf. A001597, A076438, A076439.

A076439 Numbers k which appear to have a unique representation as the difference of two perfect powers where those powers are both 2; that is, there is only one solution to Pillai's equation a^x - b^y = k, with a > 0, b > 0, x > 1, y > 1 and that solution has x = y = 2.

Original entry on oeis.org

29, 43, 52, 59, 173, 181, 263, 283, 317, 332, 347, 349, 361, 379, 383, 419, 428, 436, 461, 467, 479, 484, 491, 509, 523, 529, 569, 571, 607, 613, 619, 641, 643, 653, 661, 677, 691, 709, 733, 773, 787, 788, 811, 827, 839, 853, 877, 881, 883, 907, 911, 941

Offset: 1

T. D. Noe, Oct 12 2002

There are two types of unique solutions. See A076438 for the general case. The k for which the unique solution can be written as k = a^2 - b^2 seems to have the following properties: (1) b = a-1 for odd k and b = a-2 for even k and (2) k = 4^r * p^s, where r is in {0,1}, p is an odd prime and s is in {1,2}. This sequence was found by examining all perfect powers (A001597) less than 2^63-1. By examining a larger set of perfect powers, we may discover that some of these numbers do not have a unique representation.

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

M. E. Bennett, On Some Exponential Equations Of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
T. D. Noe, Unique solutions to Pillai's Equation requiring only squares for n <= 1000.
Eric Weisstein's World of Mathematics, Pillai's Conjecture.

Cf. A001597, A076438, A076440.

A253237 Conjectured largest perfect power k such that k+n is also a perfect power, or 0 if no such k exists.

Original entry on oeis.org

8, 25, 125, 121, 27, 0, 32761, 97336, 64000, 2187, 3364, 2197, 4900, 0, 1295029, 128, 143384152904, 343, 503284356, 196, 100, 2187, 2025, 542939080312, 144, 6436343, 216, 131044, 196, 6859, 225, 7744, 256, 0, 1296, 1728, 14348907, 1331, 10609, 2704, 400, 0, 441, 125, 9216

Offset: 1

Eric Chen, Apr 04 2015

Only a(1) is proven, all other terms (even including a(2)) are only conjectured.
These terms are searched up to 10^18, and no terms are greater than 10^12.
a(n) = A103953(n) for n in A076438.
See A076427 for further information. - M. F. Hasler, Apr 09 2015

Eric Weisstein's World of Mathematics, Pillai's Conjecture.
Wikipedia, Catalan's conjecture.
Wikipedia, Hall's conjecture.

Cf. A001597, A074981, A076427, A076438, A103953.

a(A074981(n)) = 0.

cp's OEIS Frontend

A074981 Conjectured list of positive numbers which are not of the form r^i - s^j, where r,s,i,j are integers with r>0, s>0, i>1, j>1.

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A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

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A207079 The only nonunique differences between powers of 3 and 2.

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A219551 Number of positive integer solutions to the equation |2^x - 3^y| = n.

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A076440 Numbers k which appear to have a unique representation as the difference of two perfect powers where one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = k, with a > 0, b > 0, x > 1, y > 1 and that solution has odd x or odd y (or both odd).

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A076439 Numbers k which appear to have a unique representation as the difference of two perfect powers where those powers are both 2; that is, there is only one solution to Pillai's equation a^x - b^y = k, with a > 0, b > 0, x > 1, y > 1 and that solution has x = y = 2.

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A253237 Conjectured largest perfect power k such that k+n is also a perfect power, or 0 if no such k exists.

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