Sequence A023394, Prime Factors of Fermat Numbers  

This sequence, Sloane's A023394, gives the prime factors of Fermat numbers Fm, arranged in increasing order by the value of the factor.

They are commonly expressed in the form k×2n+1, because of the results of Euler and Lucas showing that all Fermat factors have that form, where n >= m+2.

Here is the beginning of the sequence, giving the Fermat number that it is a factor of and the values of k and n for each one:

Factor Fm k 2n
3 F0 1 21
5 F1 1 22
17 F2 1 24
257 F3 1 28
641 F5 5 27
65537 F4 1 216
114689 F12 7 214
274177 F6 32×7×17 28
319489 F11 3×13 213
974849 F11 7×17 213
2424833 F9 37 216
6700417 F5 3×17449 27
13631489 F18 13 220
26017793 F12 397 216
45592577 F10 11131 212
63766529 F12 7×139 216
167772161 F23 5 225
825753601 F16 32×52×7 219
1214251009 F15 3×193 221
6487031809 F10 32×29×37×41 214
70525124609 F19 33629 221
190274191361 F12 5×11×211153 214
646730219521 F19 32×5×7×11×89 221
2710954639361 F13 5×11×752107 216

According to the OEIS entry A023394, all terms less than 258 (which is about 2.88×1017) are known.

Here are the first 12 Fermat numbers, the ones for which factorizations are known. P62, P99, etc. refer to prime numbers with 62, 99, etc. digits:

F0 = 3

F1 = 5

F2 = 17

F3 = 257

F4 = 65537

F5 = 641 × 6700417

F6 = 274177 × 67280421310721

F7 = 59649589127497217 × 5704689200685129054721

F8 = 1238926361552897 × P62

F9 = 2424833 × 7455602825647884208337395736200454918783366342657 × P99

F10 = 45592577 × 6487031809 × 4659775785220018543264560743076778192897 × P252

F11 = 319489 × 974849 × 167988556341760475137 × 3560841906445833920513 × P564

The first Fermat number whose factorization is as yet unknown is F12 = 2212+1 {~=} 1.04438888142×101233. Here are all the digits of F12, plus its currently known factors:

F12 = 10443888814131525066917527107166243825799642490473837803842334832839
53907971557456848826811934997558340890106714439262837987573438185793
60726323608785136527794595697654370999834036159013438371831442807001
18559462263763188393977127456723346843445866174968079087058037040712
84048740118609114467977783598029006686938976881787785946905630190260
94059957945343282346930302669644305902501597239986771421554169383555
98852914863182379144344967340878118726394964751001890413490084170616
75093668333850551032972088269550769983616369411933015213796825837188
09183365675122131849284636812555022599830041234478486259567449219461
70238065059132456108257318353800876086221028342701976982023131690176
78006675195485079921636419370285375124784014907159135459982790513399
61155179427110683113409058427288427979155484978295432353451706522326
90613949059876930021229633956877828789484406160074129456749198230505
71642377154816321380631045902916136926708342856440730447899971901781
46576347322385026725305989979599609079946920177462481771844986745565
92501783290704731194331655508075682218465717463732968849128195203174
57002440926616910874148385078411929804522981857338977648103126085903
00130241346718972667321649151113160292078173803343609024380470834040
3154190337
= 114689 × 26017793 × 63766529 × 190274191361 × 1256132134125569 × (an 1187-digit composite number)

Current status on factoring Fermat numbers is maintained by Wilfrid Keller, on this page

I have conjectured that this sequence contains all factors of sequence A094358 (which contains all N satisfying the relation 2↑↑N ≡ 1 mod N).


Some other sequences are discussed here.



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