Sequence A023394, Prime Factors of Fermat Numbers

This sequence, Sloane's A023394, gives the prime factors of Fermat numbers F_m, 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	F_m
3	F₀	1
5	F₁	1
17	F₂	1
257	F₃	1
641	F₅	5
65537	F₄	1
114689	F₁₂	7
274177	F₆
319489	F₁₁	3×13
974849	F₁₁	7×17
2424833	F₉	37
6700417	F₅	3×17449
13631489	F₁₈	13
26017793	F₁₂	397
45592577	F₁₀	11131
63766529	F₁₂	7×139
167772161	F₂₃	5
825753601	F₁₆
1214251009	F₁₅	3×193
6487031809	F₁₀
70525124609	F₁₉	33629
190274191361	F₁₂
646730219521	F₁₉
2710954639361	F₁₃

cellpadding="2" hspace="30px"> k 2ⁿ 2¹ 2² 2⁴ 2⁸ 2⁷ 2¹⁶ 2¹⁴ 3²×7×17 2⁸ 2¹³ 2¹³ 2¹⁶ 2⁷ 2²⁰ 2¹⁶ 2¹² 2¹⁶ 2²⁵ 3²×5²×7 2¹⁹ 2²¹ 3²×29×37×41 2¹⁴ 2²¹ 5×11×211153 2¹⁴ 3²×5×7×11×89 2²¹ 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 = 2^2¹²+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.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2012 Jan 20.

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