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.{{citation |author=Richard Crandall and Carl Pomerance |title=Prime Numbers: A Computational Perspective |publisher=Springer |year=2005}} 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 {{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 x¹ + 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).

Leyland primes

A Leyland prime is a Leyland number that is prime. The first such primes are:

:17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, ... {{OEIS|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 x^y + y^x |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 }}

One can also fix the value of y and consider the sequence of x values that gives Leyland primes, for example x² + 2^x 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 5122⁶⁷⁵³ + 6753⁵¹²² with {{val|25050}} digits. From January 2011 to April 2011, 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 |access-date=2011-04-03}} In December 2012, this was improved by proving the primality of the two numbers 3110⁶³ + 63³¹¹⁰ (5596 digits) and 8656²⁹²⁹ + 2929⁸⁶⁵⁶ ({{val|30008}} digits), the latter of which surpassed the previous record.{{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}} In February 2023, 104824⁵ + 5¹⁰⁴⁸²⁴ ({{val|73269}} digits) was proven to be prime,{{cite web | title = Leyland prime of the form 104824⁵+5¹⁰⁴⁸²⁴ | publisher = Prime Wiki | url = https://www.rieselprime.de/ziki/Leyland_prime_P_104824_5 | access-date = 2023-11-26}} and it was also the largest prime proven using ECPP, until three months later a larger (non-Leyland) prime was proven using ECPP.{{cite web | title = Elliptic Curve Primality Proof | publisher = Prime Pages | url = https://t5k.org/top20/page.php?id=27 | access-date = 2023-11-26}} There are many larger known probable primes such as 314738⁹ + 9³¹⁴⁷³⁸,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 x^y + y^x for 1 < y < x < 151 |url=http://www.primefan.ru/xyyxf/default.html |publisher=Andrey Kulsha |access-date=2008-06-24}}

Leyland number of the second kind

A Leyland number of the second kind is a number of the form

: $x^y - y^x$

where x and y are integers greater than 1. The first such numbers are:

: 0, 1, 7, 17, 28, 79, 118, 192, 399, 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.

References

External links

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

Category:Eponymous numbers in mathematics

Category:Integer sequences