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.

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

1,1

Comments

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.

References

R. K. Guy, Unsolved Problems in Number Theory, D9.

T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

Links

Table of n, a(n) for n=1..52.

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.

Crossrefs

Cf. A001597, A076438, A076440.

Sequence in context: A344515 A066502 A125870 * A341658 A168474 A341666

Adjacent sequences: A076436 A076437 A076438 * A076440 A076441 A076442

Keyword

hard,nonn

Author

T. D. Noe, Oct 12 2002

Status

approved