A Leyland number is an integer of the form , where *x* and *y* are integers greater than 1. This condition is very important as, without it, every positive integer would be a Leyland number of the form *x*^{1} + 1^{x}.

They are named after Paul Leyland, a British number theorist who studied the factorisation of integers and primality testing.

Leyland numbers are of interest as some of them are very large primes.

### Leyland Primes

A Leyland prime is a Leyland number that is also prime. The first of such primes are:

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, …

which correspond to:

3^{2}+2^{3}, 9^{2}+2^{9}, 15^{2}+2^{15}, 21^{2}+2^{21}, 33^{2}+2^{33}, 24^{5}+5^{24}, 56^{3}+3^{56}, 32^{15}+15^{32}, …

The largest known Leyland prime is .

M x

You may find reading about exponential Diophantine Equations useful : https://en.wikipedia.org/wiki/Catalan's_conjecture

Just think: How will you solve for x and y so that Leyland number is a prime.

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