64ordle.au

Fermat numbers F0 — F32

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 F33, which is currently being tested with Mlucas 21 using Pollard’s p–1 algorithm, 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.

  F0 = 3  F1 = 5  F2 = 17  F3 = 257Number of  F4 = 65,537prime 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 · c1,133 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 · c2,391 4  F14 = 116,928,085,873,074,369,829,035,993,834,596,371,340,386,703,423,373,313 · c4,880 1  F15 = 1,214,251,009 · 2,327,042,503,868,417 · 168,768,817,029,516,972,383,024,127,016,961 · c9,808 3  F16 = 825,753,601 · 188,981,757,975,021,318,420,037,633 · c19,694 2  F17 = 31,065,037,602,817 · 7,751,061,099,802,522,589,358,967,058,392,886,922,693,580,423,169 · c39,395 2  F18 = 13,631,489 · 81,274,690,703,860,512,587,777 · c78,884 2  F19 = 70,525,124,609 · 646,730,219,521 · 37,590,055,514,133,754,286,524,446,080,499,713 · c157,770 3   F20 = C315,653 0   F21 = 4,485,296,422,913 · c631,294 1  F22 = 64,658,705,994,591,851,009,055,774,868,504,577 · c1,262,577 1  F23 = 167,772,161 · c2,525,215 1   F24 = C5,050,446 0   F25 = 25,991,531,462,657 · 204,393,464,266,227,713 · 2,170,072,644,496,392,193 · c10,100,842 3  F26 = 76,861,124,116,481 · c20,201,768 1  F27 = 151,413,703,311,361 · 231,292,694,251,438,081 · c40,403,531 2  F28 = 1,766,730,974,551,267,606,529 · c80,807,103 1  F29 = 2,405,286,912,458,753 · c161,614,233 1  F30 = 640,126,220,763,137 · 1,095,981,164,658,689 · c323,228,467 2   F31 = 46,931,635,677,864,055,013,377 · u646,456,971* 1  F32 = 25,409,026,523,137 · u1,292,913,974* 1   F33 = U2,585,827,973 0
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     Factor   B1 B2  σ   g    10k37·212+1 2,000,00014,152,2674,659,775,785,220,018,543,260,885,870,817,648,693,860
= 22 · 32 · 5 · 149 · 163 · 197 · 7,187 · 18,311 · 123,677 · 226,133 · 314,263 · 4,677,583  11k17·214+1 16,000not given k18·213+116,000not given  12k50·215+1 43,000,000199,103,726,6501,428,526,317568,630,647,535,356,955,169,033,412,162,316,189,313,022,429,279,110,256
= 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  13k14·217+1 k13·219+1 k21·219+1100,000
100,000
500,000B3 = 485,301
3,542,000
stage 2 un-neededsee [BCDH2000]
4,009,189
8,020,345319,546,020,820,518,229,496,965,479 = 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  14k49·216+1 110,000,00011,000,000,0008,585,974,330,888,598116,928,085,873,074,369,829,035,993,372,903,082,226,176,977,098,089,788
= 22 · 3 · 53 · 107 · 3,433 · 37,087 · 110,323 · 128,321 · 1,738,307 · 9,338,881 · 74,968,979 · 783,277,631  15k28·217+1 10,000,000500,000,000253,301,772168,768,817,029,516,992,836,491,975,362,208
= 25 · 3 · 4,889 · 5,701 · 9,883 · 11,777 · 5,909,317 · 91,704,181  16k21·220+1 400,000
200,00020,000,000
10,000,0001,944,934,539
125,546,653188,981,757,975,005,913,471,235,500 = 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  17k44·219+1 44,000,0004,400,000,00010,717,701,036,7737,705,935,574,284,557,194,893,456,578,699,598,773,615,181,874,764
= 22 · 3 · 541 · 2,713 · 5,153 · 23,773 · 152,363 · 239,387 · 19,359,383 · 22,095,751 · 230,254,627  18k16·223+1 100,0004,000,000731,185,96881,274,690,704,014,912,758,776 = 23 · 3 · 59 · 367 · 389 · 3,613 · 50,101 · 2,221,069  19k28·222+1 3,000,000300,000,0007,121,198,363,696,30737,590,055,514,133,754,043,447,773,966,186,464
= 25 · 3 · 11 · 181 · 263 · 4,217 · 38,867 · 244,451 · 1,779,623 · 10,487,459  22k28·224+1 1,000,000100,000,0008,776,953,345,765,66864,658,705,994,591,850,983,958,649,757,139,080
= 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; A. Schindel 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 Various inc. 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; 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. King; A. Kruppa; G. Childers 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 Euler; 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 10835 post 1 (2011); cofactor compositeness tests appear in replies at 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)
[B1878] V. Bouniakowsky, Nouveau cas de divisibilité des nombres de la forme 22m+1, trouvé par le révérend père J. Pervouchine, Bulletins de l’Académie des sciences de Saint-Pétersbourg 24 (1878), 559.
[B1879] V. Bouniakowsky, Encore un nouveau cas de divisibilité des nombres de la forme 22m+1, Bulletins de l’Académie des sciences de Saint-Pétersbourg 25 (1879), 63–64
[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
[CMP2003] R. E. Crandall, E. W. Mayer, J. S. Papadopoulos, The twenty-fourth Fermat number is composite, Math. Comp. 72 (2003), 1555–1572. MR 1972753, DOI 10.1090/S0025-5718-02-01479-5
[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 9605; announcement, post 1; cofactor compositeness test, 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, in mersenneforum.org thread 8842; compositeness test of c10,100,842 | F25 cofactor, post 51 (2009); compositeness test of c20,201,768 | F26 cofactor, post 62 (2009); compositeness test of c40,403,531 | 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
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