# Full text of "History Of The Theory Of Numbers - I"

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```CHAP. XV]                    FEBMAT NUMBBES Fw=22n+l.                            377
and if a<2n~1, the prime divisors of Fn are of the form* 2*g+l, wher k = a+l [cf. Lucas18].   He noted (p. 238) that a necessary condition that F be a prime is that the residue modulo Fn of the term of rank 2n—1 in this series is zero.   He verified (p. 292) that F5 has the factor 641 and again stated that 30 hours would suffice to test F6.
F. Proth23 stated that, if & = 2n, 2*+1 is a prime if and only if it divides m = 32fc~"+1. He24 indicated a proof by use of the series of Lucas defined by 1*0 = 0,^ = 1, . . . , t^=3wn_i+l and the facts that up^i is divisible by the prime p, while m~u&/ud*-i. Cf. Lucas.26
E. Gelin25 asked if the numbers (1) are all primes. Catalan25 noted that the first four are.
E. Lucas26 noted that Froth's23 theorem is the case k =3 of Pepin's.19 Pervouchine27 announced, February 1878, that F23 has the prime factor
.                          5-225+l
W. Simerka28 gave a simple verification of the last result and the fact (Pervouchine20) that 7-214+l divides F12.
F. Landry,29 when of age 82 and after several months' labor, found that
F6 * 274177-67280421310721,
the first factor being a prime. He and Le Lasseur and G6rardin29a each proved that the last factor is a prime (cf. Lucas31).
K. Broda30 sought a prime factor p of a32+l by considering
n=(a32~l)(a64+D(a612+a384+a256+a128+l).
Multiply by u = (a32+l)/p. Thus nu = (a640 - l)/p. But a640== 1 (mod 641). Since each factor of n is prime to p, we take a=2 and see that 232+1 is divisible by 641.
E. Lucas31 stated that he had verified that FQ is composite by his22 test, before Landry found the factors.
P. Seelhoff32 gave the factor 5-239+l of F3Q and commented on Beguelin.12
"Lucas wrote fc=2tt+1 in error, as noted by R. D. Carmichael on the proof-sheets of this History.
"Comptes Rendus Paris, 87, 1878, 374.
^Nouv. Corresp. Math., 4, 1878, 210-1; 5, 1879, 31.
**Ibid., 4, 1878, 160.
MJ6id., 5, 1879, 137.
"Bull. Ac. St. Pe"tersbourg, (3), 25, 1879, 63 (presented by V. Bouniakowsky); Melanges math, astr. ac. St. PStersbourg, 5,1874r-81, 519. Cf. Nouv. Corresp. Math., 4, 1878, 284-5; 5, 1879, 22.
2»Casopis, Prag, 8,1879, 36,187-8.   F. J. Studnicka, ibid., 11,1881,137.
"Comptes Rendus Paris, 91, 1880, 138; Bull. Bibl. Storia Sc. Mat., 13, 1880, 470; Nouv. Corresp. Math., 6, 1880, 417; Les Mondes, (2), 52,1880. Q£. Seelhoff, Archiv Math. Phys., (2), 2,. 1885, 329; Lucas, Amer. Jour. Math. 1, 1878, 292; RScrSat. Math., 2, 1883, 235; I'intermeMiaire des math., 16, 1909, 200.
»°Sphinx-Oedipe, 6, 1910, 37-42.
«°Archiv Math. Phys., 68, 1882, 97.
BlR6cr6ations Math., 2, 1883, 232-5.   Lucas,** 354r-5.
"Zeitschr. Math. Phys., 31,1886, 172-4, 380. For Ft, p. 329. French transl., Sphinx-Oedipe, 1912, 84-90.```