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'''CRUS Liskovets-Gallot''' is a [[Conjectures 'R Us]] (CRUS) subproject aiming to prove the [[Liskovets-Gallot conjectures]], which relate to the smallest [[Riesel prime|Riesel]] and [[Proth prime|Proth]] {{Vk}}-values, divisible by 3, with no primes for {{Vn}}-values of a given parity.
 
'''CRUS Liskovets-Gallot''' is a [[Conjectures 'R Us]] (CRUS) subproject aiming to prove the [[Liskovets-Gallot conjectures]], which relate to the smallest [[Riesel prime|Riesel]] and [[Proth prime|Proth]] {{Vk}}-values, divisible by 3, with no primes for {{Vn}}-values of a given parity.
  
 
==Explanations==
 
==Explanations==
 
:''Main article: [[Liskovets-Gallot conjectures]]''
 
:''Main article: [[Liskovets-Gallot conjectures]]''
 +
[[Valery Liskovets]] first observed in 2001 that some {{Vk}}-values, divisible by 3, had few prime {{Vn}}-values of a given parity. He then conjectured that there existed {{Vk}}-values (initially for [[Proth prime]]s, then also for [[Riesel prime]]s), divisible by 3, that had no primes of a given parity. This was proven by [[Yves Gallot]], who provided examples for all four cases (Riesel/Proth, even/odd). Gallot further conjectured that these four examples are the smallest such {{Vk}}-values of each type, not including algebraic factorizations.<ref>[https://www.primepuzzles.net/problems/prob_036.htm Problem 36] "The Liskovets-Gallot numbers" from [https://www.primepuzzles.net/index.shtml PP&P connection] by [[Carlos Rivera]]</ref>
  
==Status==
+
This subproject is attempting to prove the latter set of conjectures by finding primes for {{Vn}}-values of the required sign (Riesel/Proth) and parity (even/odd). The process is the same as [[PrimeGrid|PrimeGrid's]] subprojects for [[PrimeGrid The Riesel Problem|The Riesel Problem]] and [[PrimeGrid Seventeen or Bust|Seventeen or Bust]].
 +
 
 +
==History==
 +
This subproject was founded by Conjectures 'R Us in January 2008, as an extension of the base 4 Riesel and Sierpiński problems. The initial search was led by [[Jean Penné]] and [[Gary Barnes]], and [https://www.rieselprime.de/Related/LiskovetsGallot.htm a page for the effort] was created on the [[Riesel and Proth Prime Database]] on January 11.
 +
 
 +
The even Proth conjecture was proven on 2015-08-02, by Penné, and the discovery was made public a day later after a personal double-check.<ref name="Even Proth proof">[https://www.mersenneforum.org/showpost.php?p=407149 A Liskovets-Gallot theorem proven!] by [[Jean Penné]], 2015-08-02</ref>
 +
 
 +
==Current status==
  
 
===Riesel values===
 
===Riesel values===
*[[Riesel prime 2 9519|{{Vk}}=9519, even {{Vn}}]], done to {{Vn}}=16777216, even {{Vn}}, not reserved
+
[[Multi Reservation:5|Multi Reservation 5]]: '''The current {{Vn}}<sub>max</sub> = {{Num|{{#expr:({{Multi Reservation:5-NMax}}*2)}}}} as of {{Multi Reservation:5-Date}} unless otherwise specified.'''
*[[Riesel prime 2 14361|{{Vk}}=14361, even {{Vn}}]]
+
 
*[[Riesel prime 2 39687|{{Vk}}=39687, odd {{Vn}}]]
+
====Even {{Vn}}'s====
*[[Riesel prime 2 103947|{{Vk}}=103947, odd {{Vn}}]]
+
*[[Riesel prime 4 9519|{{Vk}}=9519]], done to {{Vn}}={{Num|{{#expr:({{GP|Riesel prime 4 9519|RMaxn}}*2)}}}} as of {{GP|Riesel prime 4 9519|RDate}}.
*[[Riesel prime 2 154317|{{Vk}}=154317, odd {{Vn}}]]
+
*[[Riesel prime 4 14361|{{Vk}}=14361]]
*[[Riesel prime 2 163503|{{Vk}}=163503, odd {{Vn}}]]
+
 
 +
====Odd {{Vn}}'s====
 +
*[[Riesel prime 4 79374|{{Vk}}=39687]]
 +
*[[Riesel prime 4 207894|{{Vk}}=103947]]
 +
*[[Riesel prime 4 308634|{{Vk}}=154317]]
 +
*[[Riesel prime 4 327006|{{Vk}}=163503]]
  
 
===Proth values===
 
===Proth values===
*[[Proth prime 2 9267|{{Vk}}=9267, odd {{Vn}}]], reserved by [[Jean Penné]]
+
====Even {{Vn}}'s====
*[[Proth prime 2 32247|{{Vk}}=32247, odd {{Vn}}]]
+
:''The even {{Vn}} conjecture was proven in August 2015.''<ref name="Even Proth proof"/>
*[[Proth prime 2 53133|{{Vk}}=53133, odd {{Vn}}]]
 
  
==History==
+
====Odd {{Vn}}'s====
{{HistF|2015-06-13|{{NRi|19401|3086450}}|Gary Barnes|403997}}
+
[[Multi Reservation:6|Multi Reservation 6]]: '''The current {{Vn}}<sub>max</sub> = {{Num|{{#expr:({{Multi Reservation:6-NMax}}*2)}}}} as of {{Multi Reservation:6-Date}} unless otherwise specified.
{{HistF|2011-07-29|{{NRi|148323|1973319}}|Max Dettweiler|267896}}
+
 
{{HistF|2008-12-03|{{NRi|30003|613463}}|Karsten Bonath|151777}}
+
*[[Proth prime 4 18534|{{Vk}}=9267]] done to {{Vn}}={{Num|{{#expr:({{GP|Proth prime 4 18534|PMaxn}}*2+1)}}}} as of {{GP|Proth prime 4 18534|PDate}}.
{{HistF|2008-05-22|{{NRi|6927|743481}}|Gary Barnes|134050}}
+
*[[Proth prime 4 64494|{{Vk}}=32247]]
 +
*[[Proth prime 4 106266|{{Vk}}=53133]]
 +
 
 +
==Primes found==
 +
===Riesel===
 +
====Even {{Vn}}'s====
 +
The data file can be found [[:File:Liskovets-Gallot Riesel even.csv|here]].
 +
{{#get_web_data:url=https://www.rieselprime.de/z/images/2/23/Liskovets-Gallot_Riesel_even.csv|format=csv with header|delimiter=,|data=date=date,mforum=mforum,user=user,k=k,n=n}}
 +
<div style="width:38em; height:670px; overflow:auto;">
 +
{| class="wikitable sortable" style="height: auto"
 +
! data-sort-type="isoDate" class="fixhead" | Date
 +
! data-sort-type="text" class="fixhead" | Finder
 +
! data-sort-type="number" class="fixhead" | Number
 +
! data-sort-type="number" class="fixhead" | {{Vk}}
 +
! data-sort-type="number" class="fixhead" | {{Vn}}
 +
{{#for_external_table:<nowiki/>
 +
{{!}}-
 +
{{!}} style="text-align:right" data-sort-value={{{date}}} {{!}} [https://www.mersenneforum.org/showpost.php?p={{{mforum}}} {{{date}}}]
 +
{{!}} style="text-align:left" {{!}} [[{{{user}}}]]
 +
{{!}} style="text-align:right" data-sort-value={{{k}}} {{!}} [[Riesel prime 2 {{{k}}}|{{Kbn|{{{k}}}|{{{n}}}}}]]
 +
{{!}} style="text-align:right" {{!}} {{{k}}}
 +
{{!}} style="text-align:right" {{!}} {{{n}}}
 +
}}
 +
|}</div>
 +
 
 +
====Odd {{Vn}}'s====
 +
The data file can be found [[:File:Liskovets-Gallot Riesel odd.csv|here]].
 +
{{#get_web_data:url=https://www.rieselprime.de/z/images/8/88/Liskovets-Gallot_Riesel_odd.csv|format=csv with header|delimiter=,|data=date=date,mforum=mforum,user=user,k=k,n=n}}
 +
<div style="width:38em; height:670px; overflow:auto;">
 +
{| class="wikitable sortable" style="height: auto"
 +
! data-sort-type="isoDate" class="fixhead" | Date
 +
! data-sort-type="text" class="fixhead" | Finder
 +
! data-sort-type="number" class="fixhead" | Number
 +
! data-sort-type="number" class="fixhead" | {{Vk}}
 +
! data-sort-type="number" class="fixhead" | {{Vn}}
 +
{{#for_external_table:<nowiki/>
 +
{{!}}-
 +
{{!}} style="text-align:right" data-sort-value={{{date}}} {{!}} [https://www.mersenneforum.org/showpost.php?p={{{mforum}}} {{{date}}}]
 +
{{!}} style="text-align:left" {{!}} [[{{{user}}}]]
 +
{{!}} style="text-align:right" data-sort-value={{{k}}} {{!}} [[Riesel prime 2 {{{k}}}|{{Kbn|{{{k}}}|{{{n}}}}}]]
 +
{{!}} style="text-align:right" {{!}} {{{k}}}
 +
{{!}} style="text-align:right" {{!}} {{{n}}}
 +
}}
 +
|}</div>
 +
 
 +
===Proth===
 +
====Even {{Vn}}'s====
 +
The data file can be found [[:File:Liskovets-Gallot Proth even.csv|here]].
 +
{{#get_web_data:url=https://www.rieselprime.de/z/images/e/ee/Liskovets-Gallot_Proth_even.csv|format=csv with header|delimiter=,|data=date=date,mforum=mforum,user=user,k=k,n=n}}
 +
<div style="width:38em; height:670px; overflow:auto;">
 +
{| class="wikitable sortable" style="height: auto"
 +
! data-sort-type="isoDate" class="fixhead" | Date
 +
! data-sort-type="text" class="fixhead" | Finder
 +
! data-sort-type="number" class="fixhead" | Number
 +
! data-sort-type="number" class="fixhead" | {{Vk}}
 +
! data-sort-type="number" class="fixhead" | {{Vn}}
 +
{{#for_external_table:<nowiki/>
 +
{{!}}-
 +
{{!}} style="text-align:right" data-sort-value={{{date}}} {{!}} [https://www.mersenneforum.org/showpost.php?p={{{mforum}}} {{{date}}}]
 +
{{!}} style="text-align:left" {{!}} [[{{{user}}}]]
 +
{{!}} style="text-align:right" data-sort-value={{{k}}} {{!}} [[Proth prime 2 {{{k}}}|{{Kbn|+|{{{k}}}|{{{n}}}}}]]
 +
{{!}} style="text-align:right" {{!}} {{{k}}}
 +
{{!}} style="text-align:right" {{!}} {{{n}}}
 +
}}
 +
|}</div>
 +
 
 +
====Odd {{Vn}}'s====
 +
The data file can be found [[:File:Liskovets-Gallot Proth odd.csv|here]].
 +
{{#get_web_data:url=https://www.rieselprime.de/z/images/3/3f/Liskovets-Gallot_Proth_odd.csv|format=csv with header|delimiter=,|data=date=date,mforum=mforum,user=user,k=k,n=n}}
 +
<div style="width:38em; height:670px; overflow:auto;">
 +
{| class="wikitable sortable" style="height: auto"
 +
! data-sort-type="isoDate" class="fixhead" | Date
 +
! data-sort-type="text" class="fixhead" | Finder
 +
! data-sort-type="number" class="fixhead" | Number
 +
! data-sort-type="number" class="fixhead" | {{Vk}}
 +
! data-sort-type="number" class="fixhead" | {{Vn}}
 +
{{#for_external_table:<nowiki/>
 +
{{!}}-
 +
{{!}} style="text-align:right" data-sort-value={{{date}}} {{!}} [https://www.mersenneforum.org/showpost.php?p={{{mforum}}} {{{date}}}]
 +
{{!}} style="text-align:left" {{!}} [[{{{user}}}]]
 +
{{!}} style="text-align:right" data-sort-value={{{k}}} {{!}} [[Proth prime 2 {{{k}}}|{{Kbn|+|{{{k}}}|{{{n}}}}}]]
 +
{{!}} style="text-align:right" {{!}} {{{k}}}
 +
{{!}} style="text-align:right" {{!}} {{{n}}}
 +
}}
 +
|}</div>
 +
 
 +
{{reflist}}
  
 
==Links==
 
==Links==

Latest revision as of 14:05, 2 June 2024

CRUS Liskovets-Gallot is a Conjectures 'R Us (CRUS) subproject aiming to prove the Liskovets-Gallot conjectures, which relate to the smallest Riesel and Proth k-values, divisible by 3, with no primes for n-values of a given parity.

Explanations

Main article: Liskovets-Gallot conjectures

Valery Liskovets first observed in 2001 that some k-values, divisible by 3, had few prime n-values of a given parity. He then conjectured that there existed k-values (initially for Proth primes, then also for Riesel primes), divisible by 3, that had no primes of a given parity. This was proven by Yves Gallot, who provided examples for all four cases (Riesel/Proth, even/odd). Gallot further conjectured that these four examples are the smallest such k-values of each type, not including algebraic factorizations.[1]

This subproject is attempting to prove the latter set of conjectures by finding primes for n-values of the required sign (Riesel/Proth) and parity (even/odd). The process is the same as PrimeGrid's subprojects for The Riesel Problem and Seventeen or Bust.

History

This subproject was founded by Conjectures 'R Us in January 2008, as an extension of the base 4 Riesel and Sierpiński problems. The initial search was led by Jean Penné and Gary Barnes, and a page for the effort was created on the Riesel and Proth Prime Database on January 11.

The even Proth conjecture was proven on 2015-08-02, by Penné, and the discovery was made public a day later after a personal double-check.[2]

Current status

Riesel values

Multi Reservation 5: The current nmax = 10,000,000 as of 2023-07-11 unless otherwise specified.

Even n's

Odd n's

Proth values

Even n's

The even n conjecture was proven in August 2015.[2]

Odd n's

Multi Reservation 6: The current nmax = 10,000,000 as of 2022-03-04 unless otherwise specified.

Primes found

Riesel

Even n's

The data file can be found here.

Date Finder Number k n

2008-01-19 Jean Penné 2181•237890-1 2181 37890
2008-01-19 Jean Penné 6549•25076-1 6549 5076
2008-01-19 Jean Penné 8181•28018-1 8181 8018
2008-01-19 Jean Penné 8961•230950-1 8961 30950
2008-01-19 Jean Penné 11379•232252-1 11379 32252
2008-01-19 Jean Penné 12849•29788-1 12849 9788
2008-01-19 Jean Penné 14859•211228-1 14859 11228
2008-01-19 Jean Penné 15639•266328-1 15639 66328
2008-01-19 Jean Penné 16431•24198-1 16431 4198
2008-01-19 Jean Penné 17889•210628-1 17889 10628
2015-06-13 Gary Barnes 19401•23086450-1 19401 3086450
2011-06-02 Ian M. Gunn 20049•21687252-1 20049 1687252
2008-01-19 Jean Penné 21501•27286-1 21501 7286
2008-01-19 Jean Penné 26091•24198-1 26091 4198
2008-01-19 Jean Penné 26511•2167154-1 26511 167154
2008-01-19 Jean Penné 26601•246246-1 26601 46246
2008-01-19 Jean Penné 30171•276286-1 30171 76286
2008-01-19 Jean Penné 31431•216942-1 31431 16942
2008-01-19 Jean Penné 31749•25040-1 31749 5040
2008-01-19 Jean Penné 31959•219704-1 31959 19704
2008-01-19 Jean Penné 35259•210540-1 35259 10540

Odd n's

The data file can be found here.

Date Finder Number k n

2008-01-19 Jean Penné 903•210227-1 903 10227
2008-01-19 Jean Penné 4887•24289-1 4887 4289
2008-01-19 Jean Penné 5007•26765-1 5007 6765
2008-01-19 Jean Penné 5163•26183-1 5163 6183
2008-05-23 Gary Barnes 6927•2743481-1 6927 743481
2008-01-19 Jean Penné 7977•231265-1 7977 31265
2008-01-28 Gary Barnes 8367•2313705-1 8367 313705
2008-01-19 Jean Penné 9087•24741-1 9087 4741
2008-01-19 Jean Penné 10113•214535-1 10113 14535
2008-01-19 Jean Penné 15213•220311-1 15213 20311
2008-01-19 Jean Penné 19377•218677-1 19377 18677
2008-01-19 Jean Penné 21813•24283-1 21813 4283
2008-01-19 Jean Penné 22863•2101135-1 22863 101135
2008-01-19 Jean Penné 27957•221477-1 27957 21477
2008-12-03 Karsten Bonath 30003•2613463-1 30003 613463
2008-01-19 Jean Penné 30357•265361-1 30357 65361
2008-01-19 Jean Penné 32937•28473-1 32937 8473
2008-01-19 Jean Penné 33837•24273-1 33837 4273
2008-01-19 Jean Penné 34533•232899-1 34533 32899
2008-01-19 Jean Penné 35193•212483-1 35193 12483
2008-01-19 Jean Penné 44283•24439-1 44283 4439
2008-01-19 Jean Penné 46107•24277-1 46107 4277
2008-01-22 Jean Penné 46923•265175-1 46923 65175
2008-01-21 Jean Penné 48927•235861-1 48927 35861
2008-01-19 Jean Penné 52137•226309-1 52137 26309
2008-01-21 Karsten Bonath 53973•2198575-1 53973 198575
2008-01-19 Jean Penné 55983•29851-1 55983 9851
2008-01-19 Jean Penné 56493•26891-1 56493 6891
2008-01-21 Jean Penné 59655•243825-1 59655 43825
2008-01-19 Jean Penné 59763•24611-1 59763 4611
2008-01-19 Jean Penné 61833•24651-1 61833 4651
2008-01-19 Jean Penné 63153•260295-1 63153 60295
2008-01-19 Jean Penné 64023•211431-1 64023 11431
2008-01-19 Jean Penné 67737•24437-1 67737 4437
2008-01-19 Jean Penné 70743•249387-1 70743 49387
2008-01-19 Jean Penné 72327•217125-1 72327 17125
2008-01-19 Jean Penné 72993•223319-1 72993 23319
2008-01-19 Jean Penné 75093•215371-1 75093 15371
2008-01-22 Jean Penné 75363•2120595-1 75363 120595
2008-01-19 Jean Penné 75387•25181-1 75387 5181
2008-01-22 Jean Penné 75873•262419-1 75873 62419
2008-01-19 Jean Penné 78933•211443-1 78933 11443
2008-01-22 Jean Penné 79437•235093-1 79437 35093
2008-01-19 Jean Penné 84807•247389-1 84807 47389
2008-03-28 Jean Penné 86613•2356967-1 86613 356967
2008-01-19 Jean Penné 87735•24551-1 87735 4551
2008-01-19 Jean Penné 88623•213251-1 88623 13251
2008-01-19 Jean Penné 88743•24619-1 88743 4619
2008-01-19 Jean Penné 90567•26577-1 90567 6577
2008-01-19 Jean Penné 91671•28795-1 91671 8795
2008-01-19 Jean Penné 93507•25449-1 93507 5449
2008-01-19 Jean Penné 97323•252207-1 97323 52207
2008-03-31 Karsten Bonath 99363•2268879-1 99363 268879
2008-01-19 Jean Penné 100053•228459-1 100053 28459
2008-01-19 Jean Penné 100353•25147-1 100353 5147
2008-01-28 Jean Penné 100377•2231813-1 100377 231813
2008-01-19 Jean Penné 101823•24519-1 101823 4519
2008-01-19 Jean Penné 102993•248975-1 102993 48975
2008-01-19 Jean Penné 105123•25555-1 105123 5555
2008-01-19 Jean Penné 105837•25913-1 105837 5913
2008-11-17 Karsten Bonath 106377•2475569-1 106377 475569
2008-01-22 Jean Penné 114249•248469-1 114249 48469
2008-01-19 Jean Penné 115167•28685-1 115167 8685
2008-01-19 Jean Penné 117303•24451-1 117303 4451
2008-01-19 Jean Penné 117867•24513-1 117867 4513
2008-01-19 Jean Penné 120387•25645-1 120387 5645
2008-01-19 Jean Penné 121557•211817-1 121557 11817
2008-01-19 Jean Penné 129747•218657-1 129747 18657
2008-01-24 Jean Penné 130383•2104123-1 130383 104123
2008-03-31 Karsten Bonath 130467•2273437-1 130467 273437
2008-01-24 Jean Penné 131727•2169621-1 131727 169621
2008-01-19 Jean Penné 132507•24485-1 132507 4485
2008-01-19 Jean Penné 133023•29087-1 133023 9087
2008-12-05 Jean Penné 133977•2811485-1 133977 811485
2008-01-19 Jean Penné 134037•24421-1 134037 4421
2008-01-24 Jean Penné 135567•268325-1 135567 68325
2008-01-19 Jean Penné 142683•222371-1 142683 22371
2008-02-18 Jean Penné 144117•2224977-1 144117 224977
2008-01-19 Jean Penné 144393•26567-1 144393 6567
2008-01-19 Jean Penné 144957•26473-1 144957 6473
2008-04-07 Jean Penné 145257•2443077-1 145257 443077
2009-01-19 Karsten Bonath 147687•2843689-1 147687 843689
2008-01-19 Jean Penné 148227•25997-1 148227 5997
2011-07-29 Max Dettweiler 148323•21973319-1 148323 1973319
2008-01-19 Jean Penné 148803•225019-1 148803 25019
2008-01-19 Jean Penné 152907•24365-1 152907 4365
2008-01-19 Jean Penné 154827•29113-1 154827 9113
2014-11-10 Gary Barnes 155877•22273465-1 155877 2273465
2008-01-19 Jean Penné 157383•244059-1 157383 44059
2008-01-28 Jean Penné 161583•2138711-1 161583 138711
2008-01-19 Jean Penné 167007•24901-1 167007 4901
2008-01-19 Jean Penné 167997•218705-1 167997 18705
2008-01-19 Jean Penné 169527•29329-1 169527 9329
2008-01-19 Jean Penné 169743•223791-1 169743 23791
2008-01-19 Jean Penné 170223•24187-1 170223 4187
2008-01-19 Jean Penné 170733•27307-1 170733 7307
2008-01-19 Jean Penné 171783•26759-1 171783 6759
2008-04-02 Karsten Bonath 172167•2282649-1 172167 282649

Proth

Even n's

The data file can be found here.

Date Finder Number k n

2008-01-16 Jean Penné 2379•28114+1 2379 8114
2005-02-01 Harsh Aggarwal 8139•225954+1 8139 25954
2008-01-16 Jean Penné 9609•25422+1 9609 5422
2008-01-16 Jean Penné 10281•27444+1 10281 7444
2008-01-16 Jean Penné 11709•26882+1 11709 6882
2008-01-16 Jean Penné 12711•25092+1 12711 5092
2005-01-27 Jean Penné 14661•291368+1 14661 91368
2005-01-27 Darren Wallace 15441•220584+1 15441 20584
2008-01-16 Jean Penné 17169•26450+1 17169 6450
2005-01-27 Darren Wallace 21069•223006+1 21069 23006
2005-01-27 Darren Wallace 21699•272874+1 21699 72874
2015-08-03 Jean Penné 23451•23739388+1 23451 3739388
2005-01-29 Kenneth J. Brazier 23799•2105890+1 23799 105890
2008-01-16 Jean Penné 23901•211292+1 23901 11292
2005-01-28 (Mark) 30579•248594+1 30579 48594
2005-01-29 Jean Penné 33771•2178200+1 33771 178200
2005-06-20 Darren Wallace 33879•2378022+1 33879 378022
2008-01-16 Jean Penné 35889•27770+1 35889 7770
2008-01-16 Jean Penné 39231•213716+1 39231 13716
2008-01-16 Jean Penné 39759•24594+1 39759 4594
2008-01-16 Jean Penné 40269•28458+1 40269 8458
2008-01-16 Jean Penné 41289•213514+1 41289 13514
2005-01-29 Geoffrey Reynolds 41709•280594+1 41709 80594
2005-10-14 Jean Penné 42717•2905792+1 42717 905792
2008-01-16 Jean Penné 44469•213134+1 44469 13134
2005-01-31 Thomas Masser 51171•293736+1 51171 93736
2008-01-16 Jean Penné 52419•24578+1 52419 4578
2008-01-16 Jean Penné 52701•26976+1 52701 6976
2005-01-27 Thomas Masser 52839•232558+1 52839 32558
2008-01-16 Jean Penné 53979•27590+1 53979 7590
2005-01-29 Harsh Aggarwal 55611•240212+1 55611 40212
2008-01-16 Jean Penné 56019•28094+1 56019 8094
2008-01-16 Jean Penné 56139•24858+1 56139 4858
2005-01-27 (Mystwalker) 58791•279420+1 58791 79420
2014-02-14 Jean Penné 60849•23067914+1 60849 3067914
2005-01-29 Jean Penné 60891•240144+1 60891 40144
2008-01-16 Jean Penné 61371•212576+1 61371 12576
2008-01-16 Jean Penné 62391•25472+1 62391 5472
2005-01-31 Darren Wallace 63411•272064+1 63411 72064

Odd n's

The data file can be found here.

Date Finder Number k n

2008-01-15 Jean Penné 93•220917+1 93 20917
2005-02-06 Jean Penné 2943•2108041+1 2943 108041
2008-01-15 Jean Penné 5193•24277+1 5193 4277
2008-01-15 Jean Penné 5703•25149+1 5703 5149
2008-01-15 Jean Penné 5823•28105+1 5823 8105
2008-01-15 Jean Penné 6807•24415+1 6807 4415
2008-01-15 Jean Penné 6843•214753+1 6843 14753
2008-01-15 Jean Penné 7233•24277+1 7233 4277
2008-01-15 Jean Penné 9777•218975+1 9777 18975
2008-01-15 Jean Penné 10923•26801+1 10923 6801
2008-01-15 Jean Penné 14397•24347+1 14397 4347
2008-01-15 Jean Penné 16917•212799+1 16917 12799
2005-01-28 Tyler Cadigan 17457•229563+1 17457 29563
2005-01-28 Tyler Cadigan 17937•253927+1 17937 53927
2008-01-15 Jean Penné 20997•28191+1 20997 8191
2005-01-27 (Mark) 22653•228969+1 22653 28969
2005-05-25 Darren Wallace 24693•2357417+1 24693 357417
2005-01-27 (Mark) 25083•224981+1 25083 24981
2008-01-15 Jean Penné 25917•29671+1 25917 9671
2008-01-15 Jean Penné 26613•289749+1 26613 89749
2008-01-15 Jean Penné 30933•24433+1 30933 4433
2008-01-15 Jean Penné 35787•236639+1 35787 36639
2008-02-14 Gary Barnes 37953•2298913+1 37953 298913
2008-01-15 Gary Barnes 38463•258753+1 38463 58753
2008-01-20 Gary Barnes 39297•2169495+1 39297 169495
2008-01-15 Jean Penné 40857•25383+1 40857 5383
2008-01-15 Jean Penné 42993•216165+1 42993 16165
2008-01-15 Jean Penné 43167•29795+1 43167 9795
2008-01-15 Jean Penné 46623•279553+1 46623 79553
2008-01-15 Jean Penné 49563•25813+1 49563 5813
2008-01-20 Gary Barnes 50433•2156597+1 50433 156597
2008-01-15 Jean Penné 60273•27421+1 60273 7421
2011-06-01 Max Dettweiler 60357•21676907+1 60357 1676907
2008-01-15 Gary Barnes 60963•273409+1 60963 73409
2008-01-20 Gary Barnes 61137•2162967+1 61137 162967
2008-01-15 Gary Barnes 62307•244559+1 62307 44559
2008-01-15 Jean Penné 63357•24211+1 63357 4211
2008-01-15 Jean Penné 65223•24189+1 65223 4189
2008-01-15 Jean Penné 65253•210301+1 65253 10301
2008-01-15 Jean Penné 67917•213079+1 67917 13079
2008-01-15 Jean Penné 69963•25205+1 69963 5205
2008-02-11 Gary Barnes 70467•2268503+1 70467 268503
2008-01-15 Jean Penné 72537•215771+1 72537 15771
2008-01-15 Jean Penné 73023•217965+1 73023 17965
2008-01-15 Gary Barnes 75183•235481+1 75183 35481
2008-01-15 Jean Penné 78543•210089+1 78543 10089
2008-01-15 Gary Barnes 78753•263761+1 78753 63761
2008-03-31 Gary Barnes 80463•2468141+1 80463 468141
2008-01-15 Jean Penné 80517•25423+1 80517 5423
2008-01-15 Jean Penné 81147•217615+1 81147 17615
2008-01-15 Jean Penné 82197•25079+1 82197 5079
2014-10-20 Gary Barnes 84363•22222321+1 84363 2222321
2011-06-02 Lennart Vogel 85287•21890011+1 85287 1890011
2008-01-15 Jean Penné 88863•29825+1 88863 9825
2008-01-15 Jean Penné 91383•215333+1 91383 15333
2008-01-20 Gary Barnes 91437•2161615+1 91437 161615
2008-01-15 Jean Penné 93033•230473+1 93033 30473
2008-01-15 Gary Barnes 93477•263251+1 93477 63251
2008-01-15 Gary Barnes 93663•282317+1 93663 82317

Notes

  1. Problem 36 "The Liskovets-Gallot numbers" from PP&P connection by Carlos Rivera
  2. 2.0 2.1 A Liskovets-Gallot theorem proven! by Jean Penné, 2015-08-02


Links