Editing Fermat number (section)

==Generalized Fermat numbers==
Numbers of the form <math>\frac{a^{2^n}+b^{2^n}}{gcd(a+b,2)}</math> with ''a'', ''b'' any [[coprime]] integers, {{nowrap|''a'' > ''b'' > 0}}, are called '''generalized Fermat numbers'''. An odd prime ''p'' is a generalized Fermat number if and only if ''p'' is congruent to [[Pythagorean prime|1 (mod 4)]]. (Here we consider only the case {{nowrap|''n'' > 0}}, so {{nowrap|1=3 = <math>2^{2^{0}} \!+ 1</math>}} is not a counterexample.)

An example of a [[probable prime]] of this form is 200<sup>262144</sup> + 119<sup>262144</sup> (found by Kellen Shenton).<ref>[http://www.primenumbers.net/prptop/searchform.php?form=x%5E262144%2By%5E262144&action=Search PRP Top Records, search for x^262144+y^262144], by  Henri &amp; Renaud Lifchitz.</ref>

By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form <math>a^{2^{ \overset{n} {}}} \!\!+ 1</math> as ''F''<sub>''n''</sub>(''a''). In this notation, for instance, the number 100,000,001 would be written as ''F''<sub>3</sub>(10). In the following we shall restrict ourselves to primes of this form, <math>a^{2^{ \overset{n} {}}} \!\!+ 1</math>, such primes are called "Fermat primes base ''a''". Of course, these primes exist only if ''a'' is [[Parity (mathematics)|even]].

If we require {{nowrap|''n'' > 0}}, then [[Landau's problems|Landau's fourth problem]] asks if there are infinitely many generalized Fermat primes ''F<sub>n</sub>''(''a'').

===Generalized Fermat primes of the form F<sub>''n''</sub>(''a'')===
Because of the ease of proving their primality, generalized Fermat primes have become in recent years a topic for research within the field of number theory. Many of the largest known primes today are generalized Fermat primes.

Generalized Fermat numbers can be prime only for [[even number|even]] {{mvar|a}}, because if {{mvar|a}} is [[odd number|odd]] then every generalized Fermat number will be divisible by 2.  The smallest prime number <math>F_n(a)</math> with <math>n>4</math> is <math>F_5(30)</math>, or 30<sup>32</sup> + 1. Besides, we can define "half generalized Fermat numbers" for an odd base, a half generalized Fermat number to base ''a'' (for odd ''a'') is <math>\frac{a^{2^n} \!+ 1}{2}</math>, and it is also to be expected that there will be only finitely many half generalized Fermat primes for each odd base.

In this list, the generalized Fermat numbers (<math>F_n(a)</math>) to an even {{mvar|a}} are <math>a^{2^n} \!+ 1</math>, for odd {{mvar|a}}, they are <math>\frac{a^{2^n} \!\!+ 1}{2}</math>. If {{mvar|a}} is a [[perfect power]] with an odd exponent {{OEIS|id=A070265}}, then all generalized Fermat number can be algebraic factored, so they cannot be prime.

See<ref>{{cite web|url=http://jeppesn.dk/generalized-fermat.html|title=Generalized Fermat Primes|website=jeppesn.dk|access-date=7 April 2018}}</ref><ref>{{cite web|url=http://www.noprimeleftbehind.net/crus/GFN-primes.htm|title=Generalized Fermat primes for bases up to 1030|website=noprimeleftbehind.net|access-date=7 April 2018}}</ref> for even bases up to 1000, and<ref>{{cite web|url=http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt|title=Generalized Fermat primes in odd bases|website=fermatquotient.com|access-date=7 April 2018}}</ref> for odd bases. For the smallest number <math>n</math> such that <math>F_n(a)</math> is prime, see {{oeis|id=A253242}}.

{|class="wikitable"
!<math>a</math>
!numbers <math>n</math><br/>such that<br/><math>F_n(a)</math> is prime
!<math>a</math>
!numbers <math>n</math><br/>such that<br/><math>F_n(a)</math> is prime
!<math>a</math>
!numbers <math>n</math><br/>such that<br/><math>F_n(a)</math> is prime
!<math>a</math>
!numbers <math>n</math><br/>such that<br/><math>F_n(a)</math> is prime
|-
|2
|0, 1, 2, 3, 4, ...
|18
|0, ...
|34
|2, ...
|50
|...
|-
|3
|0, 1, 2, 4, 5, 6, ...
|19
|1, ...
|35
|1, 2, 6, ...
|51
|1, 3, 6, ...
|-
|4
|0, 1, 2, 3, ...
|20
|1, 2, ...
|36
|0, 1, ...
|52
|0, ...
|-
|5
|0, 1, 2, ...
|21
|0, 2, 5, ...
|37
|0, ...
|53
|3, ...
|-
|6
|0, 1, 2, ...
|22
|0, ...
|38
|...
|54
|1, 2, 5, ...
|-
|7
|2, ...
|23
|2, ...
|39
|1, 2, ...
|55
|...
|-
|8
| {{CNone|(none)}}
|24
|1, 2, ...
|40
|0, 1, ...
|56
|1, 2, ...
|-
|9
|0, 1, 3, 4, 5, ...
|25
|0, 1, ...
|41
|4, ...
|57
|0, 2, ...
|-
|10
|0, 1, ...
|26
|1, ...
|42
|0, ...
|58
|0, ...
|-
|11
|1, 2, ...
|27
| {{CNone|(none)}}
|43
|3, ...
|59
|1, ...
|-
|12
|0, ...
|28
|0, 2, ...
|44
|4, ...
|60
|0, ...
|-
|13
|0, 2, 3, ...
|29
|1, 2, 4, ...
|45
|0, 1, ...
|61
|0, 1, 2, ...
|-
|14
|1, ...
|30
|0, 5, ...
|46
|0, 2, 9, ...
|62
|...
|-
|15
|1, ...
|31
|...
|47
|3, ...
|63
|...
|-
|16
|0, 1, 2, ...
|32
| {{CNone|(none)}}
|48
|2, ...
|64
| {{CNone|(none)}}
|-
|17
|2, ...
|33
|0, 3, ...
|49
|1, ...
|65
|1, 2, 5, ...
|}

For the smallest even base {{mvar|a}} such that <math>F_n(a)</math> is prime, see {{oeis|id=A056993}}.

The generalized Fermat prime ''F''<sub>14</sub>(71) is the largest known generalized Fermat prime in bases ''b'' ≤ 1000, it is proven prime by [[elliptic curve primality proving]].<ref>[https://factordb.com/index.php?id=1100000000213085670 The entry of the generalized Fermat prime ''F''<sub>14</sub>(71) in the online factor database]</ref>

{|class="wikitable"
!<math>n</math>
!bases {{mvar|a}} such that <math>F_n(a)</math> is prime (only consider even {{mvar|a}})
![[OEIS]] sequence
|-
|0
|2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, ...
|{{OEIS link|id=A006093}}
|-
|1
|2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, ...
|{{OEIS link|id=A005574}}
|-
|2
|2, 4, 6, 16, 20, 24, 28, 34, 46, 48, 54, 56, 74, 80, 82, 88, 90, 106, 118, 132, 140, 142, 154, 160, 164, 174, 180, 194, 198, 204, 210, 220, 228, ...
|{{OEIS link|id=A000068}}
|-
|3
|2, 4, 118, 132, 140, 152, 208, 240, 242, 288, 290, 306, 378, 392, 426, 434, 442, 508, 510, 540, 542, 562, 596, 610, 664, 680, 682, 732, 782, ...
|{{OEIS link|id=A006314}}
|-
|4
|2, 44, 74, 76, 94, 156, 158, 176, 188, 198, 248, 288, 306, 318, 330, 348, 370, 382, 396, 452, 456, 470, 474, 476, 478, 560, 568, 598, 642, ...
|{{OEIS link|id=A006313}}
|-
|5
|30, 54, 96, 112, 114, 132, 156, 332, 342, 360, 376, 428, 430, 432, 448, 562, 588, 726, 738, 804, 850, 884, 1068, 1142, 1198, 1306, 1540, 1568, ...
|{{OEIS link|id=A006315}}
|-
|6
|102, 162, 274, 300, 412, 562, 592, 728, 1084, 1094, 1108, 1120, 1200, 1558, 1566, 1630, 1804, 1876, 2094, 2162, 2164, 2238, 2336, 2388, ...
|{{OEIS link|id=A006316}}
|-
|7
|120, 190, 234, 506, 532, 548, 960, 1738, 1786, 2884, 3000, 3420, 3476, 3658, 4258, 5788, 6080, 6562, 6750, 7692, 8296, 9108, 9356, 9582, ...
|{{OEIS link|id=A056994}}
|-
|8
|278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332, ...
|{{OEIS link|id=A056995}}
|-
|9
|46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992, ...
|{{OEIS link|id=A057465}}
|-
|10
|824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, ...
|{{OEIS link|id=A057002}}
|-
|11
|150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654, ...
|{{OEIS link|id=A088361}}
|-
|12
|1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696, ...
|{{OEIS link|id=A088362}}
|-
|13
|30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894, 176206, 180924, 201170, 212724, 222764, 225174, 241600, ...
|{{OEIS link|id=A226528}}
|-
|14
|67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, 422666, 426690, 449732, 462470, 468144, 498904, 506664, ...
|{{OEIS link|id=A226529}}
|-
|15
|70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, 553602, 743788, 825324, 831648, 855124, 999236, 1041870, 1074542, 1096382, 1113768, 1161054, 1167528, 1169486, 1171824, 1210354, 1217284, 1277444, 1519380, 1755378, 1909372, 1922592, 1986700, ...
|{{OEIS link|id=A226530}}
|-
|16
|48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540, 1266062, 1361846, 1374038, 1478036, 1483076, 1540550, 1828502, 1874512, 1927034, 1966374, ...
|{{OEIS link|id=A251597}}
|-
|17
|62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192, 1955556, 2194180, 2280466, 2639850, 3450080, 3615210, 3814944, 4085818, 4329134, 4893072, 4974408, ...
|{{OEIS link|id=A253854}}
|-
|18
|24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, 1828858, 2042774, 2514168, 2611294, 2676404, 3060772, 3547726, 3596074, 3673932, 3853792, 3933508, 4246258, 4489246, ...
|{{OEIS link|id=A244150}}
|-
|19
|75898, 341112, 356926, 475856, 1880370, 2061748, 2312092, 2733014, 2788032, 2877652, 2985036, 3214654, 3638450, 4896418, 5897794, 6339004, 8630170, 9332124, 10913140, 11937916, 12693488, 12900356, ...
|{{OEIS link|id=A243959}}
|-
|20
|919444, 1059094, 1951734, 1963736, 3843236, ...
|{{OEIS link|id=A321323}}
|}

The smallest even base ''b'' such that ''F''<sub>''n''</sub>(''b'') = ''b''<sup>2<sup>''n''</sup></sup> + 1 (for given ''n'' = 0, 1, 2, ...) is prime are

:2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444, ... {{OEIS|id=A056993}}

The smallest odd base ''b'' such that ''F''<sub>''n''</sub>(''b'') = (''b''<sup>2<sup>''n''</sup></sup> + 1)/2 (for given ''n'' = 0, 1, 2, ...) is prime (or [[probable prime]]) are

:3, 3, 3, 9, 3, 3, 3, 113, 331, 513, 827, 799, 3291, 5041, 71, 220221, 23891, 11559, 187503, 35963, ... {{OEIS|id=A275530}}

Conversely, the smallest ''k'' such that (2''n'')<sup>''k''</sup> + 1 (for given ''n'') is prime are

:1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 4, 1, ... (The next term is unknown) {{OEIS|id=A079706}} (also see {{oeis|id=A228101}} and {{oeis|id=A084712}})

A more elaborate theory can be used to predict the number of bases for which <math>F_n(a)</math> will be prime for fixed <math>n</math>. The number of generalized Fermat primes can be roughly expected to halve as <math>n</math> is increased by 1.

===Generalized Fermat primes of the form F<sub>n</sub>(''a'', ''b'')===
It is also possible to construct generalized Fermat primes of the form <math>a^{2^n} + b^{2^n}</math>. As in the case where ''b''=1, numbers of this form will always be divisible by 2 if ''a+b'' is even, but it is still possible to define generalized half-Fermat primes of this type. For the smallest prime of the form <math>F_n(a,b)</math> (for odd <math>a+b</math>), see also {{oeis|id=A111635}}.

{|class="wikitable"
!<math>a</math>
!<math>b</math>
!numbers <math>n</math> such that<br/><math>F_n(a,b) = \frac{a^{2^n}+b^{2^n}}{\gcd(a+b, 2)}</math><br/>is prime<ref>{{cite web |title=Original GFN factors |url=http://www.prothsearch.com/OriginalGFNs.html |website=www.prothsearch.com}}</ref><ref name="bjorn"/>
|-
| 2
| 1
| 0, 1, 2, 3, 4, ...
|-
| 3
| 1
| 0, 1, 2, 4, 5, 6, ...
|-
| 3
| 2
| 0, 1, 2, ...
|-
| 4
| 1
| 0, 1, 2, 3, ... (equivalent to <math>F_n(2, 1)</math>)
|-
| 4
| 3
| 0, 2, 4, ...
|-
| 5
| 1
| 0, 1, 2, ...
|-
| 5
| 2
| 0, 1, 2, ...
|-
| 5
| 3
| 1, 2, 3, ...
|-
| 5
| 4
| 1, 2, ...
|-
| 6
| 1
| 0, 1, 2, ...
|-
| 6
| 5
| 0, 1, 3, 4, ...
|-
| 7
| 1
| 2, ...
|-
| 7
| 2
| 1, 2, ...
|-
| 7
| 3
| 0, 1, 8, ...
|-
| 7
| 4
| 0, 2, ...
|-
| 7
| 5
| 1, 4,
|-
| 7
| 6
| 0, 2, 4, ...
|-
| 8
| 1
| {{CNone|(none)}}
|-
| 8
| 3
| 0, 1, 2, ...
|-
| 8
| 5
| 0, 1, 2,
|-
| 8
| 7
| 1, 4, ...
|-
| 9
| 1
| 0, 1, 3, 4, 5, ... (equivalent to <math>F_n(3, 1)</math>)
|-
| 9
| 2
| 0, 2, ...
|-
| 9
| 4
| 0, 1, ... (equivalent to <math>F_n(3, 2)</math>)
|-
| 9
| 5
| 0, 1, 2, ...
|-
| 9
| 7
| 2, ...
|-
| 9
| 8
| 0, 2, 5, ...
|-
| 10
| 1
| 0, 1, ...
|-
| 10
| 3
| 0, 1, 3, ...
|-
| 10
| 7
| 0, 1, 2, ...
|-
| 10
| 9
| 0, 1, 2, ...
|-
| 11
| 1
| 1, 2, ...
|-
| 11
| 2
| 0, 2, ...
|-
| 11
| 3
| 0, 3, ...
|-
| 11
| 4
| 1, 2, ...
|-
| 11
| 5
| 1, ...
|-
| 11
| 6
| 0, 1, 2, ...
|-
| 11
| 7
| 2, 4, 5, ...
|-
| 11
| 8
| 0, 6, ...
|-
| 11
| 9
| 1, 2, ...
|-
| 11
| 10
| 5, ...
|-
| 12
| 1
| 0, ...
|-
| 12
| 5
| 0, 4, ...
|-
| 12
| 7
| 0, 1, 3, ...
|-
| 12
| 11
| 0, ...
|-
|}

===Largest known generalized Fermat primes===
The following is a list of the ten largest known generalized Fermat primes.<ref name="Top Twenty's Generalized Fermat Primes">{{cite web|title=The Top Twenty: Generalized Fermat|url=http://primes.utm.edu/top20/page.php?id=12|work=The Prime Pages|first=Chris K.|last=Caldwell|access-date=5 October 2024}}</ref> The whole top-10 is discovered by participants in the [[PrimeGrid]] project.

{| class="wikitable" style="text-align:right"
|-
! Rank
! Prime number
! Generalized Fermat notation
! Number of digits
! Discovery date
! ref.
|-
| 1
| 4×5<sup>11786358</sup>&nbsp;+&nbsp;1
| ''F''<sub>1</sub>(2×5<sup>5893179</sup>)
| 8,238,312
| style="text-align:left" | Oct 2024
|<ref>[https://t5k.org/primes/page.php?id=138596 4×5<sup>11786358</sup>&nbsp;+&nbsp;1]</ref>
|-
| 2
| 3843236<sup>1048576</sup>&nbsp;+&nbsp;1
| ''F''<sub>20</sub>(3843236)
| 6,904,556
| style="text-align:left" | Dec 2024
|<ref>[https://t5k.org/primes/page.php?id=138830 3843236<sup>1048576</sup>&nbsp;+&nbsp;1]</ref>
|-
| 3
| 1963736<sup>1048576</sup>&nbsp;+&nbsp;1
| ''F''<sub>20</sub>(1963736)
| 6,598,776
| style="text-align:left" | Sep 2022
|<ref>[https://t5k.org/primes/page.php?id=134423 1963736<sup>1048576</sup>&nbsp;+&nbsp;1]</ref>
|-
| 4
| 1951734<sup>1048576</sup>&nbsp;+&nbsp;1
| ''F''<sub>20</sub>(1951734)
| 6,595,985
| style="text-align:left" | Aug 2022
|<ref>[https://t5k.org/primes/page.php?id=134298 1951734<sup>1048576</sup>&nbsp;+&nbsp;1]</ref>
|-
| 5
| 1059094<sup>1048576</sup>&nbsp;+&nbsp;1
| ''F''<sub>20</sub>(1059094)
| 6,317,602
| style="text-align:left" | Nov 2018
|<ref>[https://t5k.org/primes/page.php?id=125753 1059094<sup>1048576</sup>&nbsp;+&nbsp;1]</ref>
|-
| 6
| 919444<sup>1048576</sup>&nbsp;+&nbsp;1
| ''F''<sub>20</sub>(919444)
| 6,253,210
| style="text-align:left" | Sep 2017
|<ref>[https://t5k.org/primes/page.php?id=123875 919444<sup>1048576</sup>&nbsp;+&nbsp;1]</ref>
|-
| 7
| 81×2<sup>20498148</sup>&nbsp;+&nbsp;1
| ''F''<sub>2</sub>(3×2<sup>5124537</sup>)
| 6,170,560
| style="text-align:left" | Jun 2023
|<ref>[https://t5k.org/primes/page.php?id=136165 81×2<sup>20498148</sup>&nbsp;+&nbsp;1]</ref>
|-
| 8
| 4×5<sup>8431178</sup>&nbsp;+&nbsp;1
| ''F''<sub>1</sub>(2×5<sup>4215589</sup>)
| 5,893,142
| style="text-align:left" | Jan 2024
|<ref>[https://t5k.org/primes/page.php?id=136831 4×5<sup>8431178</sup>&nbsp;+&nbsp;1]</ref>
|-
| 9
| 4×3<sup>11279466</sup>&nbsp;+&nbsp;1
| ''F''<sub>1</sub>(2×3<sup>5639733</sup>)
| 5,381,674
| style="text-align:left" | Sep 2024
|<ref>[https://t5k.org/primes/page.php?id=138515 4×3<sup>11279466</sup>&nbsp;+&nbsp;1]</ref>
|-
| 10
| 25×2<sup>13719266</sup>&nbsp;+&nbsp;1
| ''F''<sub>1</sub>(5×2<sup>6859633</sup>)
| 4,129,912
| style="text-align:left" | Sep 2022
|<ref>[https://t5k.org/primes/page.php?id=134407 25×2<sup>13719266</sup>&nbsp;+&nbsp;1]</ref>
|}

On the [[Prime Pages]] one can find the [https://t5k.org/top20/page.php?id=12 current top 20 generalized Fermat primes] and the [https://t5k.org/primes/search.php?Comment=Generalized+Fermat&OnList=all&Number=100&Style=HTML current top 100 generalized Fermat primes].