376 History of the Theory of Numbers. [Chap, xv 
and proved in the same lengthy dull manner that the quotient is a prime. 
An anonymous writer^^ stated that 
(1) 2+1, 2-+1, 2-'+l, 22''+!,... 
are all primes and are the only primes 2*+l. See Malvy.'^ 
Joubin^^ suggested that these numbers (1) are possibly the ones really 
meant by Fermat/ evidently without having consulted all of Fermat's 
statements. 
G. Eisenstein^' set the problem to prove that there is an infinitude of 
primes F„. 
E. Lucas^^ stated that one could test the primality of F^ in 30 hours by 
means of the series 3, 17, 577, . . . , each term being one less than the double of 
the square of the preceding. Then F„ is a prime if 2""^ is the rank of the first 
term divisible by F„, composite if no term is di\'isible by F„. Finally, if a is 
the rank of the first term divisible by F„, the prime divisors of F„ are of the 
form 2''g + l, where A; = a+1 [not A: = 2°+^]. See Lucas." 
T. Pepin^^ stated that the method of Lucas^^ is not decisive when F„ 
divides a term of rank a<2''~^; for, if it does, we can conclude only that the 
prime di\'isors of F„ are of the form 2''"''^g'+l, so that we can not say whether 
or not F„ is prime if a +2 ^2""^. We may answer the question unambigu- 
ously by use of the new theorem: For n> 1, F„ is a prime if and only if it 
divides 
where k is any quadratic non-residue of Fn, as 5 or 10. To apply this test, 
take the minimum residues modulo F„ of 
2"-l 
A/, i^ ) "^ ) • • • } '^ ' 
Proof was indicated by Lucas-^ of Ch. XVII, and by Morehead.^^ 
J. Pervouchine^° (or Pervusin) announced, November 1877, that 
Fi2=0 (mod 114689 = 7-2^^+1). 
E. Lucas^^ announced the same result two months later and proved that 
every prime factor of F„ is = 1 (mod 2"'^^). 
Lucas^^ employed the series 6, 34, 1154, . . . , each term of which is 2 less 
than the square of the preceeding. Then F„ is a prime if the rank of the first 
term divisible by F^ is between 2""^ and 2" — !, but composite if no term is 
di\'isible by F„. Finally, if a is the rank of the first term divisible by F„ 
"Annales de Math. (ed. Gergonne), 19, 1828-9, 256. 
"M^moire sur le3 facteurs numdriques, Havre, 1831, note at end. 
»'Jour. fiir Math., 27, 1844, 87, Prob. 6. 
"€omptes Rendus Paris, 85, 1877, 136-9. 
'•Comptea Rendus, 85, 1877, 329-331. Reprinted, with Lucas" and Landry," Sphinx-Oedipe, 
5, 1910, 33-42. 
"Bull. Ac. St. Pdtersbourg, (3), 24, 1878, 559 (presented by V. Bouniakowsky). Melanges 
math. ast. sc St. P^tersbourg, 5, 1874-81, 505. 
"Atti R. Accad. Sc. Torino, 13, 1877-8, 271 (Jan. 27, 1878). Cf. Nouv. Corresp. Math., 4, 
1878, 284; 5, 1879, 88. See Lucas" of Ch. XVIL 
"Amer. Jour. Math., 1, 1878, 313. 
