Leyland number

In number theory, a Leyland number is a number of the form

where x and y are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are

The requirement that x and y both be greater than 1 is important, since without it every positive integer would be a Leyland number of the form x¹ + 1^x. Also, because of the commutative property of addition, the condition x ≥ y is usually added to avoid double-covering the set of Leyland numbers (so we have 1 < y ≤ x).

The first prime Leyland numbers are

corresponding to

One can also fix the value of y and consider the sequence of x values that gives Leyland primes, for example x² + 2^x is prime for x = 3, 9, 15, 21, 33, 2007, 2127, 3759, ... ( A064539).

By November 2012, the largest Leyland number that had been proven to be prime was 5122⁶⁷⁵³ + 6753⁵¹²² with 25050 digits. From January 2011 to April 2011, it was the largest prime whose primality was proved by elliptic curve primality proving. In December 2012, this was improved by proving the primality of the two numbers 3110⁶³ + 63³¹¹⁰ (5596 digits) and 8656²⁹²⁹ + 2929⁸⁶⁵⁶ (30008 digits), the latter of which surpassed the previous record. There are many larger known probable primes such as 314738⁹ + 9³¹⁴⁷³⁸, but it is hard to prove primality of large 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."