Chap. XV] FerMAT NUMBERS F„=2^" + l. 377 
and if a<2''~\ the prime divisors of F„ are of the form* 2*^+1, where 
k = a-\-l [cf . Lucas^^]. He noted (p. 238) that a necessary condition that F^ 
be a prime is that the residue modulo F^ of the term of rank 2" — 1 in this 
series is zero. He verified (p. 292) that F^ has the factor 641 and again 
stated that 30 hours would suffice to test Fq. 
F. Proth^^ stated that, if A; = 2", 2^+1 is a prime if and only if it divides 
m = 3^ ~ + 1 . He^^ indicated a proof by use of the series of Lucas defined by 
Uo = 0, Ui = l, . . . , w„ = 3if„_i + l and the facts that Up_i is divisible by 
the prime p, while m=^U2^lu2^-^. Cf. Lucas. ^^ 
E. Gelin^^ asked if the numbers (1) are all primes. Catalan^^ noted 
that the first four are. 
E. Lucas^^ noted that Proth's^^ theorem is the case A; = 3 of Pepin's. ^^ 
Pervouchine" announced, February 1878, that 7^23 lias the prime factor 
5.225+1 = 167772161. 
W. Simerka^^ gave a simple verification of the last result and the fact 
(Pervouchine^o) that 7-2'Hl divides F^^. 
F. Landry, ^^ when of age 82 and after several months' labor, found that 
7^6 = 274177-67280421310721, 
the first factor being a prime. He and Le Lasseur and G^rardin^^" each 
proved that the last factor is a prime (cf. Lucas^^). 
K. Broda^" sought a prime factor p of a^^+1 by considering 
n = (a^2-l)(a«Hl)(a'''+a^''+a2^«+a^''+l). 
Multiply by i^ = (o^2_}_l)/p. Thusnw = (a^^°-l)/p. But a^^°^l (mod641). 
Since each factor of n is prime to p, we take a = 2 and see that 2^^+ 1 is divis- 
ible by 641. 
E. Lucas^^ stated that he had verified that F^ is composite by his^^ test, 
before Landry found the factors. 
P. Seelhoff^^ gave the factor 5-2^^+1 of F^,^ and commented on Beguelin.^^ 
*Lucas wrote fc = 2"+^ in error, as noted by R. D. Carmichael on the proof-sheets of this 
History. 
23Comptes Rendus Paris, 87, 1878, 374. 
«Nouv. Corresp. Math., 4, 1878, 210-1; 5, 1879, 31. 
MUd., 4, 1878, 160. 
^HUd., 5, 1879, 137. 
"Bull. Ac. St. P6tersbourg, (3), 25, 1879, 63 (presented by V. Bouniakowsky) ; Melanges math, 
astr. ac. St. P^tersbourg, 5, 1874-81, 519. Cf. Nouv. Corresp. Math., 4, 1878, 284-5; 5, 
1879, 22. 
28Casopi3, Prag, 8, 1879, 36, 187-8. F. J. Studnicka, iUd., 11, 1881, 137, 
"Comptes Rendus Paris, 91, 1880, 138; Bull. Bibl. Storia Sc. Mat., 13, 1880, 470; Nouv. Cor- 
resp. Math., 6, 1880, 417; Lea Mondes, (2), 52, 1880. Cf. Seelhoflf, Archiv Math. Phys., 
(2), 2, 1885, 329; Lucas, Amer. Jour. Math. 1, 1878, 292; R6cr6at. Math., 2, 1883, 
235; I'intermddiaire des math., 16, 1909, 200. 
»»Sphinx-Oedipe, 5, 1910, 37-42. 
"Archiv Math. Phys., 68, 1882, 97. 
"Recreations Math., 2, 1883, 233-5. Lucas,"* 354-5. 
»2Zeitschr. Math. Phys., 31, 1886, 172-4, 380. For Ft, p. 329. French transl., Sphinx-Oedipe, 
1912, 84-90. 
