362 History of the Theory of Numbers. [Chap, xiv 
Euler^* used the idoneal number 232 to find all values of a < 300 for 
which 232a- + 1 is a prime, by excluding the values of a for which 232a^-hl 
= 232x~-\-y-,y>l. 
Euler^^ noted that N = a~ -\-\b- = x^ +\y^ imply 
N = l(\nr+n'^){\p^-\-q^), a=^x = \mp, nq; y=^h = mq, np, 
so that Xp^ + g-, or its half or quarter, is a factor of A''. He gave (p. 227) 
his^' former table of 65 idoneal numbers. Given one representation by 
cur 4-/3?/", where a/3 is idoneal, he sought a second representation. If 
A^ = 4n + 2 is idoneal, 4iV is idoneal. 
Euler^*^ called mx--\-ny- a congruent form if everj' number representable 
by it in a single way (with x, y relatively prime) is a prime, the square of a 
prime, the double of a prime, a power of 2, or the product of a prime by a 
factor of vin. Then also vinx~-\-y~ is a congruent form and conversely. 
The product mn is called an idoneal or congruent number. His table of 65 
idoneal numbers is reproduced (§18, p. 253). He stated rules for deducing 
idoneal numbers from given idoneal numbers. He factored numbers 
expressed in two ways by ax^+/3i/-, where a/S is idoneal, and noted that a 
composite number may be expressible in a single way in that form if a^ is 
not idoneal. 
Euler^^ proved that the first five squares are the only square idoneal 
numbers. 
C. F. Kausler^- proved Euler's theorem that a prime can be expressed in 
a single way in the form mx'^-\-ny'^ ii m, n are relatively prime. To find a 
prime v exceeding a given number, see whether 38a:- +5?/^ = v has a single set 
of positive solutions x, y; or use 1848x"+?/^. As the labor is smaller the 
larger the idoneal number 38-5 or 1848, it is an interesting question if there 
be idoneal numbers not in Euler's list of 65. Cf. Cunningham. ^^ 
Euler"*^ gave the 65 idoneal numbers n (with 44 a misprint for 45) such 
that a number representable in a single way by nx~-\-y'^ {x, y relatively 
prime) is a prime. By using n = 1848, he found primes exceeding 10 million. 
N. Fuss^ stated the principles due to Euler.^^ 
E. Waring^^ stated that a number is a prime if it be expressible in a single 
way in the form ar-\-mh^ and conversely. 
A. M. Legendre"*^ would express the number A to be factored, or one of 
its multiples kA , in the form t~+air, where a is as small as possible and within 
the limits of his Tables III-YII of the linear forms of divisors of f^au^. 
»»Nova Acta Petrop., 14, 1797-8 (1778). 3; Ck)mm. Arith., 2, 215-9. 
»»/Wd., p. 11; Comm. Arith., 2, 220-242. For X = 2, Opera postuma, I, 1862, 159. 
"/Wd.. 12, 1794 (1778), 22; Comm. Arith., 2, 249-260. 
"/6wf., 15, ad annos 1799-1802 (1778), 29; Comm. Arith., 2, 261-2. 
"Ibid., 156-180. 
"Nouv. M^m. Berlin, ann6e 1776, 1779, 337; letter to Beguelin, May, 1778; Comm. Arith., 2, 
270-1. 
**Ibid., 340-6 
«Medit. Algebr., ed. 3, 1782, 352. 
«^h6orie des nombres, 1798, pp. 313-320; ed. 2, 1808, pp. 287-292. German transl. by Maser, 
1, 329-336. Cf. Sphinx-Oedipe, 1906-7, 51. 
