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Difference between revisions of "Leyland number"
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| − | There are <b>{{#expr:{{formatnum:{{PAGESINCATEGORY:Leyland prime P|pages}}|R}}-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]] | + | There are <b>{{Num|{{#expr:{{formatnum:{{PAGESINCATEGORY:Leyland prime P|pages}}|R}}-2}}}}</b> numbers: [[:Category:Leyland_prime_P_proven|{{Num|{{formatnum:{{PAGESINCATEGORY:Leyland prime P proven|pages}}|R}}}} proven primes]] and [[:Category:Leyland prime P PRP|{{Num|{{formatnum:{{PAGESINCATEGORY:Leyland prime P PRP|pages}}|R}}}} PRP's]] |
*[[:Category:Leyland prime P|Category]] | *[[:Category:Leyland prime P|Category]] | ||
*[[Leyland_prime_P_table|Table]] | *[[Leyland_prime_P_table|Table]] | ||
Revision as of 07:30, 4 August 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 1,814 numbers: 307 proven primes and 1,507 PRP's
Leyland numbers second kind
Reservation history
- x=20001-40000, y=11-200 completed by Serge Batalov, 2014-05-03
- x=15001-20000, y=1001-2000 completed by Serge Batalov, 2014-05-14
- x=40001-330000, y=11-17 completed by Serge Batalov, 2014-05-16
- x=330001-400000, y=11-17 completed by Serge Batalov, 2014-05-17
- x=400001-500000, y=11-17 completed by Serge Batalov, 2014-05-19
- x=20001-30000, y=801-1000 reserved by Dylan Delgado, 2019-07-24
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"
| General numbers |
| Special numbers |
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| Prime numbers |
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