364 History of the Theory of Numbers. [Chap, xiv 
for which N is represent able by x^-\-my-. For example, if iV = 31-2^*-f 1, 
Eliminating 19-83 between the first two, we get nN = w'^ — 7f. This with 
the third leads to factors of N. In general, when elimination of common 
factors of the 77i's has led to representations of two multiples of N by the 
same form x~+ny^, we may factor N unless it be prime. 
H. Weber^^ computed the class invariants for the 65 determinants of 
Euler and remarked that there is no known proof of the fact found by 
induction by Euler and Gauss that there are only 65 determinants such that 
all classes belonging to the determinant are ambiguous and hence each 
genus has only one class. 
T. Pepin^^ developed the theory of Gauss'^^ posthumous tables and the 
means of deducing complete tables from the given abridged tables. Pepin^* 
showed how to abridge the calculations in using the auxiliary tables of Gauss 
in factoring a" — 1, where a and n are primes. 
D. F. Seliwanoff^° noted that the factoring of numbers of the form 
t' — Du- reduces to the solution of {D/x) = \, all solutions of which are 
easily found by use of six relations by Euler on these Jacobi symbols {D/x) . 
E. Lucas" gave a clear proof of Euler's remark that a prime can not be 
expressed in two ways in the form Ax~-\-By-, if A, B are positive integers. 
S. Levanen^^ showed and illustrated by examples and tables how binary 
quadratic forms may be applied to factoring. 
G. B. ^Mathews*"^ gave an exposition of the subject. 
T. Pepin^ applied determinants — 8n — 3 for which each genus has three 
classes of quadratic forms. The paper is devoted mainly to the solution 
of X-+ (871+3)?/"" = 4A, where A is the number to be factored. 
T. Pepin^^ assumed that the given number N had been tested and found 
to have no prime factor ^p. Let Xx+1, Xy+1 be the two factors of N, 
each between p and N/p. The sum of the factors lies between 2VW and 
p+N/p. Let x—y = u, x+y=pz. Then {N — l)/\ = Xxy+x+y gives 
in which special values are assigned to p. This equation yields a quadratic 
congruence for w" with respect to an arbitrary prime modulus, used as an 
excludant. The method applies mainly to numbers a^=^l. 
E. Cahen^^ used the linear divisors of x^ + Dy'^. 
'"Math. Annalen, 33, 1889. 390-410. 
"Atti Accad. Pont. Nuovi Lincei, 48, 1889, 135-156. 
"Ibid., 49, 1890. 163-191. 
"Moscow Math. Soc, 15, 1891, 789; St. Petersburg Math. See, 12, 1899. 
•'Th6orie des nombres, 1891, 356-7. 
"^vereigt af finska Vetenskaps-Soc. forhandlingar, 34, 1892, 334-376. 
"Theory of Numbers, 1892, 261-271. French transl., Sphinx-Oedipe, 1907-8, 155-8, 161-70. 
"Memorie Accad. Pontif. Nuovi Lincei, 9, I, 1893, 46-76. Cf. Pepin," 332. 
«;6iV/., 17, 1900-1, 321-344; Atti, 54, 1901, 89-93. Cf. Meissner"*, 121-2. 
"filaments de la th^orie des nombres, 1900, 324-7. Sphinx-Oedipe, 1907-8, 149-155. 
