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Leyland number
In number theory, a Leyland number is a number of the form xy + yx, where x and y are integers greater than 1.[1] The first few Leyland numbers are
8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124 (sequence A076980 in OEIS).
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 + 1x. 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
17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEIS A094133)
corresponding to
32+23, 92+29, 152+215, 212+221, 332+233, 245+524, 563+356, 3215+1532.[2]
As of June 2008, the largest Leyland number that has been proven to be prime is 26384405 + 44052638 with 15071 digits. From July 2004 to June 2006, it was the largest prime whose primality was proved by elliptic curve primality proving.[3] There are many larger known probable primes such as 913829 + 991382,[4] 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."
There is a project called XYYXF to factor composite Leyland numbers.[5]
References
^ Richard Crandall and Carl Pomerance (2005), Prime Numbers: A Computational Perspective, Springer
^ "Primes and Strong Pseudoprimes of the form xy + yx". Paul Leyland. Retrieved 2007-01-14.
^ "Elliptic Curve Primality Proof". Chris Caldwell. Retrieved 2008-06-24.
^ Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.
^ "Factorizations of xy + yx for 1 < y < x < 151". Andrey Kulsha. Retrieved 2008-06-24.
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