Leyland number
In number theory, a Leyland number is a number of the form
where x and y are integers greater than 1.[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 + 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 ( A094133)
corresponding to
- 32+23, 92+29, 152+215, 212+221, 332+233, 245+524, 563+356, 3215+1532.[2]
One can also fix the value of y and consider the sequence of x values that gives Leyland primes, for example x2 + 2x 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.[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] There are many larger known probable primes such as 3147389 + 9314738,[5] 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.[6]
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 2011-04-03.
- ↑ "Mihailescu's CIDE". mersenneforum.org. 2012-12-11. Retrieved 2012-12-26.
- ↑ Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.
- ↑ "Factorizations of xy + yx for 1 < y < x < 151". Andrey Kulsha. Retrieved 2008-06-24.
External links
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