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# 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.^{[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 (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 *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 ( A094133)

corresponding to

- 3
^{2}+2^{3}, 9^{2}+2^{9}, 15^{2}+2^{15}, 21^{2}+2^{21}, 33^{2}+2^{33}, 24^{5}+5^{24}, 56^{3}+3^{56}, 32^{15}+15^{32}.^{[2]}

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, ... ( A064539).

By November 2012, the largest Leyland number that had been proven to be prime was 5122^{6753} + 6753^{5122} 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 3110^{63} + 63^{3110} (5596 digits) and 8656^{2929} + 2929^{8656} (30008 digits), the latter of which surpassed the previous record.^{[4]} There are many larger known probable primes such as 314738^{9} + 9^{314738},^{[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]}

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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

Leyland Numbers - Numberphile on YouTube

1 Rising and falling factorials

2 Identities and relations

3 Table of values

4 See also

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