Generalized Fermat Primes sorted by base

A prime number of the form b^(2^n) + 1 is called a generalized Fermat prime. Here we give a list of such primes sorted by base b.

Exponents n
b=2 0, 1, 2, 3, 4, ...
b=4 Please see under b=2
b=6 0, 1, 2, ...
b=8 Impossible since 8=2^3
b=10 0, 1, ...
b=12 0, ...
b=14 1, ...
b=16 Please see under b=2
b=18 0, ...
b=20 1, 2, ...
b=22 0, ...
b=24 1, 2, ...
b=26 1, ...
b=28 0, 2, ...
b=30 0, 5, ...
b=32 Impossible since 32=2^5
b=34 2, ...
b=36 Please see under b=6
b=38 ...
b=40 0, 1, ...
b=42 0, ...
b=44 4, ...
b=46 0, 2, 9, ...
b=48 2, ...
b=50 ...
b=52 0, ...
b=54 1, 2, 5, ...
b=56 1, 2, ...
b=58 0, ...
b=60 0, ...
b=62 ...
b=64 Impossible since 64=4^3
b=66 0, 1, ...
b=68 ...
b=70 0, ...
b=72 0, ...
b=74 1, 2, 4, ...
b=76 4, ...
b=78 0, ...
b=80 2, ...
b=82 0, 2, ...
b=84 1, ...
b=86 ...
b=88 0, 2, ...
b=90 1, 2, ...
b=92 ...
b=94 1, 4, ...
b=96 0, 5, ...
b=98 ...
b=100 Please see under b=10
b=102 0, 6, ...
b=104 ...
b=106 0, 2, ...
b=108 0, ...
b=110 1, ...
b=112 0, 5, ...
b=114 5, ...
b=116 1, ...
b=118 2, 3, ...
b=120 1, 7, ...
b=122 ...
b=124 1, ...
b=126 0, 1, ...
b=128 Impossible since 128=2^7
b=130 0, 1, ...
b=132 2, 3, 5, ...
b=134 1, ...
b=136 0, ...
b=138 0, ...
b=140 2, 3, ...
b=142 2, ...
b=144 Please see under b=12
b=146 1, ...
b=148 0, ...
b=150 0, 1, 11, ...
b=152 3, ...
b=154 2, ...
b=156 0, 1, 4, 5, ...
b=158 4, ...
b=160 1, 2, ...
b=162 0, 6, ...
b=164 2, ...
b=166 0, ...
b=168 ...
b=170 1, ...
b=172 0, ...
b=174 2, ...
b=176 1, 4, ...
b=178 0, ...
b=180 0, 1, 2, ...
b=182 ...
b=184 1, ...
b=186 ...
b=188 4, ...
b=190 0, 7, ...
b=192 0, ...
b=194 2, ...
b=196 Please see under b=14
b=198 0, 2, 4, ...
b=200 ...
b=202 ...
b=204 1, 2, ...
b=206 1, ...
b=208 3, ...
b=210 0, 1, 2, ...
b=212 ...
b=214 ...
b=216 Impossible since 216=6^3
b=218 ...
b=220 2, ...
b=222 0, ...
b=224 1, ...
b=226 0, ...
b=228 0, 2, ...
b=230 1, ...
b=232 0, ...
b=234 7, ...
b=236 1, ...
b=238 0, 2, ...
b=240 0, 1, 3, ...
b=242 2, 3, ...
b=244 ...
b=246 ...
b=248 2, 4, ...
b=250 0, 1, ...
b=252 ...
b=254 2, ...
b=256 Please see under b=2
b=258 ...
b=260 1, ...
b=262 0, ...
b=264 1, ...
b=266 2, ...
b=268 0, ...
b=270 0, 1, ...
b=272 2, ...
b=274 6, ...
b=276 0, 2, ...
b=278 2, 8, ...
b=280 0, 1, ...
b=282 0, ...
b=284 1, ...
b=286 ...
b=288 2, 3, 4, ...
b=290 3, ...
b=292 0, ...
b=294 ...
b=296 2, ...
b=298 ...
b=300 1, 6, ...
b=302 ...
b=304 ...
b=306 0, 1, 3, 4, ...
b=308 ...
b=310 0, ...
b=312 0, 2, ...
b=314 1, ...
b=316 0, ...
b=318 4, ...
b=320 2, ...
b=322 ...
b=324 Please see under b=18
b=326 1, ...
b=328 2, ...
b=330 0, 4, ...
b=332 5, ...
b=334 2, ...
b=336 0, ...
b=338 ...
b=340 1, 2, ...
b=342 5, ...
b=344 ...
b=346 0, ...
b=348 0, 4, ...
b=350 1, ...
b=352 0, 2, ...
b=354 ...
b=356 ...
b=358 0, ...
b=360 5, ...
b=362 ...
b=364 2, ...
b=366 0, ...
b=368 ...
b=370 4, ...
b=372 0, ...
b=374 2, ...
b=376 5, ...
b=378 0, 3, ...
b=380 ...
b=382 0, 4, ...
b=384 1, ...
b=386 1, ...
b=388 0, ...
b=390 ...
b=392 3, ...
b=394 ...
b=396 0, 1, 4, ...
b=398 ...
b=400 Please see under b=20
b=402 ...
b=404 ...
b=406 1, ...
b=408 0, ...
b=410 ...
b=412 6, ...
b=414 2, ...
b=416 ...
b=418 0, ...
b=420 0, 1, ...
b=422 ...
b=424 ...
b=426 3, ...
b=428 5, ...
b=430 0, 1, 2, 5, ...
b=432 0, 5, ...
b=434 3, ...
b=436 1, 2, ...
b=438 0, ...
b=440 1, ...
b=442 0, 2, 3, ...
b=444 1, ...
b=446 ...
b=448 0, 5, ...
b=450 ...
b=452 4, ...
b=454 ...
b=456 0, 4, ...
b=458 ...
b=460 0, ...
b=462 0, ...
b=464 1, ...
b=466 0, 1, 2, ...
b=468 ...
b=470 1, 4, ...
b=472 2, ...
b=474 1, 4, ...
b=476 2, 4, ...
b=478 0, 4, ...
b=480 ...
b=482 ...
b=484 Please see under b=22
b=486 0, ...
b=488 2, ...
b=490 0, 1, ...
b=492 2, ...
b=494 2, ...
b=496 1, ...
b=498 0, 2, ...
b=500 ...
b=502 0, ...
b=504 2, ...
b=506 7, ...
b=508 0, 3, ...
b=510 3, ...
b=512 Impossible since 512=8^3
b=514 ...
b=516 2, ...
b=518 ...
b=520 0, ...
b=522 0, ...
b=524 ...
b=526 2, ...
b=528 ...
b=530 ...
b=532 7, ...
b=534 ...
b=536 1, ...
b=538 ...
b=540 0, 2, 3, ...
b=542 3, ...
b=544 1, ...
b=546 0, ...
b=548 7, ...
b=550 2, ...
b=552 ...
b=554 2, ...
b=556 0, 1, 2, ...
b=558 ...
b=560 4, ...
b=562 0, 3, 5, 6, ...
b=564 ...
b=566 2, ...
b=568 0, 2, 4, ...
b=570 0, 1, ...
b=572 ...
b=574 ...
b=576 Please see under b=24
b=578 ...
b=580 ...
b=582 2, ...
b=584 1, 2, ...
b=586 0, ...
b=588 5, ...
b=590 ...
b=592 0, 6, ...
b=594 1, ...
b=596 3, ...
b=598 0, 4, ...
b=600 0, 2, ...
b=602 ...
b=604 ...
b=606 0, ...
b=608 ...
b=610 3, ...
b=612 0, ...
b=614 8, ...
b=616 0, 2, ...
b=618 0, ...
b=620 ...
b=622 ...
b=624 2, ...
b=626 ...
b=628 2, ...
b=630 0, ...
b=632 ...
b=634 1, ...
b=636 1, ...
b=638 ...
b=640 0, ...
b=642 0, 4, ...
b=644 1, ...
b=646 0, 1, ...
b=648 ...
b=650 ...
b=652 0, ...
b=654 1, ...
b=656 2, ...
b=658 0, ...
b=660 0, ...
b=662 ...
b=664 3, ...
b=666 ...
b=668 ...
b=670 ...
b=672 0, ...
b=674 1, ...
b=676 Please see under b=26
b=678 ...
b=680 1, 3, ...
b=682 0, 3, ...
b=684 ...
b=686 1, 4, ...
b=688 4, ...
b=690 0, 1, 2, 4, ...
b=692 ...
b=694 ...
b=696 1, ...
b=698 ...
b=700 0, 1, ...
b=702 2, ...
b=704 1, ...
b=706 ...
b=708 0, ...
b=710 2, ...
b=712 ...
b=714 1, ...
b=716 1, ...
b=718 0, ...
b=720 ...
b=722 ...
b=724 ...
b=726 0, 5, ...
b=728 6, ...
b=730 2, ...
b=732 0, 2, 3, ...
b=734 ...
b=736 4, ...
b=738 0, 2, 5, ...
b=740 1, ...
b=742 0, 2, ...
b=744 ...
b=746 ...
b=748 2, ...
b=750 0, 1, ...
b=752 ...
b=754 ...
b=756 0, ...
b=758 2, ...
b=760 0, 1, 2, ...
b=762 ...
b=764 1, ...
b=766 ...
b=768 0, 2, ...
b=770 ...
b=772 0, 2, ...
b=774 4, ...
b=776 4, ...
b=778 2, 4, ...
b=780 1, ...
b=782 3, ...
b=784 Please see under b=28
b=786 0, 2, ...
b=788 2, ...
b=790 4, ...
b=792 ...
b=794 ...
b=796 0, ...
b=798 2, ...
b=800 2, 3, ...
b=802 ...
b=804 5, ...
b=806 ...
b=808 0, 3, ...
b=810 0, 2, ...
b=812 ...
b=814 ...
b=816 1, ...
b=818 ...
b=820 0, ...
b=822 0, ...
b=824 10, ...
b=826 0, 1, ...
b=828 0, ...
b=830 4, ...
b=832 4, ...
b=834 4, ...
b=836 ...
b=838 0, ...
b=840 ...
b=842 ...
b=844 ...
b=846 4, ...
b=848 ...
b=850 5, ...
b=852 0, ...
b=854 ...
b=856 0, 2, ...
b=858 0, ...
b=860 1, ...
b=862 0, ...
b=864 1, ...
b=866 3, ...
b=868 ...
b=870 ...
b=872 ...
b=874 2, ...
b=876 0, 3, ...
b=878 ...
b=880 0, ...
b=882 0, ...
b=884 3, 5, ...
b=886 0, ...
b=888 ...
b=890 1, ...
b=892 3, 8, ...
b=894 2, ...
b=896 ...
b=898 8, ...
b=900 Please see under b=30
b=902 ...
b=904 ...
b=906 0, 1, ...
b=908 ...
b=910 0, 1, ...
b=912 2, ...
b=914 2, ...
b=916 3, 4, ...
b=918 0, 3, ...
b=920 1, ...
b=922 ...
b=924 ...
b=926 ...
b=928 0, 2, ...
b=930 1, 2, ...
b=932 ...
b=934 3, ...
b=936 0, 1, 2, ...
b=938 ...
b=940 0, ...
b=942 ...
b=944 ...
b=946 0, 1, 4, ...
b=948 ...
b=950 1, ...
b=952 0, 2, ...
b=954 ...
b=956 3, 4, ...
b=958 ...
b=960 1, 7, ...
b=962 2, ...
b=964 ...
b=966 0, 1, 2, ...
b=968 ...
b=970 0, ...
b=972 4, ...
b=974 ...
b=976 0, ...
b=978 ...
b=980 ...
b=982 0, 4, ...
b=984 4, ...
b=986 1, 2, ...
b=988 ...
b=990 0, 3, ...
b=992 2, ...
b=994 ...
b=996 0, 2, ...
b=998 ...
b=1000 Impossible since 1000=10^3

(Colored cells correspond to Sloane's A075090.)

Ellipsis (dots) means that no more terms are known. It does not necessarily imply that the sequence continues. One can conjecture that every row in the above table is a finite sequence.

There are several ways to define new integer sequences from the above table: Concatenate all rows. Or give number of terms in each row. Or list row numbers whose sequences are empty. None of these sequences appear in Sloane's. But since there may be terms not yet discovered in each row, it is probably best not to submit these sequences.

Greatest known values of n

The following table is inspired by Yves Gallot's pages. Here we give the greatest known values of n. Please note that new primes may have been discovered after this page was updated.

Currently known bases b OEIS
n>22   (none)
n=22 2^n=4194304 (none)
n=21 2^n=2097152 (none)
n=20 2^n=1048576 (none)
n=19 2^n=524288 75898, 341112, 356926, 475856. A243959
n=18 2^n=262144 24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, and more. A244150
n=17 2^n=131072 62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, and more. A253854
n=16 2^n=65536 48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, and more A251597
n=15 2^n=32768 70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, and more A226530
n=14 2^n=16384 67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, and more A226529

In the above table, there is no guarantee the bases shown are the smallest possible.

See:


/JeppeSN