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Difference between revisions of "Leyland number"
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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. | + | 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 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 [ | + | The first few nontrivial Leyland numbers are given by OEIS sequence {{OEIS|A076980}}. |
| + | |||
| + | 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== | ||
| − | == | + | ==Data== |
| − | [[:Category:Leyland prime P]] | + | 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 prooving if available and the program used to proove a prime. |
| + | |||
| + | ===Leyland numbers=== | ||
| + | There are <b>{{#expr:{{PAGESINCATEGORY:Leyland prime P|pages}}-2}}</b> numbers: [[:Category:Leyland_prime_P_proven|{{PAGESINCATEGORY:Leyland prime P proven|pages}} proven primes]] and [[:Category:Leyland prime P PRP|{{PAGESINCATEGORY:Leyland prime P PRP|pages}} PRP's]] | ||
| + | *[[:Category:Leyland prime P|Category]] | ||
| + | *[[Leyland_prime_P_table|Table]] | ||
| + | |||
| + | ===Leyland numbers second kind=== | ||
| + | *[[:Category:Leyland prime M|Category]] | ||
| + | |||
| + | ==Reservation history== | ||
| + | |||
| + | ==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]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
| + | |||
| + | ==External links== | ||
| + | *[[Wikipedia:Leyland_number|Leyland number]] | ||
| + | *[https://www.mersenneforum.org/forumdisplay.php?f=110 Main thread] of [[XYYXF Project]] at [[MersenneForum]] | ||
| + | *[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]] | ||
| + | *[http://www.leyland.vispa.com/numth/factorization/main.htm Homepage] of [[Paul Leyland]] | ||
| + | *[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"] | ||
| + | |||
| − | [[Category: | + | [[Category:Leyland prime| ]] |
Revision as of 23:03, 31 July 2019
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].
Contents
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 prooving if available and the program used to proove a prime.
Leyland numbers
There are Expression error: Unrecognized punctuation character ",". numbers: 307 proven primes and 1,507 PRP's
Leyland numbers second kind
Reservation history
Contribution of Leyland numbers
This graph can be found here:
References
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
- 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"
