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A '''Leyland number''' is a number that can be expressed in the form <math>x^y+y^x</math>, where x and y are positive integers. These numbers are named after [[Paul Leyland]], who first studied these numbers.
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A '''Leyland number''' is a number that can be expressed in the form <math>x^y+y^x</math>, where x and y are positive integers greater 1 and x &ge; 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 {{OEIS|A076980}}.
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The first few nontrivial Leyland numbers are given by OEIS sequence [https://oeis.org/A076980 A076980].
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 +
A '''Leyland prime''' is a Leyland number which is also a [[prime]] (see {{OEIS|l|A094133}}).
 +
 
 +
The second kind of numbers are of the form <math>x^y-y^x</math>.
  
 
==History==
 
==History==
  
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==See also==
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==Data==
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[[:Category:Leyland prime P]]
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The data tables contains for every number the x and y values, the number of digits, the Leyland number<ref>[https://www.mersenneforum.org/showpost.php?p=521992&postcount=263 Leyland number] by [[Hans Havermann ]]</ref>, dates and persons of finding and proving if available and the program used to prove a prime.
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 +
===Leyland numbers===
 +
There are <b>{{Num|{{#expr:{{PAGESINCATEGORY:Leyland prime P|pages|R}}-2}}}}</b> numbers: [[:Category:Leyland_prime_P_proven|{{Num|{{PAGESINCATEGORY:Leyland prime P proven|pages|R}}}} proven primes]] and [[:Category:Leyland prime P PRP|{{Num|{{PAGESINCATEGORY:Leyland prime P PRP|pages|R}}}} probable primes]].
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*[[:Category:Leyland prime P|Category]]
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*[[Leyland_prime_P_table|Table]]
 +
 
 +
===Leyland numbers second kind===
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*[[:Category:Leyland prime M|Category]]
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 +
==Reservation history==
 +
===First kind (plus form)===
 +
{| class="wikitable"
 +
! 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
 +
|-
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| 5201 - 5500 || || || [[Greg Childers]] || 2003-07-21 || 2003-09-17 || 17
 +
|-
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| 5501 - 7500 || || || [[Anatoly Selevich]] || 2003-06-25 || 2004-08-07 || 154
 +
|-
 +
| 20001 - 40000 || 11 - 200 || || [[Serge Batalov]] || {{HistLink|2014-05-13|373378}} || {{HistLink|2014-05-14|373406}} || 17
 +
|-
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| 15001 - 20000 || 1001 - 2000 || || [[Serge Batalov]] || {{HistLink|2014-05-14|373426}} || {{HistLink|2014-05-14|373448}} || 20
 +
|-
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| 40001 - 330000 || 11 - 17 || || [[Serge Batalov]] || {{HistLink|2014-05-15|373560}} || {{HistLink|2014-05-16|373654}} || 3
 +
|-
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| 330001 - 400000 || 11 - 17 || || [[Serge Batalov]] || {{HistLink|2014-05-16|373670}} || {{HistLink|2014-05-17|373705}} || -
 +
|-
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| 400001 - 500000 || 11 - 17 || || [[Serge Batalov]] || {{HistLink|2014-05-17|373705}} || {{HistLink|2014-05-19|373796}} || -
 +
|-
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| 20001 - 40000 || 201 - 400 || || [[Norbert Schneider]] || {{HistLink|2014-07-09|377764}} || {{HistLink|2015-08-10|407596}} ||
 +
|-
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| 12051 - 12500 || 2001 - x-1 || || [[Norbert Schneider]] || ? || {{HistLink|2014-05-26|379199}} || 4
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|-
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| 2501 - 3000 || 12501 - 15000 || || [[Vincent Gautier]] || {{HistLink|2014-11-04|386875}} || {{HistLink|2014-11-14|387631}} || 6
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|-
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| 300000 - 500000 || 18 || || [[Andrey Kulsha]] || {{HistLink|2014-12-02|388875}} || ||
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|-
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| 40001 - 50000 || 19 - 400 || || [[Norbert Schneider]] || {{HistLink|2014-12-02|388890}} || {{HistLink|2015-10-23|413431}} || -
 +
|-
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| 50001 - 500000 || 19 - 25 || || [[Norbert Schneider]] || {{HistLink|2015-01-26|393632}} || ||
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|-
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| 12501 - 13000 || 3001 - x-1 || || [[Norbert Schneider]] || {{HistLink|2015-10-23|413431}} || {{HistLink|2016-04-21|432114}} || -
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|-
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| 13000 - 15,000 || || || [[Norbert Schneider]] || {{HistLink|2019-03-27|511972}} || ||
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|-
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| 20001 - 50000 || 401 - 800 || || [[Norbert Schneider]] || {{HistLink|2016-04-21|432114}} || ||
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|-
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| 20001 - 30000 || 801 - 1000 || || [[Dylan Delgado]] || {{HistLink|2019-07-24|522221}} || ||
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|}
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 +
====Double checks====
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*{{HistLink|2014-07-27|379174}}: [[Mark Rodenkirch]] checked x ≤ 12500, y ≤ 2500
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*{{HistLink|2014-08-03|379608}}: [[Mark Rodenkirch]] checked x ≤ 12500, y ≤ 5000
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*{{HistLink|2014-07-27|379174}}: [[Mark Rodenkirch]] checked x ≤ 12500, 10000 < y ≤ 12500
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*{{HistLink|2014-08-07|379896}}: [[Mark Rodenkirch]] checked x ≤ 12500, y ≤ 7000
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*{{HistLink|2014-08-28|381597}}: [[Mark Rodenkirch]] checked x ≤ 12500, y ≤ 12500
 +
 
 +
===Second kind (minus form)===
 +
{| class="wikitable"
 +
! Range x !! Range y !! Digits !! Reserved by !! Reserved !! Completed !! Found
 +
|-
 +
| 3 - 10200 || || || [[Norbert Schneider]] || 2010 || 2015-04-03 || 1553
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|-
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| 10201 - 10400 || || || [[Norbert Schneider]] || 2015-04-05 || 2015-06-15 || 30
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|-
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| 10401 - 10600 || || || [[Norbert Schneider]] || 2015-07-23 || 2015-10-21 || 33
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|-
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| 10601 - 10800 || || || [[Norbert Schneider]] || 2015-10-29 || 2016-01-21 || 20
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|-
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| 10801 - 11000 || || || [[Norbert Schneider]] || 2016-01-27 || 2016-06-04 || 24
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|-
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| 11000 - 11300 || || || [[Norbert Schneider]] || 2016-02-19 || 2016-10-19 || 34
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|-
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| 11300 - 12400 || || || [[Mark Rodenkirch]] || 2016-03-26 || 2016-07-22 || 118
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|-
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| 12401 - 12500 || || || [[Norbert Schneider]] || 2012-08-29 || 2013-08-27 || 13
 +
|-
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| 12501 - 13000 || || || [[Norbert Schneider]] || 2016-04-10 || || 35
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|-
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| 13001 - 15000 || 41 - 1200 || || [[Norbert Schneider]] || || ||
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|-
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| 15001 - 20000 || 41 - 100 || || [[Dylan Delgado]] || 2019-08-04 || 2019-08-05|| 5
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|-
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| 30001 - 30100 || || || [[Norbert Schneider]] || 2014-08-12 ||  || 4
 +
|-
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| 70001 - 100000 || 11 - 20 || || [[Dylan Delgado]] || 2019-08-05 || 2019-08-17 || 1
 +
|-
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| 70001 - 140000 || || || [[Norbert Schneider]] ||  ||  ||
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|}
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 +
==Contribution of Leyland numbers==
 +
This graph can be found [https://www.mersenneforum.org/showpost.php?p=522753&postcount=279 here]:
 +
 
 +
[[File:Leyland P contrib.png|1000px]]
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 +
==References==
 +
<references/>
  
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[[Category:Number]]
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==External links==
 +
*[[Wikipedia:Leyland_number|Leyland number]]
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*[https://www.mersenneforum.org/forumdisplay.php?f=110 Main thread] of [[XYYXF Project]] at [[MersenneForum]]
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*[https://www.mersenneforum.org/showthread.php?t=19347 Current search] for Leyland PRP's at [[MersenneForum]]
 +
*[https://www.mersenneforum.org/showthread.php?t=19348 Prime proofs] of Leyland numbers at [[MersenneForum]]
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*[https://mersenneforum.org/showthread.php?t=21240 Search] of x^y-y^x numbers at [[MersenneForum]]
 +
*[http://primfakt.atw.hu/ Search page] of [[Norbert Schneider]]
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*[http://www.leyland.vispa.com/numth/factorization/main.htm Homepage] of [[Paul Leyland]]
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*[http://www.leyland.vispa.com/numth/primes/xyyx.htm Page] of Leyland numbers, dated 2006-10-06 by P.Leyland
 +
*[http://www.primefan.ru/xyyxf/primes.html Homepage] of [[Andrey Kulsha]], dated 2017-01-04
 +
*[https://groups.yahoo.com/neo/groups/xyyxf/info Yahoo group], 2005 to 2016
 +
*{{OEIS|l|A061119}} of n^2 + 2^n
 +
*{{OEIS|l|A253471}} of n^3 + 3^n
 +
*[https://www.youtube.com/watch?v=Lsu2dIr_c8k YouTube "Leyland Numbers - Numberphile"]
 +
{{Navbox NumberClasses}}
 +
[[Category:Leyland prime| ]]

Latest revision as of 09:59, 5 January 2023

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 greater 1 and 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].

History

Data

The data tables contains for every number the x and y values, the number of digits, the Leyland number[1], 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.

Leyland numbers second kind

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

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:

Leyland P contrib.png

References

  1. ↑ Leyland number by Hans Havermann

External links

Number classes
General numbers
Special numbers
Prime numbers
Retrieved from "https://www.rieselprime.de/z/index.php?title=Leyland_number&oldid=26551"