364 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xiv for which AT is representable by x2+my2. For example, if AT=31-22*+l, Eliminating 19-83 between tfre first two, we get jj,N=w2—7t2. This with the third leads to factors of N. In general, when elimination of common factors of the m's has led to representations of two multiples of N by the same form x2+ny2, we may factor N unless it be prime. H. Weber57 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. Pepin58 developed the theory of Gauss'53 posthumous tables and the means of deducing complete tables from the given abridged tables. Pepin69 showed how to abridge the calculations in using the auxiliary tables of Gauss hi factoring an— 1, where a and n are primes. D. F. Seliwanoff60 noted that the factoring of numbers of the form t2—Du2 reduces to the solution of (D/a;) = l, all solutions of which are easily found by use of six relations by Euler on these Jacobi symbols (D/x). E. Lucas61 gave a clear proof of Euler's remark that a prime can not be expressed in two ways in the form Ax2+By2, if A, B are positive integers. S. Levanen62 showed and illustrated by examples and tables how binary quadratic forms may be applied to factoring. G. B. Mathews63 gave an exposition of the subject. T. Pepin64 applied determinants — 8n— 3 for which each genus has three classes of quadratic forms. The paper is devoted mainly to the solution of £2+(8n-h3)i/2=4A, where A is the number to be factored. T. Pepin65 assumed that the given number N had been tested and found to have no prime factor ^p. Let Xz-f 1, \y-\-l be the two factors of N, each between p and N/p. The sum of the factors lies between 2 VN and p+N/p. Let x—y=u, x+y~pz. Then (N—l)/\=*\xy+x+y gives _ x2 - in which special values are assigned to p. This equation yields a quadratic congruence for u2 with respect to an arbitrary prime modulus, used as an excludant. The method applies mainly to numbers ax=±= 1. E. Cahen66 used the linear divisors of x2+Dy2. 57Math. Annalen, 33, 1889, 390-410. MAtti Accad. Pont. Nuovi Lincei, 48, 1889, 135-156. "Ibid., 49, 1890, 163-191. "Moscow Math. Soc., 15, 1891, 789; St. Petersburg Math. Soc., 12, 1899. 61The*orie des nombres, 1891, 356-7. M0fversigt 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, 1, 1893, 46-76. Cf. Pepin,95 332. «/Wrf., 17, 1900-1, 321-344; Atti, 54, 1901, 89-93. Cf. Meissner1", 121-2. de la the*orie des nombres, 1900, 324-7. Sphinx-Oedipe, 1907-8, 149-155.