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GFN Prime Search Status and History

This page summarises the search history for Generalised Fermat Primes b^2^n+1, for all n >= 13 which have been searched. We record all the known primes for each n, along with notes on when each range was searched and by whom. The data has been compiled by Iain Bethune from a range of sources, including the the Top 5000 database, Yves Gallot's GFN Prime Search and PrimeGrid's own records.

It will be periodically updated, especially when new primes are found.

Quick links: n=13, n=14, n=15, n=16, n=17, n=18, n=19, n=20, n=21, n=22

Range status key:

Range has been double-checked Range is currently being searched Range has been single-checked Primes have been reported in this range but the complete search history is unknown

GFN Prime Search Status: n=15

As of 27th July 2018, there are 1014 known primes of the form b^2^15+1 = b^32768+1.

The following values of b give primes:

b Range status
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
2034902
2147196
2167350
Up to b=2.2M was first searched by Yves Gallot's GFN Search. A single-pass check was made using Proth.exe, completed on 21st February 2002. 35 primes were found.

A full double-check has been carried out on BOINC by PrimeGrid using Genefer in October 2015.
2279250
2282862
2329148
2351172
2354572
2354618
2435410
2510928
2538626
2585646
2619572
2679502
2696506
The range between b=2.2M and 2.8M was searched by Heuer and Sun (and others?) using Proth.exe between August 2003 and December 2004. 13 primes were found.

A full double-check has been carried out on BOINC by PrimeGrid using Genefer in October 2015.
2973894
3000336
3042376 *
The ranges b=2.8M-3M and b=3M-3.1M were reserved by Heuer and Sun respectively, and were are least partly searched using Proth.exe. 2 primes were found.

A full double-check on BOINC by PrimeGrid using Genefer found 1 previously undiscovered prime (*) on 23rd October 2015.
3109540
3134838
3149688
3 Primes in this range were found by Ayeni using Proth.exe between April 2004 and April 2005.

A full double-check has been carried out on BOINC by PrimeGrid using Genefer in October 2015.
3391432
3394960
3417444
3467622
3472100
3532700
3642296
3664554
3756658
3769892
4004620
4042212
4053566
4108672
4184696
4197530
4323648
4396444
4403010
4465132
4487952
4578290
4698564 *
4826770
4971052
5014954
5042524
5079358
5144952
5177652
5446484
5667050
5761466
6041274 *
6068406 *
6199694 *
6275442 *
6318186 *
6412528 *
6607866
6656034
6700428
6731948
6960350
7039132
7284770
7285278
7510824
7528624
7663846
7680822
7736350
7869772
8010848
8100176
8102494
8106614
8123056
8125520
8157528
8191116
8244666
8273506
8285500
8307368
8345790
8380046
8442162
8461246
8464936
8483782
8635576
8712924
8796286
8883278
9011408
9148556
9214318
9302934
9342420
9458852
9543806
9622594
9693048
9744750
9850652
9863800
9926910
9973932
10021136
10120074
10403098
This range was first tested on PRPNet by PrimeGrid using Genefer, LLR and PFGW from January 2011 finishing at b=10428486 on 18th October 2015 , finding 85 primes.

A full double-check has been carried out on BOINC by PrimeGrid using Genefer in October and November 2015. 7 previously undiscovered primes (*) were found during this check.
10448246
10689024
10742748
10758842
10877222
11428394
11600204
11623354
11688764
11741086
11756060
11967372
12007538
12013942
12177788
12500316
12525142
12563092
12825258
12952476
13004344
13062632
13236710
13268134
13421628
13516486
13522632
13526546
13535820
13553636
13649540
13685572
13782404
13984036
14023640
14090140
14212786
14466952
14662858
14805912
14933582
14953270
14996602
Starting in November 2015, PrimeGrid ran a two-pass search on BOINC using Genefer, finding 43 primes.
15079730
15297664 *
15547296
This range was at least partly searched by Siegert using PFGW in 2011, yielding 2 primes.

The range was double-checked by PrimeGrid on BOINC using Genefer in December 2015 and found 1 additional prime (*).

All b up to 16253972 have been checked as of 23rd December 2015.
15674780
15731584
15800610
15804124
15859134
16217720
16307584
16341644
16391094
16676202
16710646
16715766
16933402
16990688
17132938
17166276
17213996
17255882
17498720
17503170
17630048
17746856
17821911
17914162
17957740
17978924
18027484
18162892
18318176
18376198
18860674
18900442
18930698
18959384
19011418
19045454
19159066
19297056
19353558
19444332
19454016
19510862
19523144
19567100
19587542
19661856
19814292
19864152
19884080
19998404
20009904
20033660
20037206
20190116
20228770
20294998
20507016
20583814
20586806
20607250
20738336
20805908
21080328
21232188
21300678
21411466
21444466
21479990
21510206
21661504
21702126
21950862
22040372
22102946
22155010
22217326
22459694
22574138
22631632
22783708
22799674
22844832
22875794
22996256
23054064
23142958
23171048
23343918
23447300
23544450
23566592
23625178
23717922
23842202
23962732
24057570
24154818
24261684
24264168
24354070
24452630
24607410
24641706
24698156
24710848
24745350
24802522
24811616
24953074
24994852
25013972
25213932
25319642
25339900
25361258
25404726
25521108
25580284
25676304
25676940
25853424
25942142
25980454
26055736
26088376
26195856
26346504
26467316
26621776
26752326
26775424
27021706
27047488
27271312
27285676
27390636
27505842
27525168
27543360
27550814
27843292
27893132
27939240
27958104
28049668
28268460
28301302
28557808
28571334
28586446
28650834
28673162
28755186
28869440
28891074
28892870
28993696
29173516
29242790
29252466
29420948
29615708
29652588
29667260
29712396
29721802
29785538
29815500
29921086
30010294
30021856
30207658
30239030
30404404
30434172
30802884
30974216
30986868
31036206
31239744
31445750
31447144
31511860
31697302
31798656
31935886
32358434
32396118
32405610
32422084
32463028
32891372
32910540
32915544
33034040
33112556
33381416
33424304
33430490
33573070
33573138
33644288
33966024
34410222
34468032
34554354
34639564
34674870
34822194
34875654
34890750
34934836
35033874
35242026
35252438
35417398
35429674
35499568
35538016
35550710
35593678
35694480
35701238
35791356
35967498
36170182
36188718
36258288
36367446
36429026
36465632
36554880
36591234
36619214
36955096
37012138
37014478
37069104
37080844
37222692
37289184
37477478
37525066
37539082
37731398
37723960
37919898
37955742
38166298
38204376
38296030
38398404
38437674
38532442
38572858
38672504
38732164
38794556
38796892
38808992
38876358
38884144
38914702
39109462
39142864
39163776
39280014
39507650
39563960
39615996
39646302
39647020
39772158
39801432
39884560
40134364
40197348
40232118
40257652
40313088
40381188
40604624
40624084
40655430
40697230
40894766
40899438
40951164
40958948
41022582
41015312
41097904
41373534
41376648
41545108
41577848
41585720
41770314
41980792
42026370
42155204
42159980
42204108
42214528
42379292
42463906
42581378
42584798
42648386
42724662
42777640
43083602
43127998
43406228
43613034
43617780
43796074
43845888
43953704
44001880
44220114
44224774
44262564
44273124
44353554
44358274
44371796
44435560
44477904
44543696
44605360
44613578
44755938
44818970
44935222
45030014
45063980
45090978
45119780
45143836
45376298
45629164
45637798
45659574
45698978
45839458
45863930
46022608
46056952
46170810
46582454
46698964
46960218
46983936
47378690
47472846
47595950
47569014
47651824
47946238
47984028
48066320
48096646
48135770
48149268
48157002
48062530
48239150
48331236
48603326
48677646
48769710
48845872
49159028
49298736
49303742
49496924
49523650
49527560
49564282
49578898
49651148
49652988
49655768
49704390
49922164
50123936
50221936
50587866
51031756
51198384
51410204
51492082
51501460
51616728
51728384
51906054
51910848
52075406
52224456
52256606
52294700
52331768
52545018
52773748
52874414
52927856
53007214
53022198
53250488
53513474
53523364
53579878
53738372
53819034
53841266
54032224
54044372
54049292
54070422
54232280
54269880
54609780
54859632
55144710
55175134
55433918
55491168
55742096
55879658
55904786
56226958
56247626
56568622
56738172
56838966
56841438
56991358
57203908
57317268
57361710
57390028
57565718
57591252
57743714
57759468
57837322
57833524
57978144
58228590
58306260
58430022
58446698
58641604
58796050
58836382
58896930
59213912
59241586
59473128
59581078
59587840
59850740
59858148
59909644
60143970
60155332
60197028
60327268
60395700
60672378
60693660
60693544
60737802
60781272
60838920
60852976
61123304
61250164
61281396
61374756
61377584
61388912
61445596
61586622
61608268
61605660
61696642
61715470
61722492
61777432
61862814
61961696
61975666
61978760
62032036
62094976
62105082
62137258
62318154
62327894
62422592
62537488
62647522
62751960
63044270
63345338
63777050
63810168
63806314
63832458
63874536
63930486
63972944
64162084
64353012
64365734
64714688
64741068
64713412
64760412
64804044
64855922
64867956
64898128
64985968
65054868
65068286
65089562
65093132
64714688
65266240
65320658
65344212
65387276
64714688
65514106
65520628
65529328
65772334
65783514
66178446
66193158
66353184
66489992
66581228
66619406
66702198
66732020
66794266
66797836
66880828
66994438
67131166
67543536
67647742
67804828
67799868
67840636
67889194
68002908
68207494
68274864
68436968
69190662
69204292
69187606
69500606
69541996
69789832
69858944
69945900
69043610
70122604
68962894
69756930
70246422
70272906
69753964
70440684
70464022
70495740
70048258
70526044
70730318
70604480
70501724
71049528
71072080
71266866
71270310
71317522
71396914
71424294
71402554
71717128
71924716
71990306
72039366
72078378
72220632
72018316
72507322
72517714
71846790
72557802
72871570
71566294
73258252
73344746
73490616
73568254
73633862
73605148
73702246
73908134
73849266
74177258
74184902
74175006
74395440
74423074
74462706
74761736
75110388
75144604
75170884
75121068
75572312
75813694
75830238
75874810
75909350
75943868
75998140
75998954
76021108
76163854
76256192
76292748
76303116
76621244
76765734
76830754
76839878
77030532
77151556
77251600
77336548
77456816
77653214
77732740
77799572
77854474
78005582
78056554
78088106
78170154
78204572
78346746
78513342
78592730
78854230
78888570
78908368
78927912
78979192
78995780
79313098
79741450
79997300
80009158
80190300
80454716
80502418
80612174
80654880
80627682
80721324
80885288
80949468
81092448
81248152
81204086
81370822
81419654
81464226
81479200
81510990
81524524
81542662
81643512
81917748
81963086
82021296
82216948
82419040
82456836
82510414
82721056
82797180
83436332
83532916
83551588
83955848
84034616
84100142
84238684
84607726
84657054
84779654
84828582
85231318
85326618
85426480
85509180
85538174
85724774
85810740
85962194
86085882
86197780
86266356
86316726
86357100
86390618
86416078
86560194
86559824
86786930
86852980
86907000
86906874
87404222
87538694
87586782
87602080
87822288
87888966
87888968
88040256
88139808
88169528
88188718
88375556
88457536
88499486
88593558
88705770
88711024
88875842
88907932
88913808
88950444
88973858
89057846
89191710
89534186
90036322
90044424
90098824
90158524
90191700
90216162
90493102
90500380
90537066
90558822
90681404
90853870
90897136
90918816
91042040
91053092
91159406
91325814
91442036
91542430
91657968
91670880
91688986
91729308
91960990
91992926
92248286
92332160
92788434
93026652
93029694
93167764
93195672
93218150
93335534
93343682
93449606
93472028
93495032
93544108
93577266
93583888
93768930
93822920
93970766
94191588
94526242
94533120
94561956
94569926
94719100
94821812
94907042
95031798
95156930
95204390
95318702
95444178
95528838
95714646
95750230
95796272
95948598
95980204
Starting in December 2015, PrimeGrid ran a two-pass search on BOINC using Genefer, finding 822 primes. All b up to 95899542 have been checked as of 27th July 2018.
Starting in December 2015, PrimeGrid is running a two-pass search on BOINC using Genefer. As of 27th July 2018, the leading edge is b=96203186.

Return to top

GFN Prime Search Status: n=16

As of 27th July 2018, there are 424 known primes of the form b^2^16+1 = b^65536+1.

The following values of b give primes:

b Range status
48594
108368
141146
189590
255694
291726
292550
357868
440846
544118
549868
671600
843832
857678
1024390
1057476
1087540 *
1266062
1361846
1374038
1478036
1483076
1540550
Searched up to b=1.86M by Yves Gallot's GFN Search, using Proth.exe. According to this page, all of the range was reserved but it is unclear if all the reserved tests were completed. 22 primes were reported up to May 2003

PrimeGrid restarted the search in Dec 2011 using PRPNet and Genefer, LLR and PFGW. This effort found 1 previously undiscovered prime (*).

A full double-check was carried out on BOINC by PrimeGrid using Genefer in October 2015.
1828502
1874512
1927034 *
2 primes were reported in this range in March 2005 and June 2008, found using Proth.exe.

The range has been covered by the PrimeGrid PRPNet search, and has been fully double-checked on BOINC using Genefer, finding 1 previously undiscovered prime (*).
1966374 *
2019300 *
2041898 *
2056292 *
2162068
2177038
2187182
3 primes were reported in this range by Ayeni between July 2008 and April 2009, using Proth.exe.

A double-check on BOINC by PrimeGrid using Genefer in November 2015 found 4 previously undiscovered primes (*).
2251082
2313394
2336976
2483590
2484264
2679102 *
2689202 *
2738848
2744940
2779470
2829122
2909834 *
3095674
3422670
3446048
3461954
3462728
3503138
3553248
3785880
3803826
3872228
3877352
3912496
3939670
3966304
This range was checked by PrimeGrid's PRPNet search using Genefer, LLR and PFGW, yielding 23 primes. A single-pass up to b=4205312 was completed on 13 October 2015.

A double-check on BOINC by PrimeGrid using Genefer was completed in December 2015. 3 previously undiscovered primes (*) were found.
4280722
4470420
4603702
4615890
4855782
4891596
5143624
5176028
5200658
5222640
5263190
5645768
5350408
5416598
5879074
5905206
5911446
6031270
6068296
6104728
6379668
6490328
6526168
6553414
6628260
6837196
6885798
6902598
6918698
6919394
7107770
7188654
7307674
7373894
7427232
7439308
7464550
7622010
7634800
7832756
7954812
8036146
8099100
8146592
8188038
8326702
8412634
8525558
8596942
8718098
8749120
8844704
9074206
9132408
9214424
9223072
9373148
9392774
9444368
9518248
9584088
9624110
9628238
9841672
9920216
10090890
10221008
10464954
10663638
10755500
10771330
10808098
10957508
11033686
11110588
11168512
11215796
11224312
11292580
11611202
11793760
11804536
11844886
11948010
12084448
12102810
12376146
12508790
12510100
12620940
12660574
12764608
12914686
12941970
13029860
13332130
13398688
13466298
13657452
13830802
13944760
13968798
14050484
14059910
14518950
14705866
14750712
14778298
14807318
14832074
14832496
14914230
14914430
14947856
15005274
15069494
15454262
15471020
15499198
15603948
15800506
15893070
15948188
16034826
16120142
16326986
16787702
16858742
17157558
17265602
17328156
17493552
17594396
17716338
17757426
17629398 *
17655444 *
17684828 *
17770454
17931126
18255116
18364744
18387422
18612352
18682542
18788582
19149988
19165998
19194688
19415886
19432158
19451454
19502212 *
19514276
19810832
19916902
20109598
20317650
20338906
20338564
20368808
20416586
20417546
20495674
20670628
20673706
20794390
20958760
21003082
21123886
21207594
21298612
21352892
21438976
21632926
21697066
21844548
21928344
21946386
21990626
22082900
22191882
22230230
22402670
22602162
22672358
22677562
22684548
22696662
22740816
22738600
22858812
22872882
22877386
23077164
23080390
23138154
23582210
23864352
23947122
24163706
24220066
24293444
24510980
24528070
24722142
24730888
24813140
24859030
25069382
25226438
25281714
25355170
25369696
25815054
25893048
25894728
25939926
25981818
25987186
26477248
26484668
26578974
26886748
26916240
26987486
27243670
27264470
27326204
27372028
27548300
27625562
27639120
27678634
27738902
27768276
28027008
28099080
27441534
28413758
28487278
28580718
28768058
29048042
29097708
29202654
29247740
29340732
29350290
29509260
29501096
29585874
29723118
29863462
30046920
30101988
30216696
30250088
30328508
30376064
30799220
30813412
30877996
30946880
31125356
31154730
31380744
31342854
31448254
31492936
31515202
31613530
31647504
31689074
31812740
31887064
31988978
32056116
32161210
32235030
32370056
32517922
32574488
32675102
32697566
32778852
32987968
33058194
33075870
33112172
33160366
33255290
33388034
33406946
33417368
33468724
33531480
33790428
33910364
33963124
34197950
34327650
34397974
34503816
34575414
34640098
34904540
35255372
35419084
35458620
35534500
35599734
35648406
35753376
36037446
36127768
36187924
36251736
36279762
36362214
36427586
36614838
36633974
36968128
37219594
37284376
37463776
37592308
37732056
37878964
37887878
38040628
38199804
38385824
38472570
38481086
38485606
38542914
38670832
38659632
38770952
38849156
38881766
38951594
39089796
39108392
39162320
39418466
39542162
39641162
39641632
39672694
39809082
39822410
39822830
40348160
40374324
40522376
40547834
40726656
40782880
40858208
40934648
40951894
41023954
Starting in December 2015, PrimeGrid is running a two-pass search on BOINC using Genefer, finding 361 primes. 4 primes (*) were previously reported by independent searchers. As of 27th July 2018, all b up to 41059584 have been checked.
Starting in December 2015, PrimeGrid is running a two-pass search on BOINC using Genefer. As of 27th July 2018, the leading edge is b=41245342.

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GFN Prime Search Status: n=17

As of 27th July 2018, there are 107 known primes of the form b^2^17+1 = b^131072+1.

The following values of b give primes:

b Range status
62722
130816
228188
386892
572186
689186
909548 *
1063730
1176694
1361244
1372930
Between 2002 and July 2004, Yves Gallot's GFN Prime Search tested ranges up to b=1.4M using Proth.exe, but left several gaps. As a result, only 6 primes were found in this initial search.

Several independent searchers, including David Underbakke continued working in the gaps, and another 4 primes were reported between 2009 and 2013.

Between June and October 2015, PrimeGrid completed a full re-check of this range on BOINC using Genefer, finding 1 previous undiscovered prime (*)
1560730
1660830
1717162
1722230
1766192
1955556
2194180
2280466
2639850
3450080
3615210
3814944
4085818
4329134
4893072
4974408
5326454
5400728
5471814
5586416
5734100
5877582
6391936
6403134
6705932
7379442
7832704
7858180
7926326
8150484
8704114
8770526
9240606
9419976
9785844
9907326
10037266
10368632
10453790
10765720
10921162
10962066
10994460
11036888
11195602
11267296
11292782
11778792
11876066
12343130
12357518
12512992
PrimeGrid ran a two-pass search of this range on BOINC using Genefer starting in October 2015, finding 52 primes.
PrimeGrid is currently running a two-pass search on BOINC using Genefer. As of 27th July 2018, the trailing edge is b=12578982 and the leading edge is b=12634984.
Between the leading edge of the previous range and b=42658772, no organised testing has been carried out, and no primes are known.
42654182
43163894
43165206
44049878
44085096
44330870
44438760
44919410
45315256
45570624
46077492
46371508
46385310
46413358
46730280
46736070
46776558
47090246
48273828
48370248
48643706
49038514
49090565
49225986
49243622
49331672
49397682
49530004
49817700
50055102
50110436
50217306
50495632
50844724
50963598
51269192
51567684
51570250
51580416
51872628
51954384
52043532
52412612
PrimeGrid is also running a two-pass search on BOINC using Genefer starting from b=42597774 (which would give a million digit prime). As of 27th July 2018, all b up to 52475486 have been checked and 45 primes have been found.
As of 27th July 2018, the leading edge of this search is 52542792.

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GFN Prime Search Status: n=18

As of 27th July 2018, there are 25 known primes of the form b^2^18+1 = b^262144+1.

The following values of b give primes:

b Range status
24518
40734
145310
361658
Yves Gallot's GFN Prime Search tested several ranges up to b=589708 using Proth.exe but did not find any primes.

In 2008, Stephen Scott reported 1 prime: 24518^262144+1, found using Proth.exe.

Starting in 2011, PrimeGrid ran a single-pass search on PRPNet, using Genefer, LLR and PFGW, and found the other 3 primes.

A full double-check of this range by PrimeGrid using Genefer on BOINC was completed in August 2015.
525094
676754
773620
This range has been searched by PrimeGrid partly on PRPNet using Genefer, LLR and PFGW and partly on BOINC using Genefer. A total of 3 primes were found

A full double-check of this range by PrimeGrid using Genefer on BOINC was completed in August 2015.
1415198
1488256
1615588
1828858
2042774
2514168
2611294
2676404
3060772
3547726
3596074
3673932
3853792
3933508
4246258
4489246
5152128
5205422
PrimeGrid ran a two-pass search of this range on BOINC using Genefer starting in November 2015, finding 18 primes.
PrimeGrid is currently running a two-pass search on BOINC using Genefer. As of 27th July 2018, the trailing edge is b=5266628 and the leading edge is b=5416030.

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GFN Prime Search Status: n=19

As of 27th July 2018, there are 6 known primes of the form b^2^19+1 = b^524288+1.

The following values of b give primes:

b Range status
75898
341112
356926
475856
1880370
2061748
Starting in 2011, PrimeGrid ran a single-pass search on PRPNet, using Genefer, LLR and PFGW, and found 1 prime. A full double-check of this range was then carried out by PrimeGrid using Genefer on BOINC.

The range was extended with a two-pass search on BOINC using Genefer, finding 5 primes.

As of 27th July 2018, all candidates up to b=2231464 have been double-checked.
A two-pass search on BOINC using Genefer is currently running. As of 27th July 2018, the trailing edge is b=229746 and the leading edge is b=254994.

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GFN Prime Search Status: n=20

As of 27th July 2018, there is 1 known prime of the form b^2^20+1 = b^1048576+1.

The following values of b give primes:

b Range status
Between November 2012 and June 2015, PrimeGrid ran a two-pass search on BOINC using Genefer, finding no primes up to b=720000.
919444 Starting in November 2015, PrimeGrid ran a two-pass search on BOINC using Genefer. As of 27th July 2018, all b up to 1000040 have been checked and 1 prime has been found.
A two-pass search on BOINC using Genefer is currently running. As of 7th July 2018, the leading edge is b=1016444. No primes have been found.

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GFN Prime Search Status: n=21

As of 27th July 2018, there are 0 known primes of the form b^2^21+1 = b^2097152+1.

b Range status
Between September and November 2015, PrimeGrid ran a two-pass search on BOINC using Genefer, finding no primes up to b=1350.
A two-pass search on BOINC using Genefer is currently running. As of 27th July 2018, the leading edge is b=229746.

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GFN Prime Search Status: n=22

As of 27th July 2018, there are 0 known primes of the form b^2^22+1 = b^4194304+1.

b Range status
Between February 2012 and November 2015, PrimeGrid ran a two-pass search on BOINC using Genefer, finding no primes up to b=54650.
A two-pass search on BOINC using Genefer is currently running. As of 27th July 2018, the trailing edge is b=120192 and the leading edge is b=125992.

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