Leyland number

Leyland number

In number theory, a Leyland number is a number of the form "x""y" + "y""x", where "x" and "y" are integers greater than 1.cite |author=Richard Crandall and Carl Pomerance |title=Prime Numbers: A Computational Perspective |publisher=Springer |date=2005] The first few Leyland numbers are

:8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124 OEIS|id=A076980.

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 "x"1 + 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

:17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEIS2C|id=A094133)

corresponding to:32+23, 92+29, 152+215, 212+221, 332+233, 245+524, 563+356, 3215+1532.cite web |title=Primes and Strong Pseudoprimes of the form xy + yx |url=http://www.leyland.vispa.com/numth/primes/xyyx.htm |publisher=Paul Leyland |accessdate=2007-01-14]

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.cite web |title=Elliptic Curve Primality Proof |url=http://primes.utm.edu/top20/page.php?id=27 |publisher=Chris Caldwell |accessdate=2008-06-24] There are many larger known probable primes such as 913829 + 991382, [Henri Lifchitz & Renaud Lifchitz, [http://www.primenumbers.net/prptop/searchform.php?form=x%5Ey%2By%5Ex&action=Search PRP Top Records search] .] 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.cite web |title=Factorizations of xy + yx for 1 < y < x < 151 |url=http://xyyxf.at.tut.by/default.html |publisher=Andrey Kulsha |accessdate=2008-06-24]

References


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