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In number theory, a 'Leyland number' is a number of the form ''x''^''y'' + ''y''^''x'', where ''x'' and ''y'' are natural numbers with 1 < ''x'' ≤ ''y''. The first few Leyland numbers are
8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124
Because of the commutative property of addition, the condition ''x'' ≤ ''y'' could be replaced with 1 < ''y'' without changing the set of Leyland numbers (so we have 1 < ''x'', 1 < ''y''). The requirement that ''x'' and ''y'' both be greater than 1, however, is important, since without it every positive integer would be a Leyland number of the form 1^''y'' + ''y''¹.
The first Leyland numbers that are also prime are listed in . As of January 2007, the largest Leyland number that has been proven to be prime is 2638⁴⁴⁰⁵ + 4405²⁶³⁸. From July 2004 to June 2006 it was the largest proof by Elliptic curve primality proving. [1] There are many larger known probable primes, but it is hard to prove primality of Leyland numbers. Paul Leyland writes on his website: "More recently still, it was realized that numbers of this form are ideal test cases for general purpose primality proving programs. They have a simple algebraic description but no obvious cyclotomic properties which special purpose algorithms can exploit."

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References

References

★ Richard Crandall and Carl Pomerance, ''Prime Numbers : A Computational Perspective'', Springer, 2005

★ Paul Leyland, Primes and Strong Pseudoprimes of the form x^y + y^x. Retrieved on January 14, 2007.