Chap. XIV] METHODS OF FaCTOKING. 361 
5p± 1. To express a number as x^+y^, subtract squares in turn and seek a 
remainder which is a square. 
N. Beguehn^^ proposed to find x such that 4pV+ 1 is a prime by exclud- 
ing the values x making the sum composite. The latter is the case if 
4pV+l=462+(2c+l)2, a;2 = ^!±^. 
Set X = q+b/p. Then b is expressed rationally in terms of c and the known 
p. Taking p = l, he derived a tentative process for finding a prime, of the 
form 4x^+1, which exceeds a given number a. 
L. Euler^° proved that 1000^+3^ is prime since not expressible as a sum of 
two squares another way. 
A. M. Legendre^"" factored numbers represented as a sum of two squares 
in two ways. 
J. P. Kulik's^"^ tables VIII and IX, relating to the ending of squares, 
serve to test if 4n+l is a sum of two squares and hence to test if it be prime 
or composite. 
Th. Harmuth^^ suggested testing a^-j-b^ for factors, where a and b are 
relatively prime, by noting that it is divisible by 5 if a= =t 1, 6= ± 2 (mod 5) , 
and similar facts for p = 13, 17, 29, 37, there being p — 1 sets of values of a, b 
for each prime p = 4n+l. 
G. Wertheim^^ explained in full Euler's^'' method of factoring. 
R. W. D. Christie and A. Cunningham^^ granted N = A^+B^ = C^+D'^ 
and showed how to find a,...,dso that N={a^+b^){c^-\-cP). Similarly, if 
N = x^-\-Py'^ in two ways. 
Factoring by Use of Binary Quadratic Forms. 
L. Euler^^ noted that a number is composite if it be expressible in two v^ /* .fi'*^ 
ways in the form/ = ax^+i(32/^. The product of two numbers of the form/ 
is of the form g = aPx^+y^; the product of a number of the form / by one of 
the form g is of the form /. If for m>2 a composite number mp is express- 
ible in a single way in the form /, there exist an infinitude of composite 
numbers mq expressible in a single way by /. He called (§34) a number 
A'' idoneal (numerus idoneus) if, for a^ = N, every number representable 
hy f = ax^+^y^ (with ax relatively prime to ^y) is a prime, the square of a 
prime, the double of a prime or a power of 2, so that a number representable 
by / in a single way is a prime. It suffices to test N+y^<4N, y prime to N. 
He gave (§39, p. 208) the 65 idoneal numbers 1, 2, . . ., 1848 less than 10000. 
"Nouv. M^m. Acad. Sc. BerUn, 1777, ann6e 1775, 300. 
soNova Acta Petrop., 10, 1792 (1778), 63; Comm. Arith., 2, 243-8. 
^'^Theorie des nombres, ed. 3, i830, I, 310. Simplification by Vuibert, Jour, de math. 6\em., 
10, p. 42. Cf. I'interm^diaire des math., 1, 1894, 167, 245; 18, 1911, 256. 
3obTafebi der Quadrat und Kubik Zahlen ... bis hundert Tausend, Leipzig, 1848. 
"Archiv Math. Phys., 67, 1882, 215-9. 
32Elemente der Zahlentheorie, 1887, 295-9. 
"Math. Quest. Educat. Times, (2), 11, 1907, 52-3, 65-7, 89-90. 
"Nova Acta Petrop., 13, 1795-6 (1778), 14; Comm. Arith., 2, 198-214. 
