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A076438
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Numbers 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.
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8
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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
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OFFSET
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1,2
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COMMENTS
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This is the classic Diophantine equation of S. S. Pillai, who conjectured that there are only a finite number of solutions for each n. 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 n. Interestingly, the unique solutions (n,a,x,b,y) fall into two groups: (A076439) those in which x and y are even numbers, so that n 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.
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, D9.
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
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LINKS
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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STATUS
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approved
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