MATHS BITE: Leyland Numbers

A Leyland number is an integer of the form x^y + y^x, 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 x1 + 1x.

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:

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

The largest known Leyland prime is Screen Shot 2016-12-26 at 10.59.12 AM.png.

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