Chap. XIV] METHODS OF FACTORING. 363 
Then the divisors of A are included among these linear forms. When 
VS is converted into a continued fraction, let (VM +/)/!) be a complete 
quotient, and p/q the corresponding convergent. Then ^D = p^ — kAq^, 
so that the divisors of A are divisors oi p^=f=D. 
C. F. Gauss'*^ stated that the 65 idoneal numbers n of Euler and no 
other numbers have the two properties that all classes of quadratic forms 
of determinant —n are ambiguous and that any two forms in the same 
genus (Geschlecht) are both properly and improperly equivalent. 
Gauss^^ gave a method of factoring a number M based on the deter- 
mination of various small quadratic residues of M. 
Gauss^^ gave a second method of factoring M based on the finding of 
representations of M by forms x^+D, where D is idoneal. 
F. Minding^° gave an exposition of the method of Legendre.^^ 
P. L. Tchebychef^^ gave a rapid process to find many forms x^^ay^ 
which represent a given number A or a multiple of A. Then a table of 
the linear forms of the divisors of x^^ay^ serves to limit the possible factors 
of A. 
Tchebychef^^ gave theorems on the limits between which lie at least 
one set of integral solutions of x^ — Dy^ = ± iV. If there are two sets of solu- 
tions within the limits, N is composite. There are given various tests for 
primality by use of quadratic forms. 
C. F. Gauss^^ left posthumous tables to facilitate factoring by use of 
his*^ second method. 
F. Grube^^ criticized and completed certain of Euler's proofs relating to 
idoneal numbers, here called Euler numbers. While Gauss^^ said it is easy 
to prove Euler's*^ criterion for idoneal numbers, Grube could prove only 
the following modification: Let Q, be the set of numbers D+n^^4Z) in 
which n is prime to D. According as all or not all numbers of 12 are of the 
form q, 2q, q"^, 2^ {q a prime), D is or is not an idoneal number. 
E. Lucas^^ proved that if p is a prime and A; is a positive integer, and 
p = x^+ky^, then pT^Xi^+ky^^ for values Xi, yi distinct from ^x, ^y. 
P. Seelhoff^^ made use of 170 determinants (including the 65 idoneal 
numbers of Euler and certain others of Legendre), such that every reduced 
form in the principal genus is of the type ax^-\-by^. To factor A^ seek 
among the numbers m of which iV is a quadratic residue several values 
<^Disq. Arith., 1801, Art. 303. 
"7btd., Arts. 329-332. 
*mid., Arts. 333-4. 
s^Anfangsgriinde der Hoheren Arith., 1832, 185-7. 
"Theorie der Congruenzen (in Russian), 1849; German transl. by H. Schapira, 1889, Ch. 8, 
pp. 281-292. 
"Jour, de Math., 16, 1851, 257-282; Oeuvres, 1, 73. 
«Werke, 2, 1863, 508-9. 
"Zeitschrift Math. Phys., 19, 1874, 492-519. 
"Nouv. Corresp. Math., 4, 1878, 36. [Euler."] 
"Archiv Math Phys., (2), 2, 1885, 329; (2), 3, 1886, 325; Zeitschrift Math. Phys., 31, 1886, 166, 
174, 306; Amer. Jour. Math., 7, 1885, 264; 8, 1886, 26-44. 
