362 HISTORY or THE THEORY OF NUMBERS. [CHAP, xiv Euler38 used the idoneal number 232 to find all values of a<300 for which 232a2+l is a prime, by excluding the values of a for which 232a2+l Euler39 noted that tf=a2+A&2=z2+X^2 imply N- %(\m2+n2)(\p2+q2), a*±x=\mp, nq; y^b^mq, np, so that \p2+cf, or its half or quarter, is a factor of N. He gave (p. 227) his87 former table of 65 idoneal numbers. Given one representation by ax2+f3y2, where a/3 is idoneal, he sought a second representation. If #=471+2 is idoneal, 4N is idoneal. Euler40 called mx2+ny2 a congruent form if every number representable by it in a single way (with z, 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 mn. Then also mnrf+y2 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 cuc2+/%/2, where a/3 is idoneal, and noted that a composite number may be expressible in a single way in that form if a/3 is not idoneal. Euler41 proved that the first five squares are the only square idoneal numbers. C. F. Kausler42 proved Euler's theorem that a prime can be expressed in a single way in the form mx2+ny2 if m, n are relatively prime. To find a prime v exceeding a given number, see whether 38£2-h5i/2 = v has a single set of positive solutions x, y, or use 1848o:2+y2. 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.69 Euler43 gave the 65 idoneal numbers n (with 44 a misprint for 45) such that a number representable in a single way by nx2+y2 (x, y relatively prime) is a prime. By using n = 1848, he found primes exceeding 10 million. N. Fuss44 stated the principles due to Euler.37 E. Waring45 stated that a number is a prime if it be expressible in a single way in the form a2 -\-rnb2 and conversely. A. M. Legendre46 would express the number A to be factored, or one of its multiples kA, in the form t2+au2, where a is as small as possible and within the limits of his Tables III-VII of the linear forms of divisors of ••Nova Acta Petrop., 14, 1797-8 (1778), 3; Comm. Arith., 2, 215-9. "JWd., 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. «/toa., 15, ad annoa 1799-1802 (1778), 29; Comm. Arith., 2, 261-2. <*/&«*., 156-180. «Nouv. M&n. Berlin, anne*e 1776, 1779, 337; letter to Beguelin, May, 1778; Comm. Arith., 2, 270-1. «I1rid., 340-6 "Medit. Algebr., ed. 3, 1782, 352. ^The'orie des nombres, 1798, pp. 313-320; ed. 2, 1808, pp. 287-292. German transl. by Maaer, 1, 329-336. Cf. Sphinx-Oedipe, 1906-7, 51.