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{{Short description|Number of the form x^y + y^x}}
In [[number theory]], a '''Leyland number''' is a number of the form
:<math>x^y + y^x</math>
where ''x'' and ''y'' are [[integer]]s greater than 1.<ref name="CP2005">{{citation |author=[[Richard Crandall]] and [[Carl Pomerance]] |title=Prime Numbers: A Computational Perspective |publisher=Springer |year=2005}}</ref> They are named after the mathematician [[Paul Leyland]]. The first few Leyland numbers are

:[[8]], [[17 (number)|17]], [[32 (number)|32]], [[54 (number)|54]], [[57 (number)|57]], [[100 (number)|100]], [[145 (number)|145]], [[177 (number)|177]], [[320 (number)|320]], [[368 (number)|368]], [[512 (number)|512]], [[593 (number)|593]], [[945 (number)|945]], [[1124 (number)|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''<sup>1</sup> + 1<sup>''x''</sup>. Also, because of the [[Commutativity|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'').

==Leyland primes==
A '''Leyland prime''' is a Leyland number that is [[Prime number|prime]]. The first such primes are:

:[[17 (number)|17]], [[593 (number)|593]], 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, ... {{OEIS|id=A094133}}

corresponding to
:3<sup>2</sup>+2<sup>3</sup>, 9<sup>2</sup>+2<sup>9</sup>, 15<sup>2</sup>+2<sup>15</sup>, 21<sup>2</sup>+2<sup>21</sup>, 33<sup>2</sup>+2<sup>33</sup>, 24<sup>5</sup>+5<sup>24</sup>, 56<sup>3</sup>+3<sup>56</sup>, 32<sup>15</sup>+15<sup>32</sup>.<ref name="Leyland">{{cite web |title=Primes and Strong Pseudoprimes of the form x<sup>y</sup> + y<sup>x</sup> |url=http://www.leyland.vispa.com/numth/primes/xyyx.htm |publisher=Paul Leyland |access-date=2007-01-14 |archive-url=https://web.archive.org/web/20070210024511/http://www.leyland.vispa.com/numth/primes/xyyx.htm |archive-date=2007-02-10 |url-status=dead }}</ref>

One can also fix the value of ''y'' and consider the sequence of ''x'' values that gives Leyland primes, for example ''x''<sup>2</sup> + 2<sup>''x''</sup> is prime for ''x'' = 3, 9, 15, 21, 33, 2007, 2127, 3759, ... ({{OEIS2C|id=A064539}}).

By November 2012, the largest Leyland number that had been proven to be prime was 5122<sup>6753</sup> + 6753<sup>5122</sup> with {{val|25050}} digits. From January 2011 to April 2011, it was the largest prime whose primality was proved by [[elliptic curve primality proving]].<ref name="ECPP">{{cite web |title=Elliptic Curve Primality Proof |url=http://primes.utm.edu/top20/page.php?id=27 |publisher=Chris Caldwell |access-date=2011-04-03}}</ref> In December 2012, this was improved by proving the primality of the two numbers 3110<sup>63</sup> + 63<sup>3110</sup> (5596 digits) and 8656<sup>2929</sup> + 2929<sup>8656</sup> ({{val|30008}} digits), the latter of which surpassed the previous record.<ref>{{cite web | title = Mihailescu's CIDE | publisher = mersenneforum.org | date = 2012-12-11 | url = http://mersenneforum.org/showthread.php?t=17554 | access-date = 2012-12-26}}</ref> In February 2023, 104824<sup>5</sup> + 5<sup>104824</sup> ({{val|73269}} digits) was proven to be prime,<ref>{{cite web | title = Leyland prime of the form 104824<sup>5</sup>+5<sup>104824</sup> | publisher = Prime Wiki | url = https://www.rieselprime.de/ziki/Leyland_prime_P_104824_5 | access-date = 2023-11-26}}</ref> and it was also the largest prime proven using ECPP, until three months later a larger (non-Leyland) prime was proven using ECPP.<ref>{{cite web | title = Elliptic Curve Primality Proof | publisher = Prime Pages | url = https://t5k.org/top20/page.php?id=27 | access-date = 2023-11-26}}</ref> There are many larger known [[probable prime]]s such as 314738<sup>9</sup> + 9<sup>314738</sup>,<ref name=PRP>Henri Lifchitz & Renaud Lifchitz, [http://www.primenumbers.net/prptop/searchform.php?form=x%5Ey%2By%5Ex&action=Search PRP Top Records search].</ref> 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 [[Integer factorization|factor]] [[Composite number|composite]] Leyland numbers.<ref name="Kulsha">{{cite web |title=Factorizations of x<sup>y</sup> + y<sup>x</sup> for 1 < y < x < 151 |url=http://www.primefan.ru/xyyxf/default.html |publisher=Andrey Kulsha |access-date=2008-06-24}}</ref>

{{Portal|Mathematics}}

==Leyland number of the second kind==
A '''Leyland number of the second kind''' is a number of the form
:<math>x^y - y^x</math>
where ''x'' and ''y'' are [[integer]]s greater than 1. The first such numbers are:

: 0, 1, [[7]], [[17 (number)|17]], [[28 (number)|28]], [[79 (number)|79]], [[118 (number)|118]], [[192 (number)|192]], [[300 (number)|399]], [[431 (number)|431]], 513, 924, 1844, 1927, 2800, 3952, 6049, 7849, 8023, 13983, 16188, 18954, 32543, 58049, 61318, 61440, 65280, 130783, 162287, 175816, 255583, 261820, ... {{OEIS|id=A045575}}

A '''Leyland prime of the second kind''' is a Leyland number of the second kind that is also prime. The first few such primes are:

:7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, ... {{OEIS|id=A123206}}. We can also consider 145 in the form of 4 to the power of 3 plus 4 to the power of 4.

For the probable primes, see Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.<ref name=PRP></ref>

==References==
{{Reflist}}

==External links==
* {{YouTube | id= Lsu2dIr_c8k | title= Leyland Numbers - Numberphile }}

{{Prime number classes|state=collapsed}}
{{Classes of natural numbers}}

{{DEFAULTSORT:Leyland Number}}
[[Category:Eponymous numbers in mathematics]]
[[Category:Integer sequences]]