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'''Carol numbers''' and '''Kynea numbers''' are numbers of the form <math>(b^n-1)^2-2</math> and <math>(b^n+1)^2-2</math>, respectively, while '''Carol primes''' and '''Kynea primes''' are [[prime]]s of those respective forms. Named and originally studied by [[Cletus Emmanuel]] starting in 1995, the search for new Carol and Kynea primes is now coordinated in a [[MersenneForum]] thread, and the results are stored here on Prime-Wiki.
 +
 
==Definitions==
 
==Definitions==
−
In the context of the Carol/Kynea prime search, a Carol number is a number of the form <math>(b^n-1)^2-2</math> and a Kynea number is a number of the form <math>(b^n+1)^2-2</math> (can be written also as 4<sup>n</sup> ± 2<sup>n+1</sup>-1). A Carol/Kynea prime is a [[prime]] which has one of the above forms. A prime of these forms must satisfy the following criteria:
+
In the context of the Carol/Kynea prime search, a Carol number is a number of the form <math>(b^n-1)^2-2</math> and a Kynea number is a number of the form <math>(b^n+1)^2-2</math>. A Carol/Kynea prime is a [[prime]] which has one of the above forms. A prime of these forms must satisfy the following criteria:
−
*b must be even, since if it is odd then <math>(b^n±1)^2-2</math> is always even, and thus can’t be prime.
+
*{{Vb}} must be even, since if it is odd then <math>(b^n±1)^2-2</math> is always even, and thus can’t be prime.
−
*n must be greater than or equal to 1. For any b, if n is 0 then (b<sup>n</sup>±1)<sup>2</sup> is equal to 1, and thus yields -1 when 2 is subtracted from it. By definition -1 is not prime. If n is negative then (b<sup>n</sup>±1)<sup>2</sup> is not necessarily an integer.
+
*{{Vn}} must be greater than or equal to 1. For any {{Vb}}, if {{Vn}} is 0 then <math>(b^n±1)^2</math> is equal to 1, and thus yields -1 when 2 is subtracted from it. By definition -1 is not prime. If {{Vn}} is negative then <math>(b^n±1)^2</math> is not necessarily an integer.
−
*b may be a perfect power of another integer. However these form a subset of another base’s primes (ex. Base 4 Carol/Kynea primes are Base 2 Carol/Kynea primes where <math>n \bmod 2 \equiv 0</math>). So it is not necessary to search these bases separately.
+
*{{Vb}} may be a perfect power of another integer. However these form a subset of another base’s primes (ex. Base 4 Carol/Kynea primes are Base 2 Carol/Kynea primes where <math>n \bmod 2 \equiv 0</math>). So it is not necessary to search these bases separately.
−
Due to the form of these numbers, they are also classified as near-square numbers (numbers of the form n<sup>2</sup>-k).
+
Due to the form of these numbers, they are also classified as near-square numbers (numbers of the form <math>n^2-k</math>).
  
 
==History==
 
==History==
 
Carol and Kynea numbers were first studied by [[Cletus Emmanuel]] in 1995<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/5099 Yahoo Group "Primenumbers", 2002-02-04]</ref>, who named them after personal acquaintances<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/14584 Yahoo Group "Primenumbers", 2004-02-23]</ref>. He searched these forms for primes up to the limit of 15000.
 
Carol and Kynea numbers were first studied by [[Cletus Emmanuel]] in 1995<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/5099 Yahoo Group "Primenumbers", 2002-02-04]</ref>, who named them after personal acquaintances<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/14584 Yahoo Group "Primenumbers", 2004-02-23]</ref>. He searched these forms for primes up to the limit of 15000.
  
−
Starting in 2004, [[Steven Harvey]] maintained a search for this form. At this time [[Multisieve]] and [[cksieve]] were used to sieve these forms and [[PFGW]] was used to test for primality. The search went dormant in 2011 and was resurrected in 2015 by [[Mark Rodenkirch]]. Initially Multisieve was used, but then later on he wrote cksieve which would later become part of [[Mtsieve]] framework.
+
Starting in 2004, [[Steven Harvey]] maintained a search for this form. At this time [[Multisieve]] and [[cksieve]] were used to sieve these forms and [[PFGW]] was used to test for primality. The search went dormant in 2011 and was resurrected in 2015 by [[Mark Rodenkirch]]. Initially Multisieve was used, but then later on he wrote cksieve which would later become part of the [[Mtsieve]] framework.
  
−
On 2015-12-26 Mark opened a thread<ref>[https://www.mersenneforum.org/showthread.php?t=20784 Thread "Carol / Kynea search (Near-power primes)"]</ref> for a coordinated search of Carol/Kynea numbers on [[MersenneForum]], which continues to this day (although now [[Gary Barnes]], maintainer of [[No Prime Left Behind|NPLB]] and [[Conjectures 'R Us|CRUS]], maintains the search).
+
On 2015-12-26 Mark opened a thread<ref>[https://www.mersenneforum.org/showthread.php?t=20784 Thread "Carol / Kynea search (Near-power primes)"]</ref> for a coordinated search of Carol/Kynea numbers on [[MersenneForum]], which continues to this day.
 +
 
 +
<b>All data are now collected here using the information from the forum posts.</b> Feel free to get an [[Help:Logging_in|account]] and editing pages by yourself.
  
 
==Top 5 Carol primes==
 
==Top 5 Carol primes==
Line 17: Line 22:
 
! Prime !! Digits !! Found by !! Date
 
! Prime !! Digits !! Found by !! Date
 
|-
 
|-
−
|{{T5000|126269|(290<sup>124116</sup>-1)<sup>2</sup>-2}}||611246||[[Karsten Bonath]]||2019-03-01
+
|{{T5000|133970|(10<sup>859669</sup>-1)<sup>2</sup>-2}}||1719339||[[Ryan Propper]]||2022-05-25
 
|-
 
|-
−
|{{T5000|121905|(2<sup>695631</sup>-1)<sup>2</sup>-2}}||418812||[[Mark Rodenkirch]]||2016-07-16
+
|{{T5000|133946|(10<sup>657559</sup>-1)<sup>2</sup>-2}}||1315119||[[Ryan Propper]]||2022-05-18
 
|-
 
|-
−
|{{T5000|121867|(2<sup>688042</sup>-1)<sup>2</sup>-2}}||414243||[[Mark Rodenkirch]]||2016-07-05
+
|{{T5000|133929|(10<sup>393063</sup>-1)<sup>2</sup>-2}}||786127||[[Ryan Propper]]||2022-05-14
 
|-
 
|-
−
|{{T5000|121680|(178<sup>87525</sup>-1)<sup>2</sup>-2}}||393937||[[Serge Batalov]]||2016-05-21
+
|{{T5000|133928|(10<sup>334568</sup>-1)<sup>2</sup>-2}}||669137||[[Ryan Propper]]||2022-05-14
 
|-
 
|-
−
|{{T5000|121779|(2<sup>653490</sup>-1)<sup>2</sup>-2}}||393441||[[Mark Rodenkirch]]||2016-06-03
+
|{{T5000|133974|(146<sup>144882</sup>-1)<sup>2</sup>-2}}||627152||[[Ryan Propper]]||2022-06-05
 
|}
 
|}
  
Line 32: Line 37:
 
! Prime !! Digits !! Found by !! Date
 
! Prime !! Digits !! Found by !! Date
 
|-
 
|-
−
|{{T5000|126558|(362<sup>133647</sup>+1)<sup>2</sup>-2}}||683928||[[Karsten Bonath]]||2019-06-17
+
|{{T5000|133993|(146<sup>276995</sup>+1)<sup>2</sup>-2}}||1199030||[[Ryan Propper]]||2022-05-29
 
|-
 
|-
−
|{{T5000|121686|(30<sup>157950</sup>+1)<sup>2</sup>-2}}||466623||[[Serge Batalov]]||2016-05-22
+
|{{T5000|133930|(10<sup>410997</sup>+1)<sup>2</sup>-2}}||821995||[[Ryan Propper]]||2022-05-14
 
|-
 
|-
−
|{{T5000|121801|(2<sup>661478</sup>+1)<sup>2</sup>-2}}||398250||[[Mark Rodenkirch]]||2016-06-18
+
|{{T5000|133980|(146<sup>180482</sup>+1)<sup>2</sup>-2}}||781254||[[Ryan Propper]]||2022-06-05
 
|-
 
|-
−
|(1968<sup>58533</sup>+1)<sup>2</sup>-2||385619||[[Clint Stillman]]||2017-11-30
+
|{{T5000|126558|(362<sup>133647</sup>+1)<sup>2</sup>-2}}||683928||[[Karsten Bonath]]||2019-06-17
 
|-
 
|-
−
|(2<sup>621443</sup>+1)<sup>2</sup>-2||374146||[[Mark Rodenkirch]]||2016-05-30
+
|{{T5000|136056|(2634<sup>88719</sup>+1)<sup>2</sup>-2}}||606948||[[Mischa Rodermond]]||2023-05-18
 
|}
 
|}
  
Line 48: Line 53:
 
!Base!!Carol!!Kynea
 
!Base!!Carol!!Kynea
 
|-
 
|-
−
|[[Carol-Kynea_prime_2|2]]||{{OEIS|s|A091515}}||{{OEIS|s|A091513}}
+
|[[Carol-Kynea prime 2|2]]||{{OEIS|s|A091515}}||{{OEIS|s|A091513}}
 
|-
 
|-
−
|[[Carol-Kynea_prime_6|6]]||{{OEIS|s|A100901}}||{{OEIS|s|A100902}}
+
|[[Carol-Kynea prime 6|6]]||{{OEIS|s|A100901}}||{{OEIS|s|A100902}}
 
|-
 
|-
−
|[[Carol-Kynea_prime_10|10]]||{{OEIS|s|A100903}}||{{OEIS|s|A100904}}
+
|[[Carol-Kynea prime 10|10]]||{{OEIS|s|A100903}}||{{OEIS|s|A100904}}
 
|-
 
|-
−
|[[Carol-Kynea_prime_14|14]]||{{OEIS|s|A100905}}||{{OEIS|s|A100906}}
+
|[[Carol-Kynea prime 12|12]]||{{OEIS|s|A364076}}||{{OEIS|s|A364077}}
 
|-
 
|-
−
|[[Carol-Kynea_prime_22|22]]||{{OEIS|s|A100907}}||{{OEIS|s|A100908}}
+
|[[Carol-Kynea prime 14|14]]||{{OEIS|s|A100905}}||{{OEIS|s|A100906}}
 +
|-
 +
|[[Carol-Kynea prime 18|18]]||{{OEIS|s|A364078}}||{{OEIS|s|A364079}}
 +
|-
 +
|[[Carol-Kynea prime 20|20]]||{{OEIS|s|A364080}}||{{OEIS|s|A364081}}
 +
|-
 +
|[[Carol-Kynea prime 22|22]]||{{OEIS|s|A100907}}||{{OEIS|s|A100908}}
 
|}
 
|}
  
Line 88: Line 99:
  
 
===Remaining data===
 
===Remaining data===
−
All data not yet given by an own page can be found <b>here</b>.
+
All bases <= 3000 not listed yet on their own page are tested to n=2000 and listed in a file.
 +
 
 +
The data file can be found [[:File:Carol-Kynea prime remaining.csv|here]].
 +
{{#get_web_data:url=https://www.rieselprime.de/z/images/5/50/Carol-Kynea_prime_remaining.csv|format=csv with header|delimiter=;|data=b=b,C=C,K=K}}
 +
<div style="width:46em; height:500px; overflow:auto;">
 +
{| class="wikitable sortable" style="height: 200px"
 +
! data-sort-type="number" class="fixhead" | Base
 +
! class="fixhead unsortable"| Carol primes
 +
! class="fixhead unsortable"| Kynea primes
 +
{{#for_external_table:<nowiki/>
 +
{{!}}-
 +
{{!}} style="text-align:right" {{!}} {{{b}}}
 +
{{!}} style="text-align:left" {{!}} {{{C}}}
 +
{{!}} style="text-align:left" {{!}} {{{K}}}
 +
}}
 +
|}</div>
  
 
==How to participate?==
 
==How to participate?==
Line 96: Line 122:
 
===Sieving===
 
===Sieving===
 
*Use [[cksieve]] (from [[Mtsieve]]) and
 
*Use [[cksieve]] (from [[Mtsieve]]) and
−
**run a new sieve by calling <code>cksieve -b 12 -n 1 -N 10000 -P 1000000000</code> (for base=12, n-range=1-10000, max prime factor 10<sup>9</sup>).
+
**run a new sieve by calling <code>cksieve -b 12 -n 1 -N 10000 -P 1000000000</code> (for base=12, n-range=1-10000, max prime factor 10<sup>9</sup>). The sieve file will be written to ck_12.pfgw.
 
**rerun an old sieve by calling <code>cksieve -P 1e12 -i ck_12.pfgw -o ck_12.pfgw -f factors.txt</code> (for base=12, max prime factor 10<sup>12</sup>, input/output files given, storing factors to "factors.txt").
 
**rerun an old sieve by calling <code>cksieve -P 1e12 -i ck_12.pfgw -o ck_12.pfgw -f factors.txt</code> (for base=12, max prime factor 10<sup>12</sup>, input/output files given, storing factors to "factors.txt").
  
 
===PRP testing===
 
===PRP testing===
−
*Use [[PFGW]] calling <code>pfgw64.exe -l 0 ck_12.pfgw</code> (running candidates file for base 12, no further factoring).
+
*Use [[PFGW]] calling <code>pfgw64.exe -f0 ck_12.pfgw</code> (running candidates file for base 12, no further factoring).
  
 
===Prime testing===
 
===Prime testing===
 
After testing with PFGW higher probable primes will be written in "pfgw.log". These have to be checked prime by calling like <code>pfgw64 -tp -q"(12^68835-1)^2-2"</code>.
 
After testing with PFGW higher probable primes will be written in "pfgw.log". These have to be checked prime by calling like <code>pfgw64 -tp -q"(12^68835-1)^2-2"</code>.
 +
 +
===Reporting===
 +
Once you have completed your range, report any primes found in this [https://www.mersenneforum.org/showthread.php?t=21251 thread]. Then report the completed range in the reservation thread and specify whether you will continue with the base or release it.
  
 
==References==
 
==References==
Line 110: Line 139:
 
==External links==
 
==External links==
 
===Current===
 
===Current===
−
*[http://www.noprimeleftbehind.net/Carol-Kynea-prime-search.htm Search] maintained by [[Gary Barnes]]
+
*[https://www.mersenneforum.org/showthread.php?t=21216 "Carol / Kynea Coordinated Search - Reservations"] at [[MersenneForum]] up to post #284 (2021-05-19)
−
*[https://www.mersenneforum.org/showthread.php?t=21216 Reservation thread]
+
*[https://www.mersenneforum.org/showthread.php?t=21251 "Carol / Kynea Primes"] at [[MersenneForum]] up to post #250 (2021-05-19 )
−
*[https://www.mersenneforum.org/showthread.php?t=21251 Primes and results thread]
+
*[https://www.mersenneforum.org/showthread.php?t=26780 2021 project goals]
 +
 
 
===Others===
 
===Others===
 
*[http://mathworld.wolfram.com/Near-SquarePrime.html Near-Square primes]
 
*[http://mathworld.wolfram.com/Near-SquarePrime.html Near-Square primes]
−
*[[Wikipedia:Carol_number|Carol number]]
 
−
*[[Wikipedia:Kynea_number|Kynea number]]
 
 
*[https://www.mersenneforum.org/showthread.php?t=20784 Old thread]
 
*[https://www.mersenneforum.org/showthread.php?t=20784 Old thread]
−
*[https://www.rieselprime.de/Others/CarolKynea.htm More data] for bases ≤ 3000 and n ≤ 2000
 
 
*[http://harvey563.tripod.com/Carol_Kynea.txt Old project by S.Harvey]
 
*[http://harvey563.tripod.com/Carol_Kynea.txt Old project by S.Harvey]
 
{{Navbox NumberClasses}}
 
{{Navbox NumberClasses}}
 
[[Category:Carol-Kynea prime| ]]
 
[[Category:Carol-Kynea prime| ]]

Latest revision as of 00:38, 6 July 2023

Carol numbers and Kynea numbers are numbers of the form [math]\displaystyle{ (b^n-1)^2-2 }[/math] and [math]\displaystyle{ (b^n+1)^2-2 }[/math], respectively, while Carol primes and Kynea primes are primes of those respective forms. Named and originally studied by Cletus Emmanuel starting in 1995, the search for new Carol and Kynea primes is now coordinated in a MersenneForum thread, and the results are stored here on Prime-Wiki.

Definitions

In the context of the Carol/Kynea prime search, a Carol number is a number of the form [math]\displaystyle{ (b^n-1)^2-2 }[/math] and a Kynea number is a number of the form [math]\displaystyle{ (b^n+1)^2-2 }[/math]. A Carol/Kynea prime is a prime which has one of the above forms. A prime of these forms must satisfy the following criteria:

  • b must be even, since if it is odd then [math]\displaystyle{ (b^n±1)^2-2 }[/math] is always even, and thus can’t be prime.
  • n must be greater than or equal to 1. For any b, if n is 0 then [math]\displaystyle{ (b^n±1)^2 }[/math] is equal to 1, and thus yields -1 when 2 is subtracted from it. By definition -1 is not prime. If n is negative then [math]\displaystyle{ (b^n±1)^2 }[/math] is not necessarily an integer.
  • b may be a perfect power of another integer. However these form a subset of another base’s primes (ex. Base 4 Carol/Kynea primes are Base 2 Carol/Kynea primes where [math]\displaystyle{ n \bmod 2 \equiv 0 }[/math]). So it is not necessary to search these bases separately.

Due to the form of these numbers, they are also classified as near-square numbers (numbers of the form [math]\displaystyle{ n^2-k }[/math]).

History

Carol and Kynea numbers were first studied by Cletus Emmanuel in 1995[1], who named them after personal acquaintances[2]. He searched these forms for primes up to the limit of 15000.

Starting in 2004, Steven Harvey maintained a search for this form. At this time Multisieve and cksieve were used to sieve these forms and PFGW was used to test for primality. The search went dormant in 2011 and was resurrected in 2015 by Mark Rodenkirch. Initially Multisieve was used, but then later on he wrote cksieve which would later become part of the Mtsieve framework.

On 2015-12-26 Mark opened a thread[3] for a coordinated search of Carol/Kynea numbers on MersenneForum, which continues to this day.

All data are now collected here using the information from the forum posts. Feel free to get an account and editing pages by yourself.

Top 5 Carol primes

Prime Digits Found by Date
(10859669-1)2-2 1719339 Ryan Propper 2022-05-25
(10657559-1)2-2 1315119 Ryan Propper 2022-05-18
(10393063-1)2-2 786127 Ryan Propper 2022-05-14
(10334568-1)2-2 669137 Ryan Propper 2022-05-14
(146144882-1)2-2 627152 Ryan Propper 2022-06-05

Top 5 Kynea primes

Prime Digits Found by Date
(146276995+1)2-2 1199030 Ryan Propper 2022-05-29
(10410997+1)2-2 821995 Ryan Propper 2022-05-14
(146180482+1)2-2 781254 Ryan Propper 2022-06-05
(362133647+1)2-2 683928 Karsten Bonath 2019-06-17
(263488719+1)2-2 606948 Mischa Rodermond 2023-05-18

OEIS sequences

These are available OEIS sequences:

Base Carol Kynea
2 A091515 A091513
6 A100901 A100902
10 A100903 A100904
12 A364076 A364077
14 A100905 A100906
18 A364078 A364079
20 A364080 A364081
22 A100907 A100908

Data

All bases

All bases with their own page are listed here: There are 382 sequences.

Bases which are a power of

There are 22 sequences.

Bases without a Carol prime

There are 62 sequences.

Bases without a Kynea prime

There are 61 sequences.

Bases without a Carol and Kynea prime

There are 1 sequences.

Remaining data

All bases <= 3000 not listed yet on their own page are tested to n=2000 and listed in a file.

The data file can be found here.

Base Carol primes Kynea primes

366 15, 43, 61, 339 32, 149, 828, 947
370 2, 23, 24, 194, 244 1, 3, 86, 164, 590, 791, 1613
372 1, 2, 7, 51 6, 7, 31, 73, 103, 409, 1408
374 735 22, 48
376 10, 1545 2, 3, 90
378 4, 10, 21, 48, 132, 763, 1277 3, 4, 7, 18, 173, 427
380 4 2, 3, 6, 44, 707, 1220, 1278, 1347, 1962
382 5, 1491 75, 588, 696
384 321 13
386 2, 3, 8, 9, 12, 105, 399 1, 13, 254
388 1, 9, 50, 63 282
390 3 1, 13, 79, 200, 441, 1645
392 1, 5, 15, 223 2, 32, 203
396 3 3, 13, 69, 130, 147
398 512 1, 2, 4, 749
402 6, 21, 135 6, 136
404 460 1, 44, 74, 230
406 1, 21, 49, 495, 1531 5, 6, 9, 79
408 72, 168, 1581, 1660 3, 275
410 3, 5 3, 58, 655
412 2, 3, 15, 33, 280, 503, 544, 663 32
414 2, 220 1, 121, 297
416 1, 2, 12 4, 5, 287
418 3, 4, 5, 6 270
420 8, 345 1, 645, 775
422 1, 6, 31, 42 647
424 9 1, 4, 48, 220, 700, 828
428 20, 200 1, 2, 7, 8, 292, 1466
430 1, 2, 3, 18, 56 4, 30
434 2, 15, 141 1
436 1, 6, 162 2, 6, 8, 1944
438 2, 8, 201, 1731, 1826 85, 106
440 2, 231 1, 2, 28, 35, 50, 98, 233
442 1, 97, 236, 397 1, 3, 28, 232
444 1, 2, 3, 8, 109 21, 82, 657, 1630
446 5, 172, 300 1, 2, 14, 44, 642, 769
450 1, 78 45, 120, 783
452 78 72, 88, 124
454 2, 11, 16, 22, 31 264, 1547
456 7, 34 3
458 9, 33, 114 12, 77
460 3, 5, 11, 1133, 1253, 1636 85, 288, 1687
462 5, 44, 113, 135, 421, 703, 1254 15, 911
464 133, 630 2, 665, 1646
466 11, 68 1, 3, 6, 135
468 1, 6, 8, 400, 603, 998 1, 2, 5, 56, 61, 62, 1774
470 1, 2, 3, 15, 128, 451 2, 106, 162, 288, 586
474 33, 231 3, 57
476 2, 8, 48 8, 12, 1030, 1692
478 3, 45, 900 18, 140, 1611
480 2, 50, 239 1, 3, 10, 13, 64, 153, 315, 535
482 1 2, 42, 129, 132
486 45, 138, 170 4, 15, 203, 1432
488 12, 111 1, 13, 85
490 1, 4, 13, 84, 210, 1885 1, 2, 7, 183, 545, 1359, 1895
492 1, 370, 1518 8, 555, 774
494 323 1, 16, 303
496 1, 440, 690 1, 4, 6, 26, 116, 134, 751
498 1, 5, 24, 92, 99, 1123 4, 9, 11, 216, 347
502 3 6
504 2, 10, 117, 821 1, 7, 34, 220, 490
506 1, 6, 136, 445 140, 404
508 2, 102, 816 30
510 4, 9, 254, 305 216, 231, 1546
514 1, 30, 117 3
516 100 2
518 3, 19, 177 10, 23, 55, 723, 819
520 3, 6 134, 1371
522 2, 1059 1, 3, 517
524 1, 8, 109, 156, 166, 244, 471 1, 3
526 1, 39 2, 3, 5, 154
528 3, 1059 3, 4, 9, 1985
530 5, 84, 1049 1, 93
532 1, 17, 128, 716, 1982 2
534 13, 510 2, 12, 170, 1476
536 3, 357, 653, 1472 7, 10, 70
538 319 2, 6, 120, 868, 1374
540 6, 55, 547, 752, 827, 1877 1, 2, 212
544 24, 84, 94 1, 38, 90, 230
546 1, 3, 51, 196, 426, 1196 11, 21
548 24, 1218 6, 9, 33, 138, 162, 200, 444, 1496
550 30, 81, 168, 330 4, 181, 319
554 3, 72, 645 6, 138, 1887
556 3, 595 405
558 162 2, 314
560 2, 4, 8, 706 1, 2, 7, 20, 33, 68, 135
562 1, 4, 567, 1053 330
564 110, 1016, 1050 1, 76, 756
566 1, 77 56, 636, 643
568 11, 15, 72, 212, 515, 1075, 1208 1, 11, 12, 844
570 1, 2 36, 1061
572 15, 18, 293 1, 535
574 1, 3, 166, 651 1, 12, 144, 212, 298, 1190, 1353, 1951
578 9, 539 19
580 2, 3, 14, 53, 131, 167, 365, 559 2, 19, 68
582 46 1, 152, 301
584 1 9, 406, 591
586 21, 465 1, 2, 3, 12
588 1, 5, 789 11, 13, 28
590 235, 937 23, 155, 762
592 393, 582 12, 1855
594 3 1, 55, 107, 237, 280, 924, 1585
596 1, 2, 18, 25, 66, 185, 414 79, 95, 1434
598 2, 20 57, 129, 845
602 5, 79, 221, 454 640
604 13, 15, 48 38, 159, 1356
606 2, 23, 32, 60, 288 1, 100, 1129
608 1, 3, 13, 26, 326 1, 2, 3, 8, 315, 513
610 1, 9, 10, 78, 1165, 1252 19, 456
612 2, 6, 42 66, 288, 359
614 838 1, 3, 31, 135, 252, 276
616 1, 19, 1208 3, 78, 89
618 4, 16, 24, 618 3
620 11, 21, 23 1, 3, 766
622 1, 2, 4, 16, 323 118, 312, 348
624 2, 56, 83, 102, 282, 285, 286 13, 24, 32, 1589
626 8, 54, 146 69, 120, 154, 534
628 293 2, 120, 288
630 2, 5, 19, 63, 117 234, 807
636 52, 202 1, 483
638 1, 4, 1483 3, 1288
642 3, 4, 60, 1787 13, 20, 440
644 9, 393 1, 2, 5, 54, 146, 399
646 1, 4, 1884 201
648 3, 225, 759, 840 709
650 2, 6, 7, 1042, 1603 29, 100, 287
652 40, 107, 140 1, 3, 46
656 5, 6, 81, 136, 228 2, 6, 8, 30
658 60, 442, 1272 8, 20, 1015
660 43, 972 1476
662 2, 3, 137, 147, 438 1, 3
664 1, 2, 4, 50, 1010, 1600 23, 29, 44, 45, 169, 434, 483, 712, 839
666 120 1, 2, 17, 25, 142, 742
668 1, 3 16, 125, 226
670 54, 1524 1, 2, 19, 117, 660
672 1, 46, 92, 133, 941 37, 64
674 3, 52 2, 722, 1182
678 1, 21, 37, 39, 50, 280 143, 193
680 9, 12, 15, 574 2, 4, 7, 44, 521, 569
682 30 5, 9, 21
684 10, 15, 63 2
686 2, 29, 44, 172, 1170, 1260 2, 3, 800, 1866
690 413 30, 36, 681, 1162
692 57, 586, 1787 91
694 2, 6, 62, 256 2, 318, 591
696 3, 87, 1675 3, 897
700 3, 16, 60, 70, 306 29, 64, 151, 1047, 1410
704 2, 3, 5, 380 6, 7, 49
706 2, 3, 6, 1955 3, 4, 5, 10, 49, 53
708 10, 14, 21 2, 7, 162, 215, 282, 428, 538, 742
712 9, 24 1, 7, 15, 37, 42, 375, 591
714 1, 4, 84 1, 19, 81, 155
716 1, 210 2, 18, 39, 81, 1475
718 2, 48, 440 1, 3, 6, 13, 408, 1708
722 6, 86, 222 4
724 7, 216, 595 275, 393, 948
726 4, 669 1, 2, 13, 494, 1005
728 1, 4, 88, 364, 907 2, 1054, 1751
730 3, 15, 69, 91, 97, 160 27, 105
732 5, 288, 464 1, 4, 15, 27, 61, 91, 273
734 1, 4, 52, 151, 304, 620, 676 8
736 2, 52 2, 671
740 18, 47, 503 3, 50, 291
742 3, 6, 979 1, 4, 96, 585, 596, 618, 680, 1210
744 1, 49 3, 6, 14
746 2, 3, 516 1, 535
748 1, 124, 135, 179, 320 2, 14, 126
750 3, 149 1, 2, 6, 71, 381
752 1, 14, 804 60, 328
754 6 127, 372, 817
756 2, 57 1, 7, 1808
758 1, 538 2, 75, 548
760 2, 5, 38, 89 1, 3, 40, 61, 354
762 1, 12, 1341 1, 5, 39, 243, 1569
764 1, 146 115
766 93, 414, 420 95
768 42, 154, 305 2, 27, 43
770 3, 14, 77, 240, 513 2, 31, 131, 771, 1680
774 26 1, 28, 45, 81
776 1, 32, 236, 610, 808, 1537 2, 9, 11, 20, 71, 449, 973, 1432
778 8, 17 1, 2, 20
780 1, 6, 288, 936, 1200 15, 22
782 3, 40, 760 48, 171, 645
786 1 3, 8, 191, 677
788 447 1, 3, 700, 759
790 1, 2, 40, 602 4, 213
792 3, 10, 12, 112, 698 6, 51, 162, 696
794 3, 126, 290, 1232 4
796 207 1
798 1, 3, 1048 2, 689, 944
800 13, 30 5, 1506
802 2 27, 48
804 6, 257, 550 2, 4, 1660
806 2, 4, 10, 45, 51, 171, 1477, 1978 1, 8, 657, 774, 909
808 1, 3, 6, 20, 314 4, 5, 52, 214
810 5, 9, 321 1
812 1, 11, 276, 310, 376 3, 20, 43, 56, 79, 107, 760
814 45, 69, 130 530
816 6, 1910 1, 345, 1149
820 5, 255, 901 174
822 30 3, 22, 454
824 1215 1, 195
826 1, 11, 14, 32, 190, 483, 852 10, 18
828 3 465
830 375 487
834 21 45, 57, 110, 233, 730, 1294, 1626
838 33, 78 925
840 11, 54, 1595 1, 117
842 1, 19, 75, 1402 21, 33, 72, 201
844 21 33
846 5, 27, 224 3, 5, 10, 41, 116, 181
848 38 11
850 2, 12, 60, 78 6, 1815
852 3, 4, 15, 54 2, 462
854 5, 211, 522 2, 12, 152, 613
856 10, 70, 306 44
858 5 88
860 10, 64, 1052 3, 31, 72, 269
862 42, 95 1, 454
864 1, 2, 104, 993 15, 42, 1049
868 7, 357, 1166 105, 449, 517
870 385 20, 795
872 2, 5, 8, 17, 1415 3, 4, 564
874 2, 123 1, 5, 200
876 1, 21, 60, 105 4, 5
878 2, 7 180, 244
880 723, 971 1, 3, 9, 80, 121, 386
882 1, 493 842
884 3, 4, 61, 75, 252, 1213 198
886 5, 27, 33, 147 10, 84
888 11, 1505 5, 29, 124, 1063, 1568
890 2, 24 25
892 200 11, 112
896 11 1, 2, 3, 12, 376, 978
898 1, 4 6, 18
902 1 2, 449, 625, 1313, 1313
904 3, 7 14, 42, 78
906 3, 6 558
910 2, 10, 24, 49, 51, 54, 90, 200 110, 179
912 60, 208, 1638 12, 50, 1467
914 278 46, 69
916 9, 27, 57 2, 3
918 3, 21, 26, 808 8, 18
920 14, 780 221, 250
922 190, 315 3, 8
924 2, 4, 16, 288 3, 9, 18, 422
930 9, 843 1, 293, 712, 942
932 1, 10, 16, 98 46, 55, 79, 250, 260
934 3, 20, 501, 1585 65, 77
936 58, 365 3, 26, 33, 103
940 10, 55, 1788 1692
944 1, 4, 9, 12, 14, 1302 1, 3, 112, 685, 1433
946 1, 6, 7, 64, 146, 1022 10, 15, 49, 126, 140, 385
948 6, 9, 13, 21, 1923 5, 168, 1036
952 1, 245, 784 3, 17, 83, 160, 855
954 10, 120, 247, 579, 1719 84, 147
956 5, 630 6, 529
958 15, 800 1, 2, 5, 14, 21, 110
960 1, 3, 5, 252 4, 706
962 51, 282, 732 1901
972 46 1, 6, 9, 151, 225
974 1, 10, 294, 1175 2, 90
976 3 40
978 42, 53, 202 1, 1063
980 1, 2, 21, 31, 175, 634, 1354 129, 589
984 5, 14, 527 3, 10, 31
986 2 1, 2, 216, 310, 565
988 1, 6, 61 29, 275, 930
990 457 5, 6, 47
994 1, 9, 358 1, 2, 19, 35, 100
1002 2, 3 1, 13
1004 1, 27 258
1006 3, 16, 18, 786 140, 152, 254, 1111
1008 4, 77, 110, 1248, 1667 4, 74, 115
1012 14 1, 4, 22, 697, 1821, 1954
1014 1, 4, 7, 19, 1014 15, 954
1016 12, 299, 1375, 1426 11, 32, 42
1018 2, 3, 9, 15, 597, 1531 3, 35, 132, 760, 1481
1020 574, 624, 970 1, 33, 205, 450
1022 1, 17, 62, 429, 1553 1, 3, 28, 37, 1393
1026 18, 33 7, 9, 102
1028 4, 166 1, 2, 18, 53, 75, 201
1030 1, 4, 6 2, 288
1034 1440 1, 2, 6, 37, 1842
1036 1, 9, 12, 20, 240, 377, 1197 2
1040 20 10, 45
1042 10, 273, 1511 13, 141, 303, 1872
1044 2 1, 261, 1966
1046 1, 14 9, 1632
1048 108 2
1050 25, 267 1, 57
1052 1, 6, 42, 520 62, 312, 359, 627
1054 135 3
1056 18, 118, 578 1, 369, 476, 556
1058 1, 2, 27, 35, 195 5, 12, 140
1060 63, 396, 1880 70, 1362
1062 5 1814
1064 3, 5, 18, 66, 130, 1995 3, 11, 29, 52
1066 6, 22 2, 72, 128, 147
1070 1, 15, 382, 685 1
1072 1 1, 2, 122, 206
1074 1, 2, 44 12
1076 21 2, 12
1078 2, 5, 261 2, 902
1080 19, 48, 537, 1008, 1609 77, 302
1082 65, 71 144, 279
1084 4, 32, 40, 57, 408 1, 11, 381, 923
1086 1, 1201 13, 72, 203, 967
1088 9, 813 41, 1235
1090 3 1, 21
1092 1, 5, 821 2, 1150, 1316
1094 13 21, 210
1096 5, 230 10, 106, 136, 171
1098 3, 37 82, 100, 105
1100 2, 4, 229 1, 2, 3, 5, 24, 90, 182
1102 1, 36, 114, 127 5
1104 6, 15 3, 15, 25, 56
1106 2, 11, 39, 182, 672 3, 50, 218, 604
1110 36, 179 4, 52, 64, 379
1112 10 1, 3, 5, 6, 280
1114 1, 3, 4, 319, 434, 826, 1341 15, 21, 75, 81
1118 5, 63, 88 1, 57, 126
1120 1, 7, 21, 67 2, 3, 8, 525, 1112
1122 153, 1416 3, 696
1124 117 1, 216, 264, 324
1128 27, 56, 279 40, 108
1130 2, 18 15
1134 2, 13, 14, 1838 42, 154, 694
1138 14 10, 12, 426
1140 16 1, 74
1142 1, 200, 318 1
1144 1, 37 3
1146 66 1, 3, 91, 229, 459, 795, 878
1148 1, 8, 10 1, 4, 10, 21, 58, 88, 1667
1154 1, 7 2, 3, 348
1156 2, 129 1, 7, 73
1158 1, 8, 43, 66 5, 792
1160 3, 9 1, 25, 141, 1922
1162 1, 70, 125, 126, 740, 1687 3, 115, 126, 1273
1164 12, 15 9, 17
1168 36 6, 108, 322
1170 75, 121, 381, 420 12, 37, 661
1172 2, 3, 6, 14 17, 34
1174 120, 837 1
1176 1, 2, 7, 112, 212, 1672 1, 11, 12, 66, 180
1178 1, 4, 117, 426 2, 1131
1180 62, 72, 597 16, 21, 70, 1827
1182 7 2, 5, 14, 23, 27, 70, 731
1184 4, 15, 55, 56, 149, 410 2, 4, 6, 7, 16, 26, 31, 279, 286, 505
1186 3, 21, 54, 104, 254, 1272, 1899 627
1190 352, 1196 2, 6, 7, 15, 141, 256, 259
1192 34 3, 5, 8, 1239
1194 117, 120, 711 63, 70
1196 4, 20, 27 1, 180, 901
1198 1, 6, 59 4, 25, 307, 633
1200 3 3, 11, 75, 136
1202 832 2, 7, 18, 37, 50, 1266
1204 17 442
1206 3, 61, 274 3, 5, 561, 609
1208 3, 9, 548 6, 13
1212 1 3, 98
1214 397 12, 1408
1220 30, 76 2, 15, 249, 252, 722
1224 1, 6, 13, 24, 25, 64, 188, 197, 270, 453 2, 672, 684
1226 12, 94, 185, 251, 534 3, 16, 416
1228 129 3, 767
1230 65 1, 7, 67, 355
1232 1, 2, 7, 8, 118 1, 9, 125, 202
1234 1, 94, 1683 99, 1406
1238 2, 876 1, 464, 1796, 1900
1240 1, 111, 175, 518 1, 165, 176, 1010
1242 1, 896 4, 347
1244 12, 436 6, 9, 158
1246 3, 28 5, 93, 202, 1707
1248 3, 4 786
1250 2, 57, 348 1, 352, 379, 579
1252 1, 9, 25, 32 1, 12, 1058
1254 1, 1218, 1686 5, 14, 175, 1646
1256 2, 3, 21, 228 4, 5, 23
1258 39, 155, 245, 749, 843, 1260 3, 14, 859
1260 5, 20, 121, 165 1, 5, 10
1262 1, 165 11, 26, 66, 113
1264 41, 56, 425 1, 72
1266 1, 3, 4, 54, 152, 813 1, 2, 446
1268 1, 132, 651, 1530 5, 37
1270 444, 680 1253
1272 30, 45 20, 1034
1274 342 15, 26
1276 63, 118, 1077 6, 111
1278 1145 6, 7, 783
1280 2, 3, 43, 176, 272, 400 2, 5, 1546
1282 358, 404 3
1284 3, 224 201, 1895
1288 1, 2, 160 1, 2, 6, 7, 14, 142, 1713
1292 11, 41 37
1294 3, 53, 98 375
1296 5, 77, 532, 2257, 9665 1, 3, 14, 94, 108, 1942, 8392, 12700
1298 1, 120, 312 6, 15, 108
1300 4, 10, 65 3, 7, 12, 36, 56, 183
1302 231 2, 6, 12
1304 4, 75, 195, 297, 889 14, 60, 132
1306 2, 9, 42, 236 1, 6, 19, 682
1308 1, 9 6
1310 66, 78, 391, 459 1, 98
1312 1, 2, 295, 799 100, 189, 810
1314 3, 29 2, 12, 108
1316 110 131, 235, 871
1320 168, 255, 1548 1, 60, 426
1322 1, 21, 621, 1150 435
1324 2, 3, 4 6, 17
1328 46 2, 3, 30, 49
1330 12, 20, 27, 119 1, 2, 5, 52, 128, 427
1332 1, 16, 43 182, 1304
1336 59, 387, 516 756
1338 2, 547 5, 138, 179, 334, 893, 1419
1340 13, 18 5, 192, 700
1342 4, 143, 334, 654 1, 2
1344 1 12, 15, 354, 534, 780, 1033
1348 741 3
1350 2, 10, 422, 488 3, 46, 700, 1224
1352 3, 4, 39, 549 9, 53, 93, 163
1354 39, 54, 57 4, 40
1358 1, 405 1, 4, 14, 16, 204, 265, 614
1362 65 1, 1083
1364 1, 109, 315 1, 25, 990
1366 1, 8, 39, 1087, 1436 1, 3, 15, 127
1368 1, 69, 876 279
1370 3 24
1372 3, 4, 1740 1248, 1574
1374 160, 456, 618 99, 1706
1376 2, 21, 147, 1098 9, 58
1378 19, 43, 227, 619 1
1380 1, 1466 15, 100
1382 355 664
1386 4, 64, 604, 725 5, 57, 103
1390 102, 272 1, 10, 25
1392 1, 4, 6, 597 1
1394 1, 43, 59, 67 7, 9, 408, 880, 1738
1396 840 168, 522
1398 5, 6, 52 6, 18, 25, 90, 1369
1400 7 5
1402 10, 90, 519, 1105, 1915 6
1406 2, 3 17, 50
1408 2, 31, 93, 312, 338, 1166 1, 2, 15, 32, 36, 72
1410 1, 2, 235, 915, 1021 29
1412 310 1176
1414 154 1, 8
1416 1, 3, 30, 139 6
1418 9, 768 7, 513
1420 6, 30, 781, 1965 210
1422 2, 21, 236 7, 47, 353
1424 24, 60, 87 946, 1396
1426 5, 10 39, 117, 560, 1392, 1452
1428 22, 29 1, 495
1430 1 2, 39
1434 4, 15, 56, 59, 184, 462, 748, 830, 1714 1, 127, 207, 442
1436 1, 10, 11, 705 4, 64, 103, 294
1438 1500 18
1440 5 13, 51, 102
1444 1, 280 3, 1745
1446 5, 20 1, 6
1448 1, 18, 25, 240, 1116 1195
1450 126, 154, 156, 646 15, 52
1454 6 38, 555, 1435
1456 19 4, 32
1458 541 3, 5
1460 65 1, 21, 37, 72, 184, 522
1462 1, 5, 27, 34 1, 26, 43
1464 1, 2 7, 9, 23, 120, 142, 150, 1314
1470 9 85, 732
1474 2 10, 439
1476 7, 363 17, 53, 410, 1098, 1370
1478 2, 20 17
1480 111 5, 36, 45, 177, 375
1486 3, 150 1046
1488 14, 548 90
1490 8, 168, 236 1, 2
1492 1, 146, 383 10, 50
1494 9, 20 5, 6, 156, 1402
1496 15, 27, 382, 1097 1128
1498 4 1, 27, 420, 682, 1700, 1792
1500 1, 190 2, 39
1502 2, 14, 905 87, 100, 1566
1504 1303, 1308, 1743 3, 120, 1307
1506 4, 196 2, 3
1508 24, 38, 1332 3, 6, 65, 106, 903
1510 4, 6, 34, 125 26, 92, 540
1512 6, 29, 145, 493 28, 621
1514 3, 37, 520, 1363, 1623, 1980 1176
1518 1, 3, 34, 514, 1010 3, 6, 36, 94, 198, 948, 1624
1520 5, 19, 26, 52 1, 130, 944
1522 1 5, 10, 101
1526 1, 2, 6, 13, 106, 334 2, 15, 36, 227, 694
1528 3, 4, 42, 51 365
1530 36, 257 265, 657
1532 3, 119, 166, 781 5, 11, 14
1536 1633 12
1538 9 42, 79, 183, 1128
1544 2, 62 1, 87, 433
1546 1, 2, 3, 5, 104, 116 14, 30, 234, 599
1548 31, 388 1, 14, 32, 101
1550 1, 216 11, 30, 125
1552 6, 996 32, 302
1554 26, 52, 216 564, 904
1556 4, 31 240, 263, 330
1558 51, 270 9, 16
1560 4, 27, 40, 53 1
1562 1, 53 1, 34, 84, 331
1564 1, 8, 1527 3, 288, 777
1568 1, 4 5, 16
1572 11, 62 1083
1574 2, 26 4, 35, 40, 114
1576 3, 1100 907
1578 18, 763 3, 4, 47, 1913
1580 3, 34, 233 2, 25, 1278
1582 2, 4, 8, 555 7, 45, 1493
1584 9, 19, 72 42
1588 13, 867 7, 103, 124
1590 5, 155, 355, 369 10, 48, 86
1594 46 1119
1596 305 1, 2, 4, 6, 149, 1012
1598 1, 10, 16 122, 1688
1600 2, 69, 2042 1, 72, 1428, 1498, 2583, 3912, 4696, 20389
1602 1, 49, 354, 1038 1, 6, 26
1604 1, 10, 17, 18, 32, 186, 1614 55, 88
1606 237 11, 42, 168, 198
1608 45 13, 60, 138
1610 2, 15, 70, 383 1, 315, 730, 1484
1612 1 95, 854
1616 3, 44, 960, 1059, 1231 1, 5, 80, 815
1618 1, 618, 1273, 1843 1, 13, 89, 390, 1217
1620 1, 2, 74 3, 693
1622 540 6, 13, 1916
1624 3 2, 32
1626 81, 334, 1639 114, 126, 140, 168
1628 50 3, 522, 810
1630 61, 139 2, 26, 1222
1632 23, 37, 1460 37, 707, 1131
1634 3, 21, 37, 606 4, 6, 22, 147, 148, 1665
1636 491 18, 96
1638 2, 9, 25, 160, 317, 660, 1233 1, 192
1640 1, 186, 229, 1723 80, 108, 324
1642 2, 32 15, 94
1644 5, 99 21, 67, 949
1646 70, 432 2, 3, 20, 26, 288
1648 2, 660 5, 6
1652 14, 49, 442, 878, 1257 1
1654 1, 294, 510 5, 36
1658 2, 25, 142, 982, 1053 3, 10, 18, 40, 737, 896
1660 12, 101, 167 2, 6, 26, 120, 182, 400
1662 7, 1257 6, 89
1666 11, 26 20, 233, 243, 247, 1126
1670 2, 5 3
1672 2, 11, 14 1, 2, 34, 67, 689
1674 1, 15 12, 29
1676 13, 14 6, 111
1678 5, 52, 83, 516, 1380 1, 2
1680 1, 429, 864 1
1682 1, 39, 102, 120 17, 24, 44, 680
1686 4, 561, 951, 1334 1, 39, 494
1688 1, 2, 7, 12, 397 2, 158, 432
1692 174 1, 7, 25, 345, 366
1694 1, 2 1, 12
1696 1 2, 165
1698 2, 1170 4, 321
1700 2, 33, 566 1, 13, 47
1702 1, 3, 60, 95 1, 7, 31, 183, 266
1704 1, 2 9, 10, 315
1708 183, 720 9, 531, 794
1710 4, 7 26
1712 140 934
1714 859, 1627 1, 6, 198, 850
1716 1, 3, 553 1, 2
1718 1, 2, 50, 1064 184
1720 3, 366 1, 888
1722 1 2
1728 1, 17, 2608, 5152, 7538, 22945 20, 171, 349, 2502, 26286
1730 1095, 1653 6, 84, 223, 1727
1732 384, 519 1067
1734 3, 102, 1011, 1889 105
1736 8, 29, 104 1, 9, 17
1738 1, 3 9, 417
1740 2, 329 1, 16
1742 1, 16, 328 2, 5, 17, 86, 1384
1744 26 13, 128
1746 170, 806 4, 6, 30
1748 28, 684 1, 56
1750 1, 2, 123, 175, 765 1, 5
1752 1, 1182, 1522 3
1754 3, 596 64, 66, 150
1756 176 47
1758 12, 153, 1340 2, 4
1762 360 2, 8, 207
1764 3, 7, 39, 136, 1263, 2426 2, 1023, 1233, 2010, 4212, 12501
1766 4, 34, 1227 5, 11, 24, 1613
1768 17, 36, 59, 231 15, 46
1770 42, 68, 390, 780 2, 5, 9, 31, 69
1772 1445 2, 14, 756, 795
1776 23, 393 8, 12, 27
1778 2, 14, 36 2, 3, 134, 198, 1456
1780 6 2, 36
1782 8, 50, 698 1
1784 1, 2, 1845 1, 44, 119, 386
1786 1, 15, 71, 72 2, 5
1790 420, 462 2, 36, 170
1792 2, 318 5, 54
1794 9, 28, 87 11, 14, 1824
1796 2, 6, 195 1, 9
1798 1, 486, 1128 4, 9, 11, 20, 244
1800 30, 32, 264, 1280 104, 180
1802 61, 264 22, 538
1804 3, 437 175, 1743
1808 1, 9, 15, 25, 61, 1044, 1803 20, 68
1810 6, 21, 26, 39, 45, 114, 194, 290 7, 10, 15, 16, 499, 520
1812 1317 32, 96
1814 4, 16, 60, 63, 236, 1341 6, 270
1816 2, 54 534
1818 101, 156 1, 2, 7, 8
1820 1, 165, 1090 5, 665
1822 3 6, 408, 1143, 1868
1824 2, 1530 1069
1826 3, 22, 363, 968, 1816 1, 5, 11, 1508
1828 1, 304 34
1830 2 4
1832 5, 100 50, 60, 115, 187, 559
1834 2, 16, 234, 299 2, 271, 1006
1836 64, 493 14, 17, 125, 1205
1838 305 1, 808
1840 1, 427, 540 1, 4, 75
1842 1, 2, 6, 13, 61, 280 15, 64, 159, 208, 1612, 1992
1844 2, 200 141, 363
1846 10 267, 276
1848 9, 205, 1032, 1650 1, 5, 6
1850 1, 16 26, 33
1854 5, 18, 42, 45, 374, 449, 985, 1779 2, 1376
1856 2, 18, 55, 118, 1500, 1800 63, 1016
1858 374 18, 89, 90, 124, 1157
1860 76 1, 2, 3, 1873
1862 1, 4, 9, 61, 337, 732, 978 5, 25, 43, 50, 126, 310
1864 997 6, 12, 114, 506
1866 3, 65 1, 79
1868 1, 2, 7, 74, 371 8, 9, 22
1870 5, 175, 752, 1190 11
1872 3, 30, 62, 115, 824 17, 946
1874 10, 264 1, 153
1876 1, 16, 18 2, 41, 402, 659, 953
1878 3, 1228 9, 78
1880 6, 20 12
1882 22 6, 16, 1393
1884 4, 8 1, 3, 95, 150, 494, 598
1886 1, 15 4, 23, 58, 123, 478, 910
1888 717 3
1890 2, 3, 4, 6, 14, 79, 149 20, 388
1892 3, 22 27, 29
1894 63 1, 12, 36
1896 1 1, 7
1898 1, 99, 100 4
1900 2, 25 11, 23
1902 6, 22, 141 15, 20, 1298
1904 4, 70, 193, 404, 1550 2, 16, 1772
1906 10, 309, 496 2, 3, 15, 23, 71, 158
1908 2, 1409 1, 9, 226, 1702
1910 1, 4, 24 1, 2, 104, 480
1912 1, 2, 3, 4, 7, 10, 36 1, 20, 97
1914 1, 2, 8, 116, 188, 1664 6, 47
1916 4, 6, 120, 779 164
1918 20, 44, 68 2, 5, 549, 1488, 1619
1920 4, 12 2
1922 20, 53, 87, 417 1, 3, 88
1924 1, 4, 6, 117, 431 1, 42, 654, 1537, 1597
1926 1 14
1930 1246 1, 126, 1218
1932 1, 4 171
1934 3, 18 107
1936 15, 45, 644, 12893 741, 4105, 10251, 30106, 47970
1938 4, 14, 154, 536, 1026 2, 29, 302, 309
1940 1881 3, 6, 20, 57, 131
1942 206, 506, 1087 1827
1944 6 24
1946 21, 394, 888 1, 2, 5, 43, 64, 69, 95
1948 1, 7, 739 2
1952 7, 86 1, 3, 993, 1852
1954 1, 3, 180 11, 1776
1956 27 21, 911
1958 10, 15, 69 96, 192
1960 11, 423 50, 133
1964 2, 89, 96, 134, 300, 1089 1, 3, 31, 211
1966 1, 355 15, 753
1970 1, 7 276, 436
1972 11, 90 211, 284, 1179, 1746
1974 2, 25, 326 1, 5, 6, 8, 14
1976 1, 951 50
1978 11, 1986 1
1980 1, 5, 7, 11, 378, 1775 3, 5, 6, 286, 780
1982 2, 4, 1810 52, 74, 126
1984 6 540
1988 87, 667, 1754 1, 11, 12, 243, 370, 1789
1990 1, 4, 61, 127, 669, 828 30, 131, 731
1994 2, 37 3, 5, 27, 114, 854
1996 32, 57, 960 1, 3, 666
1998 1, 2, 15, 55, 150 15
2000 562 1, 2, 27, 284
2002 1, 2, 4, 62, 223, 295, 341, 648, 882, 886 6, 26, 201, 299, 385, 687
2004 3 15, 144
2006 5, 6, 773, 1521 12, 19, 66, 1204, 1717
2008 8, 11, 22, 1003, 1174 3, 18, 30, 41, 78
2012 6, 956 4, 689
2014 5 19
2016 18, 78, 253, 1670 29, 608
2020 6, 9 1
2022 1, 6, 22, 771 351, 697
2024 85, 635, 1470 5, 276, 411, 424
2028 78, 99, 279 27
2030 30 1, 7, 108, 277, 309, 424
2032 1 1160
2036 1, 62 1
2038 1, 78, 120 5, 10, 12, 66, 604
2042 4, 15, 501 1
2044 1, 5, 12, 119 15, 50, 296
2046 282 21, 336, 1011
2048 5, 12, 1299 22, 150
2050 216, 1511 2, 20
2052 265, 1119, 1144 1, 3, 10, 77, 113, 1048, 1608, 1710
2054 1, 9, 12, 24, 674, 1659, 1982 40
2056 5, 813 1, 115
2058 1, 6, 105, 416 1, 3, 231, 989, 1977
2060 1, 16, 21 162
2062 2, 5, 18 1029
2066 2, 12, 192, 203 2, 132
2068 105 1551
2070 310, 472 9, 253
2072 10, 17, 109 1, 17, 25, 138, 531
2074 1, 144 2, 3, 275, 996
2078 1, 5, 8, 31, 38, 114, 360 25, 143, 317, 1891
2080 40, 310 1, 2, 6, 20, 23, 30
2082 1 4, 29, 148
2084 3 2, 63, 75, 110
2086 141, 572, 1326 1, 2, 8, 288
2088 1 2, 5, 23, 62
2090 45, 93, 258 1, 57, 1588
2092 1, 2 1, 16, 30, 83
2094 1, 14, 16, 110, 277, 314, 1400 1, 16, 31, 58
2096 1, 132 4, 5
2098 3, 16, 34, 1881 6
2100 33, 35, 92, 181, 190 1, 47, 1252
2102 1, 9, 19, 34, 79, 84, 162, 1678 63, 111
2106 1, 119, 148, 700, 1620 1, 109
2108 1, 40, 115 1, 4, 50
2110 1, 2, 6, 162, 302 3, 5, 234, 413
2112 17, 33 1, 8, 9, 15, 44, 1327
2114 1, 2, 280 8
2116 6, 652, 2586 27, 1020
2118 17, 65, 122 25, 31, 220
2120 85 2, 317, 394, 458
2122 55 81
2124 44, 87 660
2130 1, 19 8
2134 1, 1189 3, 10, 1090
2136 26 1, 6, 7, 104
2138 1, 2, 13, 92, 390, 578 28, 39, 1121
2142 24 831
2144 82, 189 3, 77, 156, 482, 1182
2148 1323 17, 724
2150 40, 137, 510 1, 3, 273
2152 1, 2 3, 9, 47, 525
2154 411 2, 24, 33, 120
2156 137, 491 3, 345
2158 193 2, 14, 222, 327
2160 3, 9, 212, 266 10, 58
2162 16, 80, 560, 689 1, 524
2164 1, 172 10, 74, 128, 132, 344
2166 368 27
2170 9, 119 1, 2, 875, 1238
2172 1, 40 12
2174 5 106, 744
2176 15, 125 1, 6, 73
2178 1, 3, 4, 25 1, 2, 3, 543
2180 1, 8, 15, 78, 1014 5, 6, 10
2184 1, 8, 64, 715, 722 45, 60
2186 561, 1035, 1350 2, 1284
2188 6 51
2190 110, 114, 440 49, 1824
2192 201 8, 36, 47
2194 25 15, 1245
2196 57, 100, 612 110, 198
2198 417, 552, 562 3, 10, 91, 425
2200 4, 6, 37 5, 6, 200
2202 33, 200, 330, 627 15, 49, 280, 1024
2206 8 519, 1181, 1701
2208 37, 795 3, 11
2210 4, 347, 437 1, 18, 1110, 1326
2212 1, 2, 6, 15, 62 9, 53, 108
2214 3, 16, 394, 1768 35, 89
2216 17, 50 58
2218 233 5, 114, 647, 828
2220 5, 78, 122, 699 2, 15, 169, 204
2222 2, 13, 242 17, 135, 334
2224 286 1, 97
2226 1 1, 2, 3, 79, 107
2228 1, 6, 10, 13, 378 2, 26
2230 3, 228 13
2232 8, 59, 1874 2, 316
2234 9, 12, 48 7, 97
2236 511 147
2238 273 1
2240 1, 2, 3, 1646 1, 8, 13, 61, 101, 383
2242 1, 3, 312 45, 65, 113, 429
2244 5, 885 46
2246 2, 3, 4, 1134, 1919 1
2248 1, 2, 8, 50, 94 7
2252 1293 50, 122, 181
2254 14 2, 5
2256 555 3
2258 1215 4
2260 4, 454 1, 4, 8, 15, 119
2262 1, 2, 78, 98, 235, 1745 3, 53, 110, 154, 271
2268 2 1, 30, 54, 141, 286, 376
2270 1, 24, 388 795, 1274, 1500
2272 6, 1175, 1736 1, 9, 415
2274 1, 79, 826 5, 13, 69
2276 33, 36, 69, 820 1, 29, 113, 903
2278 1, 108 18, 46, 99, 129, 233, 345
2282 2, 3, 5, 33, 70, 1309, 1327 1, 385
2284 1 18, 275, 1478
2286 20 1, 6, 433, 1416, 1590
2288 1, 193 9
2292 7, 32, 60 6, 258
2294 4 87, 350, 481
2296 15, 25, 85, 535 1, 2, 6, 10, 26, 140, 1485
2302 3, 107 1, 3, 84, 216, 292, 323
2304 1, 2, 3, 6, 1941 6637, 12989
2306 69, 165 29
2310 1, 2, 1308 3, 4, 1252
2314 1257 1, 52
2316 1, 13, 73 43, 148, 387
2318 38, 39, 299, 306 12, 88, 94, 325, 375
2320 2, 63, 200, 252 15, 78
2322 3, 5, 6, 22, 90, 154, 1762 2, 67
2324 2, 5, 37, 813 28, 39, 237
2326 120 833
2328 51 3, 6
2332 3 109, 156, 1089
2336 15, 333 1, 7, 432
2338 1, 1286 204, 759
2340 4, 7, 10, 24, 106 14, 1665
2344 4 1, 10, 239, 635
2346 1, 14, 141 9, 12, 1930
2348 13 4, 149
2350 3, 106, 267, 453 3
2352 2, 29, 57, 944, 1270 4, 8, 12
2354 154 20
2356 140 45, 1948
2358 5, 207, 605 1, 28, 98, 1735
2360 1, 2, 11, 108 7, 182, 286, 1113
2362 939 53, 1422
2364 3, 6, 405 151, 1646
2366 24 7, 109, 370, 625
2368 156, 864 2, 29
2370 2, 3, 80, 959, 1475 12
2372 7, 13 4, 171
2374 4 1, 200, 214
2376 1, 9, 84, 378 54
2378 3, 18 2
2380 346 1, 435, 521, 1723
2386 20, 36, 40, 180, 196 24, 112, 1034
2388 110 1, 4, 6
2390 1, 14, 18 36, 69, 402
2392 21 1, 8, 384
2394 1, 205, 905 3, 4, 9, 54, 140
2396 18, 1233 11, 161
2402 1, 10, 23, 24, 661 156, 366
2406 9, 1212 2, 3, 6, 26, 153
2408 21 7, 40
2410 156, 316, 493, 1582 2, 3, 113
2412 56 1, 3, 517
2414 1, 3 3, 21, 41, 87, 647
2416 3 625
2418 2, 247, 1279, 1975 51, 57
2420 4, 900 1, 2, 21, 66, 1916
2422 1, 3, 10, 332 1, 10
2424 1, 420 827
2426 6, 51, 495 7, 193
2428 2 68
2430 3, 5, 12, 27, 122, 147, 155, 404 1, 11, 104
2432 1, 7, 510 3, 6, 30
2434 4 8, 66, 73, 434
2436 4, 19, 30, 133 590
2440 3, 15, 65, 101 3, 7, 375
2442 64, 115 33, 455
2444 26, 70, 205 34, 104, 287
2446 6, 61, 63 4, 190
2448 1240 1, 3, 25, 98
2450 1, 34, 96, 126 2, 21, 225, 423, 1344
2452 127 3, 770
2454 15 1
2456 1, 5, 17 21, 425, 561
2458 4, 17, 19, 86, 318 77
2460 13, 78 3, 6
2462 70, 1360 74
2464 3, 482, 1882 1
2466 1, 7 35, 908
2468 2, 1250 1, 9, 76, 223, 1272
2470 1, 6, 11, 14, 49, 195 5
2472 76, 1853 3, 332
2476 3, 435 42, 169, 1524
2478 3, 22 2, 6, 275
2484 713, 953 1, 207, 344, 866
2486 1, 40, 79, 217, 290 41, 298, 562
2488 9, 128, 1249 310
2490 83 1
2492 1, 2, 81, 978, 1429 2, 3, 6, 1090
2494 670, 1212 182
2496 38, 126, 495 3
2498 2, 9 33, 39, 251, 535
2500 2, 33, 215, 1273, 1337, 3180 2, 19, 60, 2198
2502 2, 3 1329
2504 16, 59 2, 20, 198, 717
2506 3, 10, 13, 1755 1, 43
2508 1, 106, 331, 721 38, 107
2510 9, 12, 1271, 1496 1, 76, 489
2512 1, 2, 10, 276, 583 1, 5, 97, 362, 759, 1193, 1289
2514 1, 14, 270 1, 7, 10, 13, 933
2516 1, 54, 144, 339 4, 159
2520 3, 10, 82, 819 3, 8
2522 90 2, 6, 101, 710, 819
2526 8, 38, 1219 2, 37
2528 18, 29, 188, 204, 434, 554, 581 1, 142
2532 4 1, 271, 381
2536 9, 646 56, 795, 1610
2538 2 10, 1081
2540 122, 306 1
2542 1, 7, 11, 538, 670, 1012 1, 19, 132, 1782
2544 1, 2, 8, 21, 368 34, 42
2546 52 18
2548 2 1, 3, 7, 44, 199
2550 1 38, 1346, 1644
2552 9 4, 61, 1455
2554 3, 4, 15, 121, 205, 231 1, 2, 182
2556 1, 6, 34, 90 57
2558 332 22, 285
2562 61 1, 2, 5, 12, 200, 995
2564 1, 93, 100, 268 5, 17, 62, 1310
2566 2, 11 114, 175, 535
2568 105, 130, 859 1, 2, 290, 1676
2570 1, 8 6, 486
2572 3 3, 47
2576 108, 119, 305, 648, 834, 1669 1, 2, 35, 196, 717, 1144, 1514
2578 1 168
2580 101, 497, 1085 55
2582 3, 27, 92, 305, 1278 3, 4, 6, 342
2584 29, 46, 67, 512 1, 405
2590 1, 12, 25, 32, 335, 347 1, 2, 33, 89, 383
2592 1, 373, 481, 756 2, 9, 287, 312
2594 200, 476 16, 28, 271, 310, 1380
2596 13, 18, 75, 82 1, 6
2598 1, 24, 104, 367, 719, 798 1, 4
2600 1, 3, 9, 39, 57 3, 130, 420
2602 77, 84, 317 1, 2, 7, 20, 87, 145
2604 1, 5, 9, 17, 79, 1220 135, 191
2608 2, 27, 32 12
2612 1, 102, 1042 4, 8
2618 81, 854 1, 29, 207, 976
2620 1, 217 209, 261
2628 115 558, 748
2630 10, 837 2
2632 7, 196, 342 3, 10, 15, 351, 1065
2636 2, 18, 81, 357 1
2638 1, 18 2, 14, 23, 46, 462, 1053, 1786
2640 2 79, 1960
2642 2 132
2644 41, 267, 1155 8, 282
2648 1, 3, 12, 34, 751 20, 1157
2652 25, 154, 173, 299 1, 2, 7, 10, 86
2654 1, 2, 354, 1139 2
2656 12 16, 293
2662 3 2, 5, 56
2664 3 1, 10, 118, 124
2666 1, 6, 12, 24, 264 1, 124
2668 1, 2, 4, 6 20, 1772
2670 12, 63 42, 113
2672 51, 575 2, 20
2674 305, 435 8
2676 280 11, 435
2680 1 1, 2, 10, 15, 50, 107
2682 1, 34, 84, 106, 118 78, 1708
2684 2, 44 5, 366
2686 257, 888 12, 42, 1112
2690 165 2
2692 143, 893 90, 1870
2694 36, 578, 1739 2, 242, 470, 944, 1118
2696 1134 2, 35, 59, 330, 372
2698 2, 73, 93 51
2700 159 1, 2, 50
2702 1 1
2704 1, 7, 12, 11617 83, 228, 7205
2706 5 1, 7
2708 1, 13, 61, 722 2, 3, 52, 77, 160
2710 21, 284, 424, 542 50
2712 103, 111 15, 102
2714 21, 149, 179, 216 1, 12, 25
2716 1, 13, 79, 99 1, 10, 50, 790
2718 1, 3, 4, 9, 57, 312, 828 9, 14, 1235
2720 24, 39, 305, 1560 1, 177, 1168
2722 1, 2, 13, 22, 65, 1032 3, 4, 33, 343, 1102
2724 1300 1, 23, 25
2726 1, 30, 43, 752 35, 58, 324
2728 10, 11 1, 6, 14, 73
2730 1, 120, 340 2, 42, 124, 200
2732 3, 4, 6 6, 27
2734 17 1, 3, 343
2736 1, 2, 16, 256 1, 18, 245, 610, 796
2738 1, 3 1, 6, 15, 1267
2740 1, 374 6, 39, 210, 522
2744 2, 15, 80, 4116, 4553 20, 24, 166, 1948, 1979
2746 28, 252 2, 45, 452, 537
2748 8, 26, 42 1, 33, 109
2750 1, 2, 10, 44, 218 2, 53, 64, 414, 526
2752 6, 57, 157, 167, 247 17, 27
2754 2, 42, 73 11, 12, 82
2756 3, 27 2, 9, 27
2758 24, 798 1, 5, 35
2760 1 98
2762 3, 234 22, 94, 1450
2764 5, 34, 1261 27, 91, 627, 644
2766 3, 5, 278 3
2772 1130 1, 63
2776 141 27, 397
2778 747 1, 2, 1349
2780 1, 4 69
2784 1386 1, 3, 51, 174, 339, 1071, 1574
2786 1 125, 137, 702, 965
2788 3, 24, 402, 435, 718, 1416 2, 3, 36, 935
2790 5 127
2792 42, 1821 37, 89, 494
2796 1, 2, 147 9, 29, 370, 1895
2800 22, 1240 20
2806 33, 133 1
2808 1, 28 1, 12, 584, 1547
2810 1, 691, 866 190, 1390
2812 9 26
2814 6, 11, 16 12, 1626
2816 453 2
2818 74 4, 15, 420
2820 21, 194, 1718 1, 5, 11, 63, 303, 710, 1180
2822 1, 8 3, 1682
2824 3, 26 117
2826 9, 400, 753 7, 50
2828 9, 113 3, 490, 542
2832 18, 72, 227, 1518 30
2834 11, 157, 223, 386, 463 6, 40, 42, 1814
2836 4 1, 3, 185
2838 1, 2, 12, 25 186, 316
2840 4, 5, 34, 658 2, 3, 111
2842 22, 196, 609 8, 29, 79, 121, 311
2844 42 24, 384
2848 8, 30, 589, 650 1, 9, 59, 636, 829
2850 1 4, 372
2852 2, 13, 63, 1040 108
2854 5 1, 97, 98
2856 1, 2 20
2858 7, 49, 393 2, 291, 576
2860 50 12
2862 7, 16, 97, 145 9, 121, 180, 265
2864 34, 58, 85, 1315 2
2866 192 21
2868 46, 108 1, 3
2870 1, 765 1, 22, 118, 1221
2872 1, 7, 24, 37, 58 5, 8
2874 11, 332 1776
2876 51, 135 12, 93
2878 4, 33, 256 44, 59, 99, 325, 765
2880 14, 243 1594
2882 681 1, 24, 57, 170
2884 1, 680, 852, 1048 4, 14, 44
2888 5, 20, 749 1
2890 1, 16, 1931 7, 501
2892 6, 802, 1788 154
2894 57, 584 30, 53, 198, 1071, 1906
2896 17, 45, 850 1, 3, 60, 206
2898 1, 11, 13, 21, 633 13
2900 40 5, 6, 9
2902 15, 29, 62, 1142 10, 103
2906 7, 243, 820 13, 29, 33, 76
2908 39, 475 3, 9
2910 4, 226, 1974 1, 2
2912 1, 34, 333 6, 8, 13
2914 4, 34 2, 113, 285, 411
2916 4034 955
2920 17, 34, 59, 136 2, 6, 688
2926 1, 2, 6, 362, 1232 5, 11, 20
2928 7, 217 17, 20, 299, 884
2930 134 1, 49, 1152, 1881
2932 1, 79, 468 1, 2, 41
2934 1, 2, 35 2
2936 12 48, 65
2938 454 31, 36
2940 3, 16, 90 1, 2, 48
2942 1, 45 3, 551
2946 2, 22, 138 20, 118
2948 31, 69, 426, 1577, 1586 66, 128
2950 21 57, 130
2952 4, 12, 420, 455 2, 207, 1920
2954 20 1, 129, 1487
2956 1, 25 5, 24, 41, 80, 839
2958 3, 18, 798 4
2960 25, 444, 1793 25, 367
2964 2, 27, 49, 129 15, 196
2966 3, 52 54
2968 1475 3, 12
2972 2, 32, 153 1, 169
2976 61, 91, 157 1, 1673
2980 1066, 1420 2
2982 2 4, 78, 82, 156, 606
2984 16, 64, 421 18
2986 3, 6 237
2988 3, 121 120, 990
2990 23, 28, 212 2, 53
2992 3, 8, 1800 6, 84, 102
2996 237 1, 4, 30, 70
2998 1, 12, 111, 378, 733 84
3000 312, 366 180

How to participate?

Reserving

  • Reserve your base(s)/range(s) in this thread.

Sieving

  • Use cksieve (from Mtsieve) and
    • run a new sieve by calling cksieve -b 12 -n 1 -N 10000 -P 1000000000 (for base=12, n-range=1-10000, max prime factor 109). The sieve file will be written to ck_12.pfgw.
    • rerun an old sieve by calling cksieve -P 1e12 -i ck_12.pfgw -o ck_12.pfgw -f factors.txt (for base=12, max prime factor 1012, input/output files given, storing factors to "factors.txt").

PRP testing

  • Use PFGW calling pfgw64.exe -f0 ck_12.pfgw (running candidates file for base 12, no further factoring).

Prime testing

After testing with PFGW higher probable primes will be written in "pfgw.log". These have to be checked prime by calling like pfgw64 -tp -q"(12^68835-1)^2-2".

Reporting

Once you have completed your range, report any primes found in this thread. Then report the completed range in the reservation thread and specify whether you will continue with the base or release it.

References

  1. ↑ Yahoo Group "Primenumbers", 2002-02-04
  2. ↑ Yahoo Group "Primenumbers", 2004-02-23
  3. ↑ Thread "Carol / Kynea search (Near-power primes)"

External links

Current

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General numbers
Special numbers
Prime numbers
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