The requirement thatx andy both be greater than 1 is important, since without it every positive integer would be a Leyland number of the formx1 + 1x. Also, because of thecommutative property of addition, the conditionx ≥y is usually added to avoid double-covering the set of Leyland numbers (so we have 1 <y ≤x).
One can also fix the value ofy and consider the sequence ofx values that gives Leyland primes, for examplex2 + 2x is prime forx = 3, 9, 15, 21, 33, 2007, 2127, 3759, ... (OEIS: A064539).
By November 2012, the largest Leyland number that had been proven to be prime was 51226753 + 67535122 with25050 digits. From January 2011 to April 2011, it was the largest prime whose primality was proved byelliptic curve primality proving.[3] 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.[4] In February 2023, 1048245 + 5104824 (73269 digits) was proven to be prime,[5] and it was also the largest prime proven using ECPP, until three months later a larger (non-Leyland) prime was proven using ECPP.[6] There are many larger knownprobable primes such as 3147389 + 9314738,[7] 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 obviouscyclotomic properties which special purpose algorithms can exploit."
