History Of The Theory Of Numbers - I

376 HlSTOEY OF THE THEOKY OF NUMBERS. [CHAP. XV
and proved in the same lengthy dull manner that the quotient is a prime. An anonymous writer15 stated that
(1) 2+1, 22+l, 222+l, 2222+l,...
are all primes and are the only primes 2*+l. See Malvy.39
Joubin16 suggested that these numbers (1) are possibly the ones really meant by Fermat,1 evidently without having consulted all of Fermat's statements.
G. Eisenstein17 set the problem to prove that there is an infinitude of primes Fn.
E. Lucas18 stated that one could test the primality of F6 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 Fn is a prime if 2n~1 is the rank of the first term divisible by Fn, composite if no term is divisible by Fn. Finally, if a is the rank of the first term divisible by Fn, the prune divisors of Fn are of the form 2fcg+l, where &=a+l [not k=2a+l]. See Lucas.22
T. Pepin19 stated that the method of Lucas18 is not decisive when Fn divides a term of rank a<2n~1; for, if it does, we can conclude only that the prime divisors of Fn are of the form 2a+2#+l, so that we can not say whether or not Fn is prune if a+2^2n""2. We may answer the question unambiguously by use of the new theorem: For n>l, Fn is a prime if and only if it divides
where k is any quadratic non-residue of Fnj as 5 or 10. To apply this test, take the minimum residues modulo Fn of
1.2 r.4 7.8 P2""1
A/, n/, A/,..., A;
Proof was indicated by Lucas29 of Ch. XVII, and by Morehead.58 J. Pervouchine20 (or Pervusm) announced, November 1877, that
F12s=0 (mod 114689 = 7-214+l).
E. Lucas21 announced the same result two months later and proved that every prime factor of Fn is s=l (mod 2n+2).
Lucas22 employed the series 6, 34, 1154, . . . , each term of which is 2 less than the square of the preceeding. Then Fn is a prime if the rank of the first term divisible by Fn is between 2n~1 and 2n— 1, but composite if no term is divisible by Fn. Finally, if a is the rank of the first term divisible by Fn
"Annales de Math. (ed. Gergonne), 19, 1828-9, 256.
16Me*moire sur lea facteurs nume'riques, Havre, 1831, note at end.
17Jour. fur Math., 27, 1844, 87, Prob. 6.
"Comptes Rendus Paris, 85, 1877, 136-9.
l*Comptea Rendus, 85, 1877, 329-331. Reprinted, with Lucas18 and Landry,29 Sphinx-Oedipe,
5, 1910, 33-42. "Bull. Ac. St. P&ersbourg, (3), 24, 1878, 559 (presented by V. Bouniakowsky) . Melanges
math. ast. sc. St. Pe"tersbourg, 5, 1874r-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 Lucas40 of Ch. XVII.
r. Jour. Math., 1, 1878, 313.