378 History of the Theory of Numbers. [Chap, xv 
J. Hermes^^ indicated a test for composite F„ by Fermat's theorem. 
R. Lipschitz^ separated all integers into classes, the primes of one class 
being Fermat numbers F„, and placed in a new light the question of the 
infinitude of primes F„. 
E. I.ucas^^ stated the result of Proth,-^ but with a misprint [Cipolla^^]. 
H. Scheffler^^ stated that Legendre believed that every F„ is a prime(!), 
and obtained artificially the factor 641 of F5. He noted (p. 167) that 
F„F„+, . . .F„_, = l+22"+22-2"+23-2''+ . . . +2^"-2". 
He repeated (pp. 173-8) the test by Pepin/^ with k = 3, and (p. 178) expressed 
his belief that the numbers (1) are all primes, but had no proof for Fie- 
W. W. R. Ball" gave references and quoted known results. 
T. M. Pervouchine^^ checked his verification that F12 and F23 are com- 
posite by comparing the residues on di\'ision by 10^ — 2. 
Mah'y^^ noted that the prime 2^+1 is not in the series (1). 
F. Klein^° stated that F7 is composite. 
A. Hurwitz"*^ gave a generalization of Proth's^^ theorem. Let F„(x) 
denote an irreducible factor of degree 4>{n) of x" — 1. Then if there exists 
an integer q such that Fp_i(g) is divisible by p, p is a prime. When 
p = 2'-+l,Fp_i(x)^=x''"' + l. 
J. Hadamard'*- gave a very simple proof of the second remark by Lucas. ^^ 
A. Cunningham^^ found that Fn has the factor 319489-974849. 
A. E. \Yestern^ found that Fg has the factor 2^^-37 + 1, Fis the factor 
2'°-13 + l, the quotient of Fn by the known factor 2^'*-74-l has the factors 
2^^-397+1 and 2^^-7-139 + l. He verified the primality of the factor 2^^-3-f 1 
of F38, found by J. Cullen and A. Cunningham. He and A. Cunningham 
found that no more F„ have factors < 10^ and similar results. 
m-2 
M. Cipolla^^ noted that, if g is a prime >(9"^ -l)/2'"+^and m>\, 
2'"g+ 1 is a prime if and only if it divides 3H 1 for k = ^•2'"+^ He^^ pointed 
out the misprint in Lucas'^^ statement. 
Nazarevsky^^ proved Proth's-^ result by using the fact that 3 is a primi- 
tive root of a prime 2''+l. 
"Archiv Math. Phys., (2), 4, 1886, 214-5, footnote. 
"Jour, fur Math., 105, 1889, 152-6; 106, 1890, 27-29. 
"Th^orie des nombres, 1891, preface, xii. 
»«Beitrage zur Zahlentheorie, 1891, 147, 151-2, 155 (bottom), 168. 
»^Math. Recreations and Problems, ed. 2, 1892, 26; ed. 4, 1905, 36-7; ed. 5, 1911, 39-40. 
"Math. Papers Chicago Congress of 1893, I, 1896, 277. 
"L'interm6diaire des math., 2, 1895, 41 (219). 
"Vortrage iiber aiisgewahlte FragenderElementar Geometric, 1895, 13; French transl., 1896, 26; 
English transl., "Famous Problems of Elementary Geometry," by Beman and Smith, 
1897, 16. 
"L'interm^diaire des math., 3, 1896, 214. 
«/6u/., p. 114. 
"Report British Assoc, 1899, 653-4. The misprint in the second factor has been corrected 
to agree with the true " value 2".7.17 + 1. 
^Cunningham and Western, Proc. Lond. Math. Soc, (2), 1, 1903, 175; Educ. Times, 1903, 270. 
«Periodico di Mat., 18, 1903, 331. 
"Also in Annali di Mat., (3), 9, 1904, 141. 
*'L'interm6diaire des math., 11, 1904. 215. 
