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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].
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 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
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
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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