A076427 -id:A076427 - 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.

A076438 Numbers k 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 = k, with a > 0, b > 0, x > 1, y > 1.

Original entry on oeis.org

1, 2, 10, 29, 30, 38, 43, 46, 52, 59, 122, 126, 138, 142, 146, 150, 154, 166, 170, 173, 181, 190, 194, 214, 222, 234, 263, 270, 282, 283, 298, 317, 318, 332, 338, 342, 347, 349, 354, 361, 370, 379, 382, 383, 386, 406, 419, 428, 436, 461, 467, 479, 484, 486

Offset: 1

T. D. Noe, Oct 12 2002

This is the classic Diophantine equation of S. S. Pillai, who conjectured that there are only a finite number of solutions for each k. A generalization of Catalan's conjecture that a^x - b^y = 1 has only one solution. See A076427 for the number of solutions for each k. Interestingly, the unique solutions (k,a,x,b,y) fall into two groups: (A076439) those in which x and y are even numbers, so that k is the difference of squares, and (A076440) those requiring an odd power. 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 for n <= 1000.
Eric Weisstein's World of Mathematics, Pillai's Conjecture.

Cf. A001597, A053289, A074981, A076427, A076438, A076439, A076440, A207079.

A189117 Conjectured number of pairs of consecutive perfect powers (A001597) differing by n.

Original entry on oeis.org

1, 1, 2, 3, 1, 0, 2, 1, 3, 1, 2, 1, 3, 0, 2, 1, 5, 2, 3, 1, 1, 0, 1, 2, 1, 2, 1, 3, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 3, 1, 1, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 2, 0, 2, 1, 1, 0, 1, 2, 1, 0, 1, 0, 3, 0, 2, 2, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 5

Offset: 1

T. D. Noe, Apr 16 2011

nonn

Only a(1) is proved. Perfect powers examined up to 10^21. This is similar to A076427, but more restrictive.
Hence, through 10^21, there is only one value in the sequence: Semiprimes which are both one more than a perfect power and one less than another perfect power. This is to perfect powers A001597 approximately as A108278 is to squares. A more exact analogy would be to the set of integers such as 30^2 = 900 since 900-1 = 899 = 29 * 31, and 900+1 = 901 = 17 * 53. A189045 INTERSECTION A189047. a(1) = 26 because 26 = 2 * 13 is semiprime, 26-1 = 25 = 5^2, and 26+1 = 27 = 3^3. - Jonathan Vos Post, Apr 16 2011
Pillai's conjecture is that a(n) is finite for all n. - Charles R Greathouse IV, Apr 30 2012

			1 = 3^2 - 2^3;
2 = 3^3 - 2^5;
3 = 2^2 - 1^2 = 2^7 - 5^3;
4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2.

Cf. A023056 (least k such that k and k+n are consecutive perfect powers).

Cf. A023057 (conjectured n such that a(n)=0).

nn = 10^12; pp = Join[{1}, Union[Flatten[Table[n^i, {i, 2, Log[2, nn]}, {n, 2, nn^(1/i)}]]]]; d = Select[Differences[pp], # <= 100 &]; Table[Count[d, n], {n, 100}]

A074852 Composite n such that n and n+2 are prime powers.

Original entry on oeis.org

9, 25, 27, 81, 125, 6561, 24389, 59049, 161051, 357911, 571787, 1442897, 4782969, 5177717, 14348907, 18191447, 30080231, 73560059, 80062991, 118370771, 127263527, 131872229, 318611987, 344472101, 440711081, 461889917, 590589719

Offset: 1

Benoit Cloitre, Sep 10 2002

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006549, A053702.

list(lim)=my(v=List(),t);lim+=.5;for(e=2,log(lim)\log(3), forprime(p=3, lim^(1/e),ispower(t=p^e+2,,&t); if(isprime(t), listput(v,p^e)))); vecsort(Vec(v))
\\ Charles R Greathouse IV, Apr 30 2012

list(lim)=my(v=List());if(lim>=25,listput(v,25));lim+=.5;for(e=2, log(lim)\log(3), forprime(p=3, lim^(1/e),if(isprime(p^e+2), listput(v, p^e)))); vecsort(Vec(v))
/* This second program assumes A076427(2) = 1 but is about a hundred times faster. I proved that it is correct up to 10^20 without this assumption. */
\\ Charles R Greathouse IV, Apr 30 2012

More terms from Sascha Kurz, Jan 30 2003

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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A076438 Numbers k 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 = k, with a > 0, b > 0, x > 1, y > 1.

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A189117 Conjectured number of pairs of consecutive perfect powers (A001597) differing by n.

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A074852 Composite n such that n and n+2 are prime powers.

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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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