Leyland number

A Leyland number is a number that can be expressed in the form [math]\displaystyle{ x^y+y^x }[/math], where x and y are positive integers with the condition 1 < x ≤ y. These numbers are named after Paul Leyland, who first studied these numbers in 1994. The first few nontrivial Leyland numbers are given by OEIS sequence A076980.

A Leyland prime is a Leyland number which is also a prime (see sequence A094133 in OEIS).

The second kind of numbers are of the form [math]\displaystyle{ x^y-y^x }[/math].

Data

The data tables contains for every number the x and y values, the number of digits, the Leyland number, dates and persons of finding and proving if available and the program used to prove a prime.

Leyland numbers

There are 1,814 numbers: 307 proven primes and 1,507 probable primes.

• Category
• Table

Leyland numbers second kind

• Category

Reservation history

First kind (plus form)

Range x	Range y	Digits	Reserved by	Reserved	Completed	Found
3 - 1030		~ 2 - 3000	Paul Leyland	1994	1997	88
1031 - 1050		~ 700 - 2800	Paul Leyland	2001-06-04	2001-06-04	4
1051 - 1500		~ 2000 - 4600	Andrey Kulsha	2001-06-03	2001-06-23	41
1501 - 1700		~ 2600 - 5100	Paul Leyland	2001-06-05	2001-06-11	18
1701 - 2000			Christ van Willegen	2001-06-05	2001-06-09	23
2001 - 2200			Greg Childers	2001-06-07	2001-06-09	19
2201 - 2400			Greg Childers	2001-06-09	2001-06-12	17
2401 - 2600			Peter Liaskovsky	2001-08-31	2001-09-03	22
2601 - 2700			Paul Leyland	2001-07-24	2001-08-08	13
2701 - 2800			Greg Childers	2001-07-30	2001-07-31	7
2801 - 3000			Greg Childers	2001-07-31	2001-08-05	14
3001 - 3050			Sander Hoogendoorn	2001-06-14	2001-06-19	2
3051 - 3100			Peter Liaskovsky	2001-09-03	2001-09-05	6
3101 - 3200			Alexander Kuzmich	2001-09-16	2001-10-06	12
3201 - 3379			Leonid Muraviov	2001-11-28	2002-03-28	12
3380 - 3500			Andrey Kulsha	2002-04-19	2002-05-19	7
3501 - 4400			Paul Leyland	2001-12-21	2002-06-27	70
4401 - 4500			Greg Childers	2002-07-13	2002-08-13	9
4501 - 5000			Mark Rodenkirch	2002-09-07	2002-11-19	42
5001 - 5100			Paul Leyland	2003-06-13	2003-06-23	11
5101 - 5200			Paul Leyland	2003-06-23	2003-07-16	5
5201 - 5500			Greg Childers	2003-07-21	2003-09-17	17
5501 - 7500			Anatoly Selevich	2003-06-25	2004-08-07	154
20001 - 40000	11 - 200		Serge Batalov	2014-05-13	2014-05-14	17
15001 - 20000	1001 - 2000		Serge Batalov	2014-05-14	2014-05-14	20
40001 - 330000	11 - 17		Serge Batalov	2014-05-15	2014-05-16	3
330001 - 400000	11 - 17		Serge Batalov	2014-05-16	2014-05-17	-
400001 - 500000	11 - 17		Serge Batalov	2014-05-17	2014-05-19	-
20001 - 40000	201 - 400		Norbert Schneider	2014-07-09	2015-08-10
12051 - 12500	2001 - x-1		Norbert Schneider	?	2014-05-26	4
2501 - 3000	12501 - 15000		Vincent Gautier	2014-11-04	2014-11-14	6
300000 - 500000	18		Andrey Kulsha	2014-12-02
40001 - 50000	19 - 400		Norbert Schneider	2014-12-02	2015-10-23	-
50001 - 500000	19 - 25		Norbert Schneider	2015-01-26
12501 - 13000	3001 - x-1		Norbert Schneider	2015-10-23	2016-04-21	-
13000 - 15,000			Norbert Schneider	2019-03-27
20001 - 50000	401 - 800		Norbert Schneider	2016-04-21
20001 - 30000	801 - 1000		Dylan Delgado	2019-07-24

Double checks

• 2014-07-27: Mark Rodenkirch checked x ≤ 12500, y ≤ 2500
• 2014-08-03: Mark Rodenkirch checked x ≤ 12500, y ≤ 5000
• 2014-07-27: Mark Rodenkirch checked x ≤ 12500, 10000 < y ≤ 12500
• 2014-08-07: Mark Rodenkirch checked x ≤ 12500, y ≤ 7000
• 2014-08-28: Mark Rodenkirch checked x ≤ 12500, y ≤ 12500

Second kind (minus form)

Range x	Range y	Digits	Reserved by	Reserved	Completed	Found
3 - 10200			Norbert Schneider	2010	2015-04-03	1553
10201 - 10400			Norbert Schneider	2015-04-05	2015-06-15	30
10401 - 10600			Norbert Schneider	2015-07-23	2015-10-21	33
10601 - 10800			Norbert Schneider	2015-10-29	2016-01-21	20
10801 - 11000			Norbert Schneider	2016-01-27	2016-06-04	24
11000 - 11300			Norbert Schneider	2016-02-19	2016-10-19	34
11300 - 12400			Mark Rodenkirch	2016-03-26	2016-07-22	118
12401 - 12500			Norbert Schneider	2012-08-29	2013-08-27	13
12501 - 13000			Norbert Schneider	2016-04-10		35
13001 - 15000	41 - 1200		Norbert Schneider
15001 - 20000	41 - 100		Dylan Delgado	2019-08-04	2019-08-05	5
30001 - 30100			Norbert Schneider	2014-08-12		4
70001 - 100000	11 - 20		Dylan Delgado	2019-08-05	2019-08-17	1
70001 - 140000			Norbert Schneider

Contribution of Leyland numbers

This graph can be found here:

External links

• Leyland number
• Main thread of XYYXF Project at MersenneForum
• Current search for Leyland PRP's at MersenneForum
• Prime proofs of Leyland numbers at MersenneForum
• Search of x^y-y^x numbers at MersenneForum
• Search page of Norbert Schneider
• Homepage of Paul Leyland
• Page of Leyland numbers, dated 2006-10-06 by P.Leyland
• Homepage of Andrey Kulsha, dated 2017-01-04
• Yahoo group, 2005 to 2016
• sequence A061119 in OEIS of n^2 + 2^n
• sequence A253471 in OEIS of n^3 + 3^n
• YouTube "Leyland Numbers - Numberphile"

Number classes

General numbers

Integer Natural Even Odd Rational Irrational Real Prime Probable prime

Special numbers

Mersenne 2n-1 Fermat 22n+1 Proth k•2n+1 Riesel k•2n-1 Woodall n•2n-1 Cullen n•2n+1 Double Mersenne 22p-1-1 Perfect 2n-1(2n-1) Leyland xy ± yx Gen.Repunit (b^n-1)/(b-1)

Prime numbers

Mersenne Proth Riesel Williams (b±1)bn±1 Carol/Kynea (bn±1)2-2 Multifactorial n!j±1