Leyland number

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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 (sequence A076980 in OEIS).

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 + y¹.

The first Leyland numbers that are also prime are

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (A094133)

corresponding to

2³+3², 2⁹+9², 2¹⁵+15², 2²¹+21², 2³³+33², 5²⁴+24⁵, 3⁵⁶+56³, 15³²+32¹⁵.

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 prime whose primality was proved 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."