

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).


3



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
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OFFSET

1,2


COMMENTS

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 2^631. 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..53.
M. E. Bennett, On Some Exponential Equations Of S. S. Pillai,Canad. J. Math. 53 (2001), 897922.
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


CROSSREFS

Cf. A001597, A076438, A076439.
Sequence in context: A098425 A098408 A063564 * A047198 A290461 A162524
Adjacent sequences: A076437 A076438 A076439 * A076441 A076442 A076443


KEYWORD

hard,nonn


AUTHOR

T. D. Noe, Oct 12 2002


STATUS

approved



