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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 :8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124 . 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 () corresponding to :32+23, 92+29, 152+215, 212+221, 332+233, 245+524, 563+356, 3215+1532.〔(【引用サイトリンク】url=http://www.leyland.vispa.com/numth/primes/xyyx.htm )〕 One can also fix the value of ''y'' and consider the sequence of ''x'' values that gives Leyland primes, for example ''x''2 + 2''x'' is prime for ''x'' = 3, 9, 15, 21, 33, 2007, 2127, 3759, ... (). 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.〔(【引用サイトリンク】url=http://primes.utm.edu/top20/page.php?id=27 )〕 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,〔Henri Lifchitz & Renaud Lifchitz, (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.〔(【引用サイトリンク】url=http://xyyxf.at.tut.by/default.html )〕 ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Leyland number」の詳細全文を読む スポンサード リンク
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