Fermat numbers F0

The smaller Fermat numbers F_m = 2^{2^m} + 1, m < 33

Catherine Cowie, updated to January 2024

Firstly, this collation of information, and in particular its organisation, owes an obvious, substantial debt of gratitude to Wilfrid Keller’s Fermat number factoring status page for so easily making the data on the known Fermat numbers not merely available, but coherent and logical to grasp and assimilate.

Secondly, a word or three on the comparative, smaller. The Fermat primes (up to F₄) might be regarded as small, but generally Fermat numbers are anything but small. Fermat numbers grow so quickly that beyond about F₃₂, the only tool currently available is essentially just trial division; the known k·2ⁿ+1 form of Fermat divisors allows confidence in searching for them. Other advanced methods of factoring that have proved invaluable on the smaller Fermat numbers, or primality tests such as Pépin’s theorem, become more difficult and probably impossible with current hardware and software somewhere beyond this point.

Correspondingly we have somewhat more information on the Fermat numbers that are within reach, so this page confines itself to re-organising the data on these. Several obvious points of difference from Keller’s page may be noticed, such as the colour coding to give a visual indicator to the character of the various Fermat numbers when gathering them together into a continuous list of each type: their standard notation factorisation, and k, n values of factors.

Citations and direct links to original sources are also provided where possible, and in providing dates, italics denote a ‘terminus ante quem’; if a manuscript, letter, or lecture were delivered on a given day, then the discovery contained within must naturally have occurred at a prior date (and most papers helpfully indicate both initial received dates as well as dates of revision). These italics have additional information available when the cursor hovers above them.

The section below on compositeness proofs of Fermat numbers and cofactors is a very short summary of a larger historical study of the subject available here.

Summary of the smaller Fermat numbers versus the larger Fermat numbers
Factorisations of smaller Fermat numbers F_m, m ≤ 33, in standard notation
Fermat primes, and prime factors k · 2ⁿ + 1 of smaller Fermat numbers
Elliptical curve method factorisation data
Chronology of proving composite nature of smaller Fermat numbers (known factors = 0) or cofactors (prime factors ≥ 1)
References

Summary of the smaller Fermat numbers versus the larger Fermat numbers

Status:    5 prime (0 ≤ m ≤ 4);

   324 composite; 322 completely or partially factored (5 ≤ m ≤ 18,233,954):

   7 completely factored (5 ≤ m ≤ 11);

   17 incompletely factored, with composite cofactor (12 ≤ m ≤ 30, m ≠ 20, 24);

   298 incompletely factored, with cofactor of unknown character (31 ≤ m ≤ 18,233,954);

   2 composite, but without known factors (m = 20, 24);

   infinitely many of unknown character, whether prime or composite (m ≥ 33).

Number of factors 5 ≤ m < 33 m ≥ 33 Totals

6 prime factors:    m = 12 6

5 prime factors:    m = 11 5

4 prime factors:    m = 13
   m = 10 8

3 prime factors:    m = 15, 19, 25
   m = 9    m = 52, 287 18

2 prime factors:    m = 16, 17, 18, 27, 30
   m = 5, 6, 7, 8    m = 36, 38, 39, 42, 77,
          147, 150, 284, 416, 417 38

1 prime factor:    m = 14, 21, 22, 23, 26, 28, 29
   m = 31, 32    m = 37, 40, 43, 48, … (280 more) 293

0 known factors:    m = 20, 24    m = 33, 34, 35, 41, 44, 45,
          46, 47, 49, 50, 51, … 0

total 368

Factorisations of smaller Fermat numbers F_m, m ≤ 33, in standard notation

F₀ = 3

F₁ = 5

F₂ = 17

F₃ = 257 Number of

F₄ = 65,537 prime factors

F₅ = 641 · 6,700,417 2

F₆ = 274,177 · 67,280,421,310,721 2

F₇ = 59,649,589,127,497,217 · 5,704,689,200,685,129,054,721 2

F₈ = 1,238,926,361,552,897 · p₆₂ 2

F₉ = 2,424,833 · 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 · p₉₉ 3

F₁₀ = 45,592,577 · 6,487,031,809 · 4,659,775,785,220,018,543,264,560,743,076,778,192,897 · p₂₅₂ 4

F₁₁ = 319,489 · 974,849 · 167,988,556,341,760,475,137 · 3,560,841,906,445,833,920,513 · p₅₆₄ 5

F₁₂ = 114,689 · 26,017,793 · 63,766,529 · 190,274,191,361 · 1,256,132,134,125,569 ·

568,630,647,535,356,955,169,033,410,940,867,804,839,360,742,060,818,433 · c_1,133 6

F₁₃ = 2,710,954,639,361 · 2,663,848,877,152,141,313 · 3,603,109,844,542,291,969 ·

319,546,020,820,551,643,220,672,513 · c_2,391 4

F₁₄ = 116,928,085,873,074,369,829,035,993,834,596,371,340,386,703,423,373,313 · c_4,880 1

F₁₅ = 1,214,251,009 · 2,327,042,503,868,417 ·

168,768,817,029,516,972,383,024,127,016,961 · c_9,808 3

F₁₆ = 825,753,601 · 188,981,757,975,021,318,420,037,633 · c_19,694 2

F₁₇ = 31,065,037,602,817 ·

7,751,061,099,802,522,589,358,967,058,392,886,922,693,580,423,169 · c_39,395 2

F₁₈ = 13,631,489 · 81,274,690,703,860,512,587,777 · c_78,884 2

F₁₉ = 70,525,124,609 · 646,730,219,521 ·

37,590,055,514,133,754,286,524,446,080,499,713 · c_157,770 3

F₂₀ = C_315,653 0

F₂₁ = 4,485,296,422,913 · c_631,294 1

F₂₂ = 64,658,705,994,591,851,009,055,774,868,504,577 · c_1,262,577 1

F₂₃ = 167,772,161 · c_2,525,215 1

F₂₄ = C_5,050,446 0

F₂₅ = 25,991,531,462,657 · 204,393,464,266,227,713 · 2,170,072,644,496,392,193 · c_10,100,842 3

F₂₆ = 76,861,124,116,481 · c_20,201,768 1

F₂₇ = 151,413,703,311,361 · 231,292,694,251,438,081 · c_40,403,531 2

F₂₈ = 1,766,730,974,551,267,606,529 · c_80,807,103 1

F₂₉ = 2,405,286,912,458,753 · c_161,614,233 1

F₃₀ = 640,126,220,763,137 · 1,095,981,164,658,689 · c_323,228,467 2

F₃₁ = 46,931,635,677,864,055,013,377 · u_646,456,971* 1

F₃₂ = 25,409,026,523,137 · u_{1,292,913,974}* 1

F₃₃ = U_{2,585,828,011} 0
* p, c, u denote cofactors that are prime, composite, or of unknown character.
Subscripts denote the number of decimal digits in the cofactor (between 60 and 2·10⁹)

Fermat primes, and prime factors k · 2ⁿ + 1 of smaller Fermat numbers

m k n Year Discoverer Method Citation

0 1 1 }
}
}
} Aug 1640 P. de Fermat [F1991]

1 1 2

2 1 4

3 1 8

4 1 16

5 5
52,347 7
7 } 7 Oct 1732 L. Euler trial
division [E1738]

6 1,071
262,814,145,745 8
8 } 1 Jan 1855 T. Clausen; F. Landry, 1880 (Primality
of k₁₂ cofactor: H. Le Lasseur, 1880) [W1993] [B1964, L1880]

7 116,503,103,764,643
11,141,971,095,088,142,685 9
9 } 13 Sep 1970 M. A. Morrison & J. Brillhart continued
fraction [MB1971, MB1975]

8 604,944,512,477
k₅₉ 11
11 } 18 Sep 1980 R. P. Brent & J. M. Pollard (Primality
of k₅₉ cofactor: H. C. Williams) Pollard rho [BP1981]

k₅₉ = 45,635,566,267,264,637,582,599,393,652,151,804,972,681,268,330,878,021,767,715

9 37 16 14 May 1903 A. E. Western trial [CW1904]

k₄₆
k₉₆ 11
11 } 15 Jun 1990 A. K. Lenstra, M. S. Manasse, & a large
team (Primality: A. M. Odłyżko) SNFS [LLMP1993]

k₄₆ = 3,640,431,067,210,880,961,102,244,011,816,628,378,312,190,597
k₉₆ = 362,128,936,829,849,024,182,024,
971,631,805,407,255,830,459,520,272,960,891,514,314,523,640,507,570,656,742,232,821,636,569,307

10 11,131 12 15 Aug 1953 J. L. Selfridge trial [S1953]

395,937 14 1962 J. Brillhart trial [B1963]

k₃₇ 12 20 Oct 1995 R. P. Brent ECM [B1999]

k₂₄₈ 13 1995 R. P. Brent

k₃₇ = 1,137,640,572,563,481,089,664,199,400,165,229,051
k₂₄₈ = 15,922,836,231,138,695,035,093,355,565,980,
212,884,107,486,675,001,451,682,970,617,160,257,863,311,947,248,971,452,664,548,043,591,906,237,
644,522,563,833,477,152,239,872,181,860,196,421,948,439,690,685,317,315,553,051,258,143,326,480,
945,577,516,888,976,026,564,843,006,895,573,500,498,133,825,643,594,092,555,886,322,403,200,003

11 39 13 1899 A. J. C. Cunningham trial [CW1904]

119 13 1899 A. J. C. Cunningham

10,253,207,784,531,279 14 17 May 1988 R. P. Brent ECM [B1989, B1996]

434,673,084,282,938,711 13 13 May 1988 R. P. Brent ECM

k₅₆₀ 13 20 Jun 1988 R. P. Brent (Primality: F. J. Morain) ECPP

k₅₆₀ = 21,174,615,134,173,285,574,982,784,529,334,689,743,337,627,529,744,150,958,
172,243,537,764,108,788,193,250,592,967,656,046,192,485,007,078,101,912,652,776,662,834,559,689,
734,635,521,223,667,093,019,353,364,100,169,585,433,799,507,320,937,371,688,159,076,970,887,037,
493,581,569,352,118,776,521,064,958,422,163,933,812,649,044,026,502,558,555,356,775,560,067,461,
648,993,426,750,049,061,580,191,794,744,396,103,493,131,476,781,686,200,989,377,719,638,682,976,
424,873,973,574,085,951,980,316,371,376,859,104,992,795,318,729,984,801,869,785,145,588,809,492,
038,969,317,284,320,651,500,418,425,949,345,494,944,448,110,057,412,733,268,967,446,592,534,704,
415,768,023,768,439,849,177,511,907,048,426,136,846,561,848,711,377,379,319,145,718,177,075,053

12 7 14 30 Nov 1877 I. M. Pervushin; 1878, F. É. A. Lucas trial [B1878, B1879]

397 16 14 May 1903 A. E. Western trial [CW1904]

973 16 14 May 1903 A. E. Western trial

11,613,415 14 20 May 1974 J. C. Hallyburton & J. Brillhart trial [HB1975]

76,668,221,077 14 25 Jul 1986 R. J. Baillie Pollard p–1

k₅₀ 15 27 Mar 2010 M. Vang, Zimmerman & Kruppa ECM [V2010]

k₅₀ = 17,353,230,210,429,594,579,133,099,699,123,162,989,482,444,520,899

13 41,365,885 16 20 May 1974 J. C. Hallyburton & J. Brillhart trial [HB1975]

20,323,554,055,421 17 5 Jan 1991 R. E. Crandall ECM

6,872,386,635,861 19 9 May 1991 R. E. Crandall ECM

609,485,665,932,753,836,099 19 16 Jun 1995 R. P. Brent ECM [BCDH2000]

14 k₄₉ 16 3 Feb 2010 T. Rajala, Woltman ECM [R2010]

k₄₉ = 1,784,180,997,819,127,957,596,374,417,642,156,545,110,881,094,717

15 579 21 1925 M. B. Kraïtchik trial [K1952]

17,753,925,353 17 4 Aug 1987 G. B. Gostin trial [G1995]

1,287,603,889,690,528,658,928,101,555 17 3 Jul 1997 R. E. Crandall & C. van Halewyn ECM [BCDH2000]

16 1,575 19 14 Aug 1953 J. L. Selfridge trial [S1953]

180,227,048,850,079,840,107 20 Dec 1996 R. E. Crandall & K. Dilcher ECM [BCDH2000]

17 59,251,857 19 May 1978 G. B. Gostin trial [G1980]

k₄₄ 19 15 Mar 2011 D. Bessell, Woltman ECM [B2011]

k₄₄ = 14,783,975,791,554,494,074,552,473,179,612,897,725,474,511

18 13 20 14 May 1903 A. E. Western (Prime: P. P. H. Seelhoff, 1886) trial [CW1904]

9,688,698,137,266,697 23 16 Apr 1999 R. E. Crandall, R. J. McIntosh, & C. Tardif ECM [BCDH2000]

19 33,629 21 16 Nov 1962 H. I. Riesel trial [R1963]

308,385 21 22 Aug 1963 C. P. Wrathall trial [W1963]

8,962,167,624,028,624,126,082,526,703 22 18 Jul 2009 D. Bessell, Woltman ECM [W2009]

21 534,689 23 22 Aug 1963 C. P. Wrathall trial [W1963]

22 3,853,959,202,444,067,657,533,632,211 24 26 Mar 2010 D. Bessell, Woltman ECM [D2010]

23 5 25 5 Feb 1878 I. M. Pervushin trial [B1879]

25 48,413 29 22 Aug 1963 C. P. Wrathall trial [W1963]

1,522,849,979 27 7 Aug 1985 G. B. Gostin trial [G1995]

16,168,301,139 27 16 Dec 1987 P. B. McLaughlin trial

26 143,165 29 22 Aug 1963 C. P. Wrathall trial [W1963]

27 141,015 30 22 Aug 1963 C. P. Wrathall trial [W1963]

430,816,215 29 21 Feb 1985 G. B. Gostin trial [G1995]

28 25,709,319,373 36 5 Feb 1997 T. Taura trial [CMP2003]

29 1,120,049 31 17 Oct 1980 G. B. Gostin & P. B. McLaughlin trial [GM1982]

30 149,041 32 22 Aug 1963 C. P. Wrathall trial [W1963]

127,589 33 22 Aug 1963 C. P. Wrathall trial [W1963]

31 5,463,561,471,303 33 12 Apr 2001 A. Kruppa & T. Forbes trial [CMP2003]

32 1,479 34 22 Aug 1963 C. P. Wrathall trial [W1963]

	m		k		n		Year		Discoverer	Method	Citation
	0	1	1	} } } }	Aug 1640	P. de Fermat		[F1991]
	1	1	2
	2	1	4
	3	1	8
	4	1	16
	5	5 52,347	7 7	}	7 Oct 1732	L. Euler	trial division	[E1738]
	6	1,071 262,814,145,745	8 8	}	1 Jan 1855	T. Clausen; F. Landry, 1880 (Primality of k₁₂ cofactor: H. Le Lasseur, 1880)	[W1993]	[B1964, L1880]
	7	116,503,103,764,643 11,141,971,095,088,142,685	9 9	}	13 Sep 1970	M. A. Morrison & J. Brillhart	continued fraction	[MB1971, MB1975]
	8	604,944,512,477 k₅₉	11 11	}	18 Sep 1980	R. P. Brent & J. M. Pollard (Primality of k₅₉ cofactor: H. C. Williams)	Pollard rho	[BP1981]
k₅₉ = 45,635,566,267,264,637,582,599,393,652,151,804,972,681,268,330,878,021,767,715
	9		37		16		14 May 1903		A. E. Western	trial	[CW1904]
	k₄₆ k₉₆	11 11	}	15 Jun 1990	A. K. Lenstra, M. S. Manasse, & a large team (Primality: A. M. Odłyżko)	SNFS	[LLMP1993]
k₄₆ = 3,640,431,067,210,880,961,102,244,011,816,628,378,312,190,597 k₉₆ = 362,128,936,829,849,024,182,024, 971,631,805,407,255,830,459,520,272,960,891,514,314,523,640,507,570,656,742,232,821,636,569,307
	10		11,131		12		15 Aug 1953		J. L. Selfridge	trial	[S1953]
	395,937	14	1962	J. Brillhart	trial	[B1963]
	k₃₇	12	20 Oct 1995	R. P. Brent	ECM	[B1999]
	k₂₄₈	13	1995	R. P. Brent
k₃₇ = 1,137,640,572,563,481,089,664,199,400,165,229,051 k₂₄₈ = 15,922,836,231,138,695,035,093,355,565,980, 212,884,107,486,675,001,451,682,970,617,160,257,863,311,947,248,971,452,664,548,043,591,906,237, 644,522,563,833,477,152,239,872,181,860,196,421,948,439,690,685,317,315,553,051,258,143,326,480, 945,577,516,888,976,026,564,843,006,895,573,500,498,133,825,643,594,092,555,886,322,403,200,003
	11		39		13		1899		A. J. C. Cunningham	trial	[CW1904]
	119	13	1899	A. J. C. Cunningham
	10,253,207,784,531,279	14	17 May 1988	R. P. Brent	ECM	[B1989, B1996]
	434,673,084,282,938,711	13	13 May 1988	R. P. Brent	ECM
	k₅₆₀	13	20 Jun 1988	R. P. Brent (Primality: F. J. Morain)	ECPP
k₅₆₀ = 21,174,615,134,173,285,574,982,784,529,334,689,743,337,627,529,744,150,958, 172,243,537,764,108,788,193,250,592,967,656,046,192,485,007,078,101,912,652,776,662,834,559,689, 734,635,521,223,667,093,019,353,364,100,169,585,433,799,507,320,937,371,688,159,076,970,887,037, 493,581,569,352,118,776,521,064,958,422,163,933,812,649,044,026,502,558,555,356,775,560,067,461, 648,993,426,750,049,061,580,191,794,744,396,103,493,131,476,781,686,200,989,377,719,638,682,976, 424,873,973,574,085,951,980,316,371,376,859,104,992,795,318,729,984,801,869,785,145,588,809,492, 038,969,317,284,320,651,500,418,425,949,345,494,944,448,110,057,412,733,268,967,446,592,534,704, 415,768,023,768,439,849,177,511,907,048,426,136,846,561,848,711,377,379,319,145,718,177,075,053
	12		7		14		30 Nov 1877		I. M. Pervushin; 1878, F. É. A. Lucas	trial	[B1878, B1879]
	397	16	14 May 1903	A. E. Western	trial	[CW1904]
	973	16	14 May 1903	A. E. Western	trial
	11,613,415	14	20 May 1974	J. C. Hallyburton & J. Brillhart	trial	[HB1975]
	76,668,221,077	14	25 Jul 1986	R. J. Baillie	Pollard p–1
	k₅₀	15	27 Mar 2010	M. Vang, Zimmerman & Kruppa	ECM	[V2010]
k₅₀ = 17,353,230,210,429,594,579,133,099,699,123,162,989,482,444,520,899
	13		41,365,885		16		20 May 1974		J. C. Hallyburton & J. Brillhart	trial	[HB1975]
	20,323,554,055,421	17	5 Jan 1991	R. E. Crandall	ECM
	6,872,386,635,861	19	9 May 1991	R. E. Crandall	ECM
	609,485,665,932,753,836,099	19	16 Jun 1995	R. P. Brent	ECM	[BCDH2000]
	14	k₄₉	16	3 Feb 2010	T. Rajala, Woltman	ECM	[R2010]
k₄₉ = 1,784,180,997,819,127,957,596,374,417,642,156,545,110,881,094,717
	15		579		21		1925		M. B. Kraïtchik	trial	[K1952]
	17,753,925,353	17	4 Aug 1987	G. B. Gostin	trial	[G1995]
	1,287,603,889,690,528,658,928,101,555	17	3 Jul 1997	R. E. Crandall & C. van Halewyn	ECM	[BCDH2000]
	16	1,575	19	14 Aug 1953	J. L. Selfridge	trial	[S1953]
	180,227,048,850,079,840,107	20	Dec 1996	R. E. Crandall & K. Dilcher	ECM	[BCDH2000]
	17	59,251,857	19	May 1978	G. B. Gostin	trial	[G1980]
	k₄₄	19	15 Mar 2011	D. Bessell, Woltman	ECM	[B2011]
k₄₄ = 14,783,975,791,554,494,074,552,473,179,612,897,725,474,511
	18		13		20		14 May 1903		A. E. Western (Prime: P. P. H. Seelhoff, 1886)	trial	[CW1904]
	9,688,698,137,266,697	23	16 Apr 1999	R. E. Crandall, R. J. McIntosh, & C. Tardif	ECM	[BCDH2000]
	19	33,629	21	16 Nov 1962	H. I. Riesel	trial	[R1963]
	308,385	21	22 Aug 1963	C. P. Wrathall	trial	[W1963]
	8,962,167,624,028,624,126,082,526,703	22	18 Jul 2009	D. Bessell, Woltman	ECM	[W2009]
	21	534,689	23	22 Aug 1963	C. P. Wrathall	trial	[W1963]
	22	3,853,959,202,444,067,657,533,632,211	24	26 Mar 2010	D. Bessell, Woltman	ECM	[D2010]
	23	5	25	5 Feb 1878	I. M. Pervushin	trial	[B1879]
	25	48,413	29	22 Aug 1963	C. P. Wrathall	trial	[W1963]
	1,522,849,979	27	7 Aug 1985	G. B. Gostin	trial	[G1995]
	16,168,301,139	27	16 Dec 1987	P. B. McLaughlin	trial
	26	143,165	29	22 Aug 1963	C. P. Wrathall	trial	[W1963]
	27	141,015	30	22 Aug 1963	C. P. Wrathall	trial	[W1963]
	430,816,215	29	21 Feb 1985	G. B. Gostin	trial	[G1995]
	28	25,709,319,373	36	5 Feb 1997	T. Taura	trial	[CMP2003]
	29	1,120,049	31	17 Oct 1980	G. B. Gostin & P. B. McLaughlin	trial	[GM1982]
	30	149,041	32	22 Aug 1963	C. P. Wrathall	trial	[W1963]
	127,589	33	22 Aug 1963	C. P. Wrathall	trial	[W1963]
	31	5,463,561,471,303	33	12 Apr 2001	A. Kruppa & T. Forbes	trial	[CMP2003]
	32	1,479	34	22 Aug 1963	C. P. Wrathall	trial	[W1963]

Elliptical curve method factorisation data

m Factor B₁ B₂ σ g

10 k₃₇·2¹²+1 2,000,000 14,152,267 4,659,775,785,220,018,543,260,885,870,817,648,693,860
= 2² · 3² · 5 · 149 · 163 · 197 · 7,187 · 18,311 · 123,677 · 226,133 · 314,263 · 4,677,583

11 k₁₇·2¹⁴+1 16,000 not given

k₁₈·2¹³+1 16,000 not given

12 k₅₀·2¹⁵+1 43,000,000 199,103,726,650 1,428,526,317 568,630,647,535,356,955,169,033,412,162,316,189,313,022,429,279,110,256
= 2⁴ · 3² · 7 · 17 · 293 · 349 · 8,821 · 23,753 · 65,123 · 2,413,097 · 9,027,881 · 23,759,413 · 45,947,380,867

13 k₁₄·2¹⁷+1

k₁₃·2¹⁹+1

k₂₁·2¹⁹+1 100,000
100,000
500,000 B₃ = 485,301
3,542,000
stage 2 un-needed see [BCDH2000]
4,009,189
8,020,345 319,546,020,820,518,229,496,965,479 = 3² · 7² · 13 · 31 · 3,803 · 6,037 · 9,887 · 28,859 · 274,471
317,976,969,488,002,049,294,329,728 = 2⁷ · 3 · 127 · 3,083 · 3,539 · 9,649 · 18,239 · 3,395,653
319,546,020,820,551,567,984,515,352 = 2³ · 3 · 17 · 23 · 41 · 113 · 271 · 3,037 · 10,687 · 12,251 · 68,209

14 k₄₉·2¹⁶+1 110,000,000 11,000,000,000 8,585,974,330,888,598 116,928,085,873,074,369,829,035,993,372,903,082,226,176,977,098,089,788
= 2² · 3 · 53 · 107 · 3,433 · 37,087 · 110,323 · 128,321 · 1,738,307 · 9,338,881 · 74,968,979 · 783,277,631

15 k₂₈·2¹⁷+1 10,000,000 500,000,000 253,301,772 168,768,817,029,516,992,836,491,975,362,208
= 2⁵ · 3 · 4,889 · 5,701 · 9,883 · 11,777 · 5,909,317 · 91,704,181

16 k₂₁·2²⁰+1 400,000
200,000 20,000,000
10,000,000 1,944,934,539
125,546,653 188,981,757,975,005,913,471,235,500 = 2² · 3 · 5³ · 7 · 13 · 19 · 83 · 113 · 2,027 · 386,677 · 9,912,313
188,981,757,975,004,093,943,814,852 = 2² · 3² · 7² · 109 · 761 · 2,053 · 20,297 · 101,483 · 305,419

17 k₄₄·2¹⁹+1 44,000,000 4,400,000,000 10,717,701,036,773 7,705,935,574,284,557,194,893,456,578,699,598,773,615,181,874,764
= 2² · 3 · 541 · 2,713 · 5,153 · 23,773 · 152,363 · 239,387 · 19,359,383 · 22,095,751 · 230,254,627

18 k₁₆·2²³+1 100,000 4,000,000 731,185,968 81,274,690,704,014,912,758,776 = 2³ · 3 · 59 · 367 · 389 · 3,613 · 50,101 · 2,221,069

19 k₂₈·2²²+1 3,000,000 300,000,000 7,121,198,363,696,307 37,590,055,514,133,754,043,447,773,966,186,464
= 2⁵ · 3 · 11 · 181 · 263 · 4,217 · 38,867 · 244,451 · 1,779,623 · 10,487,459

22 k₂₈·2²⁴+1 1,000,000 100,000,000 8,776,953,345,765,668 64,658,705,994,591,850,983,958,649,757,139,080
= 2³ · 3 · 5 · 11 · 23 · 193 · 4,451 · 862,231 · 886,069 · 898,769 · 3,610,351

	m		Factor	B₁		B₂		σ		g
	10	k₃₇·2¹²+1	2,000,000		14,152,267	4,659,775,785,220,018,543,260,885,870,817,648,693,860 = 2² · 3² · 5 · 149 · 163 · 197 · 7,187 · 18,311 · 123,677 · 226,133 · 314,263 · 4,677,583
	11	k₁₇·2¹⁴+1	16,000		not given
k₁₈·2¹³+1	16,000		not given
	12	k₅₀·2¹⁵+1	43,000,000	199,103,726,650	1,428,526,317	568,630,647,535,356,955,169,033,412,162,316,189,313,022,429,279,110,256 = 2⁴ · 3² · 7 · 17 · 293 · 349 · 8,821 · 23,753 · 65,123 · 2,413,097 · 9,027,881 · 23,759,413 · 45,947,380,867
	13	k₁₄·2¹⁷+1
k₁₃·2¹⁹+1
k₂₁·2¹⁹+1	100,000 100,000 500,000	B₃ = 485,301 3,542,000 stage 2 un-needed	see [BCDH2000] 4,009,189 8,020,345	319,546,020,820,518,229,496,965,479 = 3² · 7² · 13 · 31 · 3,803 · 6,037 · 9,887 · 28,859 · 274,471 317,976,969,488,002,049,294,329,728 = 2⁷ · 3 · 127 · 3,083 · 3,539 · 9,649 · 18,239 · 3,395,653 319,546,020,820,551,567,984,515,352 = 2³ · 3 · 17 · 23 · 41 · 113 · 271 · 3,037 · 10,687 · 12,251 · 68,209
	14	k₄₉·2¹⁶+1	110,000,000	11,000,000,000	8,585,974,330,888,598	116,928,085,873,074,369,829,035,993,372,903,082,226,176,977,098,089,788 = 2² · 3 · 53 · 107 · 3,433 · 37,087 · 110,323 · 128,321 · 1,738,307 · 9,338,881 · 74,968,979 · 783,277,631
	15	k₂₈·2¹⁷+1	10,000,000	500,000,000	253,301,772	168,768,817,029,516,992,836,491,975,362,208 = 2⁵ · 3 · 4,889 · 5,701 · 9,883 · 11,777 · 5,909,317 · 91,704,181
	16	k₂₁·2²⁰+1	400,000 200,000	20,000,000 10,000,000	1,944,934,539 125,546,653	188,981,757,975,005,913,471,235,500 = 2² · 3 · 5³ · 7 · 13 · 19 · 83 · 113 · 2,027 · 386,677 · 9,912,313 188,981,757,975,004,093,943,814,852 = 2² · 3² · 7² · 109 · 761 · 2,053 · 20,297 · 101,483 · 305,419
	17	k₄₄·2¹⁹+1	44,000,000	4,400,000,000	10,717,701,036,773	7,705,935,574,284,557,194,893,456,578,699,598,773,615,181,874,764 = 2² · 3 · 541 · 2,713 · 5,153 · 23,773 · 152,363 · 239,387 · 19,359,383 · 22,095,751 · 230,254,627
	18	k₁₆·2²³+1	100,000	4,000,000	731,185,968	81,274,690,704,014,912,758,776 = 2³ · 3 · 59 · 367 · 389 · 3,613 · 50,101 · 2,221,069
	19	k₂₈·2²²+1	3,000,000	300,000,000	7,121,198,363,696,307	37,590,055,514,133,754,043,447,773,966,186,464 = 2⁵ · 3 · 11 · 181 · 263 · 4,217 · 38,867 · 244,451 · 1,779,623 · 10,487,459
	22	k₂₈·2²⁴+1	1,000,000	100,000,000	8,776,953,345,765,668	64,658,705,994,591,850,983,958,649,757,139,080 = 2³ · 3 · 5 · 11 · 23 · 193 · 4,451 · 862,231 · 886,069 · 898,769 · 3,610,351

Chronology of proving composite nature of Fermat numbers (known factors = 0)
or cofactors (prime factors ≥ 1)

m    Year    known/prime
factors    Digits    Earliest prover(s)    Method(s)    Status    Citation(s)

5
6 1877 2
0 (2) 10
20 } F. É. A. Lucas Pell
sequence Clausen’s F₆ factoring was unpublished;
subsequently refactored [L1877a, L1877b]

7 1905 0 39 J. C. Morehead; A. E. Western Pépin subsequently factored [M1905, W1905]

8 1909 0 78 J. C. Morehead & A. E. Western Pépin subsequently factored [MW1909]

9 1967 1 148 J. Brillhart subsequently factored [HB1975]

10 1952 0 309 R. M. Robinson Pépin subsequently factored [R1954]

1967 2 291 J. Brillhart subsequently factored [HB1975]

11 1979 2 606 S. S. Wagstaff subsequently factored [G1980]

1988 3 584 R. P. Brent subsequently factored [B1989, B1996]

12 1979 4 1,202 S. S. Wagstaff new factor discovered [G1980]

1986 5 1,187 R. J. Baillie new factor discovered [W1987]

2010 6 1,133 M. Vang; S. Batalov yet to be factored [V2010]

13 1960 0 2,467 G. A. Paxson Pépin new factor discovered [P1961]

1979 1 2,454 S. S. Wagstaff new factor discovered [G1980]

1991 2 2,436 A. K. Lenstra, W. Keller new factor discovered

1991 3 2,417 R. E. Crandall new factor discovered

1995 4 2,391 R. P. Brent yet to be factored [BCDH2000]

14 1961 0 4,933 A. Hurwitz & J. L. Selfridge Pépin new factor discovered [SH1964]

2010 1 4,880 W. B. Lipp; R. D. Silverman; T. Rajala; P. Moore Suyama yet to be factored [R2010]

15 1984 1 9,856 H. Suyama Suyama new factor discovered [S1984]

1987 2 9,840 H. Suyama; R. J. Baillie Suyama new factor discovered [S1987, W1987]

1997 3 9,808 R. P. Brent & R. E. Crandall yet to be factored [BCDH2000]

16 1987 1 19,720 R. J. Baillie new factor discovered [W1987]

1996 2 19,694 R. P. Brent & R. E. Crandall yet to be factored [BCDH2000]

17 1987 1 39,444 R. J. Baillie new factor discovered

2011 2 39,395 D. Chia; A. T. Höglund; T. Sorbera yet to be factored [B2011]

18 1990 1 78,907 D. V. & G. V. Chudnovsky yet to be factored [W1990]

1999 2 78,884 R. E. Crandall yet to be factored

19 1993 2 157,804 R. E. Crandall, J. Doenias, C. Norrie & J. Young Suyama new factor discovered [CDNY1995]

2009 3 157,770 J. R. K.; A. Kruppa yet to be factored [W2009]

20 1987 0 315,653 J. Young & D. A. Buell Pépin no factors known [YB1988]

21 1993 1 631,294 R. E. Crandall, J. Doenias, C. Norrie & J. Young Suyama yet to be factored [CDNY1995]

22 1993 0 1,262,612 { R. E. Crandall, J. Doenias, C. Norrie & J. Young;
V. Trevisan & J. B. Carvalho Pépin new factor discovered [CDNY1995, TC1995]

2010 1 1,262,577 D. Domanov; S. Yamada Suyama yet to be factored [D2010, Y2010]

23 2000 1 2,525,215 R. E. Crandall, E. W. Mayer & J. S. Papadopoulos Suyama yet to be factored [CMP2003]

24 1999 0 5,050,446 R. E. Crandall, E. W. Mayer & J. S. Papadopoulos Pépin no factors known [CMP2003]

25 2009 3 10,100,842 S. Yamada; A. T. Höglund Suyama; Fermat-PRP yet to be factored [H2009a, Y2009]

26 2009 1 20,201,768 A. T. Höglund Fermat-PRP yet to be factored [H2009b]

27 2010 2 40,403,531 A. T. Höglund Fermat-PRP yet to be factored [H2010]

28 2022 1 80,807,103 E. W. Mayer Suyama yet to be factored [M2022]

29 2022 1 161,614,233 E. W. Mayer Suyama yet to be factored [M2022]

  30 2022 2 323,228,467 E. W. Mayer Suyama yet to be factored [M2022]

m		Year		known/prime factors		Digits		Earliest prover(s)		Method(s)		Status		Citation(s)
5 6	1877	2 0 (2)	10 20	}	F. É. A. Lucas	Pell sequence	Clausen’s F₆ factoring was unpublished; subsequently refactored	[L1877a, L1877b]
7	1905	0	39		J. C. Morehead; A. E. Western	Pépin	subsequently factored	[M1905, W1905]
8	1909	0	78	J. C. Morehead & A. E. Western	Pépin	subsequently factored	[MW1909]
9	1967	1	148	J. Brillhart		subsequently factored	[HB1975]
10	1952	0	309	R. M. Robinson	Pépin	subsequently factored	[R1954]
1967	2	291	J. Brillhart		subsequently factored	[HB1975]
11	1979	2	606	S. S. Wagstaff		subsequently factored	[G1980]
1988	3	584	R. P. Brent		subsequently factored	[B1989, B1996]
12	1979	4	1,202	S. S. Wagstaff		new factor discovered	[G1980]
1986	5	1,187	R. J. Baillie		new factor discovered	[W1987]
2010	6	1,133	M. Vang; S. Batalov		yet to be factored	[V2010]
13	1960	0	2,467	G. A. Paxson	Pépin	new factor discovered	[P1961]
1979	1	2,454	S. S. Wagstaff		new factor discovered	[G1980]
1991	2	2,436	A. K. Lenstra, W. Keller		new factor discovered
1991	3	2,417	R. E. Crandall		new factor discovered
1995	4	2,391	R. P. Brent		yet to be factored	[BCDH2000]
14	1961	0	4,933	A. Hurwitz & J. L. Selfridge	Pépin	new factor discovered	[SH1964]
2010	1	4,880	W. B. Lipp; R. D. Silverman; T. Rajala; P. Moore	Suyama	yet to be factored	[R2010]
15	1984	1	9,856	H. Suyama	Suyama	new factor discovered	[S1984]
1987	2	9,840	H. Suyama; R. J. Baillie	Suyama	new factor discovered	[S1987, W1987]
1997	3	9,808	R. P. Brent & R. E. Crandall		yet to be factored	[BCDH2000]
16	1987	1	19,720	R. J. Baillie		new factor discovered	[W1987]
1996	2	19,694	R. P. Brent & R. E. Crandall		yet to be factored	[BCDH2000]
17	1987	1	39,444	R. J. Baillie		new factor discovered
2011	2	39,395	D. Chia; A. T. Höglund; T. Sorbera		yet to be factored	[B2011]
18	1990	1	78,907	D. V. & G. V. Chudnovsky		yet to be factored	[W1990]
1999	2	78,884	R. E. Crandall		yet to be factored
19	1993	2	157,804	R. E. Crandall, J. Doenias, C. Norrie & J. Young	Suyama	new factor discovered	[CDNY1995]
2009	3	157,770	J. R. K.; A. Kruppa		yet to be factored	[W2009]
20	1987	0	315,653	J. Young & D. A. Buell	Pépin	no factors known	[YB1988]
21	1993	1	631,294	R. E. Crandall, J. Doenias, C. Norrie & J. Young	Suyama	yet to be factored	[CDNY1995]
22	1993	0	1,262,612	{	R. E. Crandall, J. Doenias, C. Norrie & J. Young; V. Trevisan & J. B. Carvalho	Pépin	new factor discovered	[CDNY1995, TC1995]
2010	1	1,262,577		D. Domanov; S. Yamada	Suyama	yet to be factored	[D2010, Y2010]
23	2000	1	2,525,215	R. E. Crandall, E. W. Mayer & J. S. Papadopoulos	Suyama	yet to be factored	[CMP2003]
24	1999	0	5,050,446	R. E. Crandall, E. W. Mayer & J. S. Papadopoulos	Pépin	no factors known	[CMP2003]
25	2009	3	10,100,842	S. Yamada; A. T. Höglund	Suyama; Fermat-PRP	yet to be factored	[H2009a, Y2009]
26	2009	1	20,201,768	A. T. Höglund	Fermat-PRP	yet to be factored	[H2009b]
27	2010	2	40,403,531	A. T. Höglund	Fermat-PRP	yet to be factored	[H2010]
28	2022	1	80,807,103	E. W. Mayer	Suyama	yet to be factored	[M2022]
29	2022	1	161,614,233	E. W. Mayer	Suyama	yet to be factored	[M2022]
30	2022	2	323,228,467	E. W. Mayer	Suyama	yet to be factored	[M2022]

References, sorted by primary author

[B2011] D. Bessell, New factor for F₁₇, mersenneforum.org thread 15358 page 1, post 1 (2011); cofactor compositeness tests appear in replies at page 1, post 6 (D. Chia) and post 9 (T. Sorbera)

[B1964] K.-R. Biermann, Thomas Clausen, Mathematiker und Astronom, J. für die reine und angewandte Mathematik 216 (1964), 159–198. DOI 10.1515/crll.1964.216.159 (see page 185, relating a letter by Clausen to C. F. Gauss, dated 1 January 1855)

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[B1989] R. P. Brent, Factorization of the eleventh Fermat number (preliminary report), Amer. Math. Soc. Abstracts 10 (1989), 89T-11-73

[B1996] R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS-96-02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1996, 25 pp.

[B1999] R. P. Brent, Factorization of the tenth Fermat number, Math. Comp. 68 (1999), 429–451. MR 1489968, DOI 10.1090/S0025-5718-99-00992-8

[BCDH2000] R. P. Brent, R. E. Crandall, K. Dilcher, C. van Halewyn, Three new factors of Fermat numbers, Math. Comp. 69 (2000), 1297–1304. MR 1697645, DOI 10.1090/S0025-5718-00-01207-2

[BP1981] R. P. Brent, J. M. Pollard, Factorization of the eighth Fermat number, Math. Comp. 36 (1981), 627–630. MR 606520, DOI 10.1090/S0025-5718-1981-0606520-5

[B1963] J. Brillhart, Some miscellaneous factorizations, Math. Comp. 17 (1963), 447–450. DOI 10.1090/S0025-5718-63-99176-2

[CDNY1995] R. E. Crandall, J. Doenias, C. Norrie, J. Young, The twenty-second Fermat number is composite, Math. Comp. 64 (1995), 863–868. MR 1277765, DOI 10.1090/S0025-5718-1995-1277765-9

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[CW1904] A. J. C. Cunningham, A. E. Western, On Fermat’s numbers, Proc. Lond. Math. Soc. (2) 1 (1904), 175. DOI 10.1112/plms/s2-1.1.175

[D2010] D. Domanov, F₂₂ factored, mersenneforum.org thread 13209; announcement, page 1, post 1; cofactor compositeness test, page 2, post 12 (2010)

[E1738] L. Euler, Observationes de theoremate quodam Fermatiano, aliisque ad numeros primos spectantibus, Commentarii academiae scientiarum Petropolitanae 6 (1738), 103–107; translated from Latin by D. Zhao (2006), J. Bell (2008)

[F1991] C. R. Fletcher, A reconstruction of the Frenicle–Fermat correspondence of 1640, Historia Mathematica 18 (1991), 344–351

[G1980] G. B. Gostin, A factor of F₁₇, Math. Comp. 35 (1980), 975–976. MR 572869, DOI 10.1090/S0025-5718-1980-0572869-7

[G1995] G. B. Gostin, New factors of Fermat numbers, Math. Comp. 64 (1995), 393–395. MR 1265015, DOI 10.1090/S0025-5718-1995-1265015-9

[GM1982] G. B. Gostin, P. B. McLaughlin, Jr., Six new factors of Fermat numbers, Math. Comp. 38 (1982), 645–649. MR 645680, DOI 10.1090/S0025-5718-1982-0645680-8

[HB1975] J. C. Hallyburton, Jr., J. Brillhart, Two new factors of Fermat numbers, Math. Comp. 29 (1975), 109–112. MR 369225, DOI 10.1090/S0025-5718-1975-0369225-1. Corrigenda ibid. 30 (1976), 198.

[H2009/10] A. T. Höglund, mersenneforum.org thread 12168; compositeness test of c_10,100,842 | F₂₅ cofactor, page 5, post 51 (2009); compositeness test of c_20,201,768 | F₂₆ cofactor, page 6, post 62 (2009); compositeness test of c_40,403,531 | F₂₇ cofactor, post 64 (2010)

[K1952] M. B. Kraïtchik, On the factorization of 2ⁿ ± 1, Scripta Math. 18 (1952), 39–52.

[L1880] F. Landry, Sur la décomposition du nombre 2⁶⁴ + 1, C. R. Acad. Sci. Paris 91 (1880), 138 (see also [W1993])

[LLMP1993] A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319–349. MR 1182953, DOI 10.1090/S0025-5718-1993-1182953-4, Addendum ibid. 64 (1995), 1357

[L1877a] F. É. A. Lucas, Sur la division de la circonférence en parties égales, Comptes rendus hebdomadaires des séances de l’Académie des Sciences de Paris 85 (1877), 136–139

[L1877b] F. É. A. Lucas, Considérations nouvelles sur la théorie des nombres premiers et sur la division géométrique de la circonférence en parties égales, Association française pour l’avancement des sciences, Comptes-rendus de la 6^e session, Le Havre (1877), 159–167

[M2022] E. W. Mayer, Pépin tests of Fermat numbers beyond F₂₄, mersenneforum.org thread 18748 (2013–22); results of Suyama tests for F₂₅ to F₃₀, “F25–F30 cofactor status, Part 2”, page 8, post 83 (2022)

[M1905] J. C. Morehead, Note on Fermat’s numbers, Bull. Amer. Math. Soc. 11 (1905), 543–545. MR 1558255, DOI 10.1090/S0002-9904-1905-01255-6

[MW1909] J. C. Morehead, A. E. Western, Note on Fermat’s numbers, Bull. Amer. Math. Soc. 16 (1909), 1–6. MR 1558828, DOI 10.1090/S0002-9904-1909-01841-5

[MB1971] M. A. Morrison, J. Brillhart, The factorization of F₇, Bull. Amer. Math. Soc. 77 (1971), 264. MR 268113 10.1090/S0002-9904-1971-12711-8

[MB1975] M. A. Morrison, J. Brillhart, A method of factoring and the factorization of F₇, Math. Comp. 29 (1975), 183–205. MR 371800, DOI 10.1090/S0025-5718-1975-0371800-5

[P1961] G. A. Paxson, The compositeness of the thirteenth Fermat number, Math. Comp. 15 (1961), 420. MR 124264, DOI 10.1090/S0025-5718-1961-0124264-0

[P1877] T. Pépin, Sur la formule 2^2ⁿ + 1, Comptes rendus hebdomadaires des séances de l’Académie des Sciences de Paris 85 (1877), 329–331

[R2010] T. Rajala, GIMPS’ second Fermat factor, mersenneforum.org thread 13051, page 1, post 1; page 2, posts 13, 18, and 19 (2010); cofactor compositeness tests, page 1, post 5 (W. B. Lipp); page 2, post 12 (R. D. Silverman); page 3, post 24 (P. Moore)

[R1963] H. I. Riesel, A factor of the Fermat number F₁₉, Math. Comp. 17 (1963), 458. DOI 10.1090/S0025-5718-63-99175-0

[R1954] R. M. Robinson, Mersenne and Fermat numbers, Proc. Amer. Math. Soc. 5 (1954), 842–846. MR 64787, DOI 10.1090/S0002-9939-1954-0064787-4

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[SH1964] J. L. Selfridge, A. Hurwitz, Fermat numbers and Mersenne numbers, Math. Comp. 18 (1964), 146–148. MR 159775, DOI 10.1090/S0025-5718-1964-0159775-8

[S1984] H. Suyama, The cofactor of F₁₅ is composite, Abstracts Amer. Math. Soc. 5 (1984), 271–272

[S1987] H. Suyama, The new cofactor of F₁₅ is still composite, written communication (23 October 1987)

[TC1995] V. Trevisan, J. B. Carvalho, The composite character of the twenty-second Fermat number, J. Supercomputing 9 (1995), 179–182. DOI 10.1007/BF01245403

[V2010] M. Vang, F₁₂ has a factor, mersenneforum.org thread 13215, page 1, post 1 and post 9 (2010); compositeness tests, page 1, post 2 (S. Batalov) and post 4

[W1987] S. S. Wagstaff, Jr., Update #5 to Factorizations of bⁿ ± 1 (1987), 6 pp.

[W1990] S. S. Wagstaff, Jr., Cover letter for Page 61 (1 October 1990)

[W1905] A. E. Western, Note on Fermat’s numbers and the converse of Fermat’s theorem, Proc. Lond. Math. Soc (2) 3 (1905), xxi–xxii. DOI 10.1112/plms/s2-3.1.1-v

[W1993] H. C. Williams, How was F₆ factored? Math. Comp. 61 (1993), 463–474. MR 1182248, DOI 10.1090/S0025-5718-1993-1182248-9

[W2009] G. F. Woltman, GIMPS’ first Fermat factor, mersenneforum.org thread 12168, page 1, post 1 (2009), announcement of new factor of F₁₉; with replies by: ‘J. R. K.’, compositeness test of c_157,770 | F₁₉ cofactor, page 1, post 5 (2009); A. Kruppa, compositeness test of F₁₉ cofactor, post 6 (2009); ‘E. P.’, compositeness test of F₁₉ cofactor, post 9 (2009)

[W1964] C. P. Wrathall, New factors of Fermat numbers, Math. Comp. 18 (1964), 324–325. MR 163868, DOI 10.1090/S0025-5718-1964-0163868-9

[Y2009/10] S. Yamada, mersenneforum.org thread 12168; Suyama test of c_10,100,842 | F₂₅ cofactor, page 4, post 40 (2009); Suyama test of c_1,262,577 | F₂₂ cofactor, page 7, post 70 (2010)

[YB1988] J. Young, D. A. Buell, The twentieth Fermat number is composite, Math. Comp. 50 (1988), 261–263. MR 917833, DOI 10.1090/S0025-5718-1988-0917833-8

Status:	5 prime (0 ≤ m ≤ 4);
	324 composite; 322 completely or partially factored (5 ≤ m ≤ 18,233,954):
	7 completely factored (5 ≤ m ≤ 11);
	17 incompletely factored, with composite cofactor (12 ≤ m ≤ 30, m ≠ 20, 24);
	298 incompletely factored, with cofactor of unknown character (31 ≤ m ≤ 18,233,954);

	2 composite, but without known factors (m = 20, 24);
	infinitely many of unknown character, whether prime or composite (m ≥ 33).

Number of factors	5 ≤ m < 33	m ≥ 33	Totals
6 prime factors:	m = 12		6
5 prime factors:	m = 11		5
4 prime factors:	m = 13 m = 10		8
3 prime factors:	m = 15, 19, 25 m = 9	m = 52, 287	18
2 prime factors:	m = 16, 17, 18, 27, 30 m = 5, 6, 7, 8	m = 36, 38, 39, 42, 77, 147, 150, 284, 416, 417	38
1 prime factor:	m = 14, 21, 22, 23, 26, 28, 29 m = 31, 32	m = 37, 40, 43, 48, … (280 more)	293
0 known factors:	m = 20, 24	m = 33, 34, 35, 41, 44, 45, 46, 47, 49, 50, 51, …	0
			total 368

F₀ = 3
F₁ = 5
F₂ = 17
F₃ = 257	Number of
F₄ = 65,537	prime factors
F₅ = 641 · 6,700,417		2
F₆ = 274,177 · 67,280,421,310,721		2
F₇ = 59,649,589,127,497,217 · 5,704,689,200,685,129,054,721		2
F₈ = 1,238,926,361,552,897 · p₆₂		2
F₉ = 2,424,833 · 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 · p₉₉		3
F₁₀ = 45,592,577 · 6,487,031,809 · 4,659,775,785,220,018,543,264,560,743,076,778,192,897 · p₂₅₂		4
F₁₁ = 319,489 · 974,849 · 167,988,556,341,760,475,137 · 3,560,841,906,445,833,920,513 · p₅₆₄		5
F₁₂ = 114,689 · 26,017,793 · 63,766,529 · 190,274,191,361 · 1,256,132,134,125,569 ·
568,630,647,535,356,955,169,033,410,940,867,804,839,360,742,060,818,433 · c_1,133		6
F₁₃ = 2,710,954,639,361 · 2,663,848,877,152,141,313 · 3,603,109,844,542,291,969 ·
319,546,020,820,551,643,220,672,513 · c_2,391		4
F₁₄ = 116,928,085,873,074,369,829,035,993,834,596,371,340,386,703,423,373,313 · c_4,880		1
F₁₅ = 1,214,251,009 · 2,327,042,503,868,417 ·
168,768,817,029,516,972,383,024,127,016,961 · c_9,808		3
F₁₆ = 825,753,601 · 188,981,757,975,021,318,420,037,633 · c_19,694		2
F₁₇ = 31,065,037,602,817 ·
7,751,061,099,802,522,589,358,967,058,392,886,922,693,580,423,169 · c_39,395		2
F₁₈ = 13,631,489 · 81,274,690,703,860,512,587,777 · c_78,884		2
F₁₉ = 70,525,124,609 · 646,730,219,521 ·
37,590,055,514,133,754,286,524,446,080,499,713 · c_157,770		3
F₂₀ = C_315,653		0
F₂₁ = 4,485,296,422,913 · c_631,294		1
F₂₂ = 64,658,705,994,591,851,009,055,774,868,504,577 · c_1,262,577		1
F₂₃ = 167,772,161 · c_2,525,215		1
F₂₄ = C_5,050,446		0
F₂₅ = 25,991,531,462,657 · 204,393,464,266,227,713 · 2,170,072,644,496,392,193 · c_10,100,842		3
F₂₆ = 76,861,124,116,481 · c_20,201,768		1
F₂₇ = 151,413,703,311,361 · 231,292,694,251,438,081 · c_40,403,531		2
F₂₈ = 1,766,730,974,551,267,606,529 · c_80,807,103		1
F₂₉ = 2,405,286,912,458,753 · c_161,614,233		1
F₃₀ = 640,126,220,763,137 · 1,095,981,164,658,689 · c_323,228,467		2
F₃₁ = 46,931,635,677,864,055,013,377 · u_646,456,971*		1
F₃₂ = 25,409,026,523,137 · u_{1,292,913,974}*		1
F₃₃ = U_{2,585,828,011}		0

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The smaller Fermat numbers Fm = 22m + 1, m < 33