Leyland number

A Leyland number is an integer of the form x^y + y^x 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 the OEIS)

Because of the commutative property of addition, it is not actually required that y be greater than or equal to x, this simply keeps the algorithm from getting bogged down with duplicates. 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.

Leyland numbers that are also prime are listed in OEIS: A094133. The largest known Leyland number that is also a prime is 2638⁴⁴⁰⁵ + 4405²⁶³⁸.

References

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