Difference between revisions of "Leyland number"
(begin populating completed reservations from Mersenneforum (Norb’s res. are tricky to follow!)) 
(formatnum) 

Line 12:  Line 12:  
===Leyland numbers===  ===Leyland numbers===  
−  There are <b>{{#expr:{{PAGESINCATEGORY:Leyland prime Ppages}}2}}</b> numbers: [[:Category:Leyland_prime_P_proven{{PAGESINCATEGORY:Leyland prime P provenpages}} proven primes]] and [[:Category:Leyland prime P PRP{{PAGESINCATEGORY:Leyland prime P PRPpages}} PRP's]]  +  There are <b>{{#expr:{{formatnum:{{PAGESINCATEGORY:Leyland prime Ppages}}R}}2}}</b> numbers: [[:Category:Leyland_prime_P_proven{{PAGESINCATEGORY:Leyland prime P provenpages}} proven primes]] and [[:Category:Leyland prime P PRP{{PAGESINCATEGORY:Leyland prime P PRPpages}} PRP's]] 
*[[:Category:Leyland prime PCategory]]  *[[:Category:Leyland prime PCategory]]  
*[[Leyland_prime_P_tableTable]]  *[[Leyland_prime_P_tableTable]] 
Revision as of 21:25, 3 August 2019
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 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]x^yy^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 1481 numbers: 291 proven primes and 1,190 PRP's
Leyland numbers second kind
Reservation history
 x=2000140000, y=11200 completed by Serge Batalov, 20140503
 x=1500120000, y=10012000 completed by Serge Batalov, 20140514
 x=40001330000, y=1117 completed by Serge Batalov, 20140516
 x=330001400000, y=1117 completed by Serge Batalov, 20140517
 x=400001500000, y=1117 completed by Serge Batalov, 20140519
 x=2000130000, y=8011000 reserved by Dylan Delgado, 20190724
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 20061006 by P.Leyland
 Homepage of Andrey Kulsha, dated 20170104
 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 
Prime numbers 
