Currently there may be errors shown on top of a page, because of a missing Wiki update (PHP version and extension DPL3). |
Topics | Help • Register • News • History • How to • Sequences statistics • Template prototypes |
Difference between revisions of "Leyland number"
(added info) |
(cond. corr.) |
||
(8 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
− | 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 | + | 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 ≥ 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}}. |
− | 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}}). | A '''Leyland prime''' is a Leyland number which is also a [[prime]] (see {{OEIS|l|A094133}}). | ||
Line 9: | Line 8: | ||
==Data== | ==Data== | ||
− | 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 | + | 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. |
===Leyland numbers=== | ===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}} | + | 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]]. |
*[[:Category:Leyland prime P|Category]] | *[[:Category:Leyland prime P|Category]] | ||
*[[Leyland_prime_P_table|Table]] | *[[Leyland_prime_P_table|Table]] | ||
Line 20: | Line 19: | ||
==Reservation history== | ==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 | ||
+ | |- | ||
+ | | 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]] || {{HistLink|2014-05-13|373378}} || {{HistLink|2014-05-14|373406}} || 17 | ||
+ | |- | ||
+ | | 15001 - 20000 || 1001 - 2000 || || [[Serge Batalov]] || {{HistLink|2014-05-14|373426}} || {{HistLink|2014-05-14|373448}} || 20 | ||
+ | |- | ||
+ | | 40001 - 330000 || 11 - 17 || || [[Serge Batalov]] || {{HistLink|2014-05-15|373560}} || {{HistLink|2014-05-16|373654}} || 3 | ||
+ | |- | ||
+ | | 330001 - 400000 || 11 - 17 || || [[Serge Batalov]] || {{HistLink|2014-05-16|373670}} || {{HistLink|2014-05-17|373705}} || - | ||
+ | |- | ||
+ | | 400001 - 500000 || 11 - 17 || || [[Serge Batalov]] || {{HistLink|2014-05-17|373705}} || {{HistLink|2014-05-19|373796}} || - | ||
+ | |- | ||
+ | | 20001 - 40000 || 201 - 400 || || [[Norbert Schneider]] || {{HistLink|2014-07-09|377764}} || {{HistLink|2015-08-10|407596}} || | ||
+ | |- | ||
+ | | 12051 - 12500 || 2001 - x-1 || || [[Norbert Schneider]] || ? || {{HistLink|2014-05-26|379199}} || 4 | ||
+ | |- | ||
+ | | 2501 - 3000 || 12501 - 15000 || || [[Vincent Gautier]] || {{HistLink|2014-11-04|386875}} || {{HistLink|2014-11-14|387631}} || 6 | ||
+ | |- | ||
+ | | 300000 - 500000 || 18 || || [[Andrey Kulsha]] || {{HistLink|2014-12-02|388875}} || || | ||
+ | |- | ||
+ | | 40001 - 50000 || 19 - 400 || || [[Norbert Schneider]] || {{HistLink|2014-12-02|388890}} || {{HistLink|2015-10-23|413431}} || - | ||
+ | |- | ||
+ | | 50001 - 500000 || 19 - 25 || || [[Norbert Schneider]] || {{HistLink|2015-01-26|393632}} || || | ||
+ | |- | ||
+ | | 12501 - 13000 || 3001 - x-1 || || [[Norbert Schneider]] || {{HistLink|2015-10-23|413431}} || {{HistLink|2016-04-21|432114}} || - | ||
+ | |- | ||
+ | | 13000 - 15,000 || || || [[Norbert Schneider]] || {{HistLink|2019-03-27|511972}} || || | ||
+ | |- | ||
+ | | 20001 - 50000 || 401 - 800 || || [[Norbert Schneider]] || {{HistLink|2016-04-21|432114}} || || | ||
+ | |- | ||
+ | | 20001 - 30000 || 801 - 1000 || || [[Dylan Delgado]] || {{HistLink|2019-07-24|522221}} || || | ||
+ | |} | ||
+ | |||
+ | ====Double checks==== | ||
+ | *{{HistLink|2014-07-27|379174}}: [[Mark Rodenkirch]] checked x ≤ 12500, y ≤ 2500 | ||
+ | *{{HistLink|2014-08-03|379608}}: [[Mark Rodenkirch]] checked x ≤ 12500, y ≤ 5000 | ||
+ | *{{HistLink|2014-07-27|379174}}: [[Mark Rodenkirch]] checked x ≤ 12500, 10000 < y ≤ 12500 | ||
+ | *{{HistLink|2014-08-07|379896}}: [[Mark Rodenkirch]] checked x ≤ 12500, y ≤ 7000 | ||
+ | *{{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 | ||
+ | |- | ||
+ | | 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== | ==Contribution of Leyland numbers== | ||
Line 34: | Line 153: | ||
*[https://www.mersenneforum.org/showthread.php?t=19347 Current search] for Leyland PRP's 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]] | *[https://www.mersenneforum.org/showthread.php?t=19348 Prime proofs] of Leyland numbers at [[MersenneForum]] | ||
+ | *[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]] | ||
*[http://www.leyland.vispa.com/numth/factorization/main.htm Homepage] of [[Paul Leyland]] | *[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.leyland.vispa.com/numth/primes/xyyx.htm Page] of Leyland numbers, dated 2006-10-06 by P.Leyland | ||
Line 41: | Line 162: | ||
*{{OEIS|l|A253471}} of n^3 + 3^n | *{{OEIS|l|A253471}} of n^3 + 3^n | ||
*[https://www.youtube.com/watch?v=Lsu2dIr_c8k YouTube "Leyland Numbers - Numberphile"] | *[https://www.youtube.com/watch?v=Lsu2dIr_c8k YouTube "Leyland Numbers - Numberphile"] | ||
− | + | {{Navbox NumberClasses}} | |
− | |||
[[Category:Leyland prime| ]] | [[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].
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
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
- 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
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:
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
- Search of x^y-y^x numbers at MersenneForum
- Search page of Norbert Schneider
- 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 |
|
Prime numbers |
|