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

Author

T. D. Noe, Oct 12 2002

Keywords

• hard
• nonn

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

• 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.