Chap. XIV] METHODS OF FACTORING. 365 
A. Cunningham" and J. Cullen listed the 188 prime numbers x^+18482/^ 
between 10"^ and 101-10^, with x prime to 1848?/. 
A. Cunningham^ ^ noted that two representations of N by ixx^-\-vy^ 
lead to factors of N under certain conditions. 
A. Cunningham^^ recalled that an idoneal number / has the property 
that, if an odd number A'' is expressible in only one way in the form 
N = mx^-\-ny^, where mn = I, and mx^ is prime to ny^, then A" is a prime or 
the square of a prime. Euler's largest I is 1848. There is no larger I under 
50000, a computation checked by J. Cullen. Cunningham noted on the 
proof-sheets of this history that this limit has been extended to 100 000. 
A. Cunningham^" noted conditions that an odd prime be expressible by 
f^qu" when q or —q'l^ idoneal. 
F. N. Cole'^^ discussed Seelhoff's^^ method of factoring. 
Al. Laparewicz^^ described and applied Gauss' ^^'^^ two methods. 
P. Meyer^^ discussed Euler's theorem that, if n is idoneal, a number 
representable only once by x^-\-ny'^ is a prime. 
R. BurgwedeF^ gave an exposition and completion of the method of 
Euler^^'^^ and an exposition of the methods of Gauss.^^'*^ 
L. Valroff stated and A. Cunningham'^^" proved that (Dx'^ — a^) (Dy"^ — a^) 
= Dz^ — a^ implies that one factor is composite unless x^ = y^ = 4: when 
o = 1, D = 2, and in the remaining cases if the two factors are distinct and > 1. 
A. Gerardin'^^ gave a method illustrated for N = a^ — 5-29^, where a = 6326. 
We shall have a second such representation N = (a-{-5xy — 5y^ if 
E=5x^+2ax+841=y^. 
Use is made of various moduli ?« = 4, 3, 7, 25, ... . On square-ruled paper, 
mark x = 0, 1, 2, . . . at the head of the columns. On the line for modulus m, 
shade the square under the heading x when x makes E a quadratic non- 
residue of m. Then examine the column in which occurs no shaded square. 
Up to x^l5, these are x = (excluded), and x = 4, which gives A = 6346^ 
— 5-227^ and the factor 99^ — 5-2^. The same diagram serves for all num- 
bers 1050 ff +671, our N being given by i7 = 38108. To apply the method 
to A = (2a;)*+1 = (4x^+1)^ — 2(2x)^, seek a second representation N=(4x^ 
+2p + lY-2{2uy. The condition is {2p-\-l)x^+Mp-j-l)=u^, solutions 
of which are found for p = 1, 8, 9, . . . , 6^, 35^, ... Or we may choose x, say 
X = 48, and find p = S, u = 198. 
s^Brit. Assoc. Reports, 1901, 552. The entry 10098201 is erroneous. 
"Proc. London Math. Soc, 33, 1900-1, 361. 
«976id., 34, 1901-2, 54. 
■"^Ibid., (2), 1, 1903, 134. 
7iBuU. Amer. Math. Soc, 10, 1903-4, 134-7. 
"Prace mat. fiz., Warsaw, 16, 1905, 45-70 (PoUsh). 
^'Beweis eines von Euler entdeckten Satzes, betreffend die Bestimmung von Primzahlen, Diss., 
Strassburg, 1906. 
'^Ueber die Eulerschen und Gausschen Methoden der Primzahlbestimnaung, Diss., Strassburg, 
1910, 101 pp. 
'*»Sphinx-Oedipe, 7, 1912, 60, 77-9. 
"Wiskundig Tijdschrift, 10, 1913, 52-62. 
