44 HISTORY or THE THEORY OP NUMBERS. [CHAP. I Multiply by 6/a and replace bfa by 2a6—ac [see case (1)]. Thus exy-bhx-bgy = b(f--l), e^bf-bgh+cgh. Thus Ls620/i-t- be(f-~ 1) is to be expressed as the product PQ of two factors and they are to be equated to ex—bg, ey—bh. The case a = 2 is unfruitful. (30 Leta = 4. Then 6=4, c=l, e=4/—3gh. The case/= 3 is excluded since it gives e = 0. For/=5, 0 = 2, 7i = 3, we again get (a) and also (/3). For/=5, 0=1, h = 6, we get only the same two pairs. For a prime /^7, no new solutions are found. For/=5-13, (51) results. (32) Let a = 8, whence 6 = 8, c=l. The cases/= 11, 13 are fruitless, while/= 17 yields (16). The least composite / yielding solutions is 11-23, giving (44), (45), (46). This fruitful case led Euler to the more convenient notations (§88) M~hP, N=gQ, L~PQ. The problem is now to resolve L J7 into two factors, M, N, such that wf/ , are integers and primes, while in r-fl = (p+l)(#+l)/J/, r is a prime. (33) Let a = 16. For/=17, we obtain the pairs (21), (22); for /=19, (23); for/=23, (17), (19), (20); for/=47, (18); for /= 17-167, (49). Cases /=31, 17-151 are fruitless [the last since 129503 has the factor 11, not noticed by Euler]. (34) For a = 33-5 or 32-7-13, 6 = 9, c = 2; the first a with/ =7 yields (30). (4) Problem 4 relates to amicable numbers agpq} ahr, where p, q} r are primes. Eventually he took also g and h as primes. We may then set g+l = km, h+l = kn. For m = l, n = 3, a = 4 or 8, no amicables are found. For m = 3, n = l, the cases a = 10, &-8 and a = 33-5, A; = 8, yield (38), (55). (5) Euler;s final problem 5 is of a new type. He discussed amicable numbers zap, zbq, where a and 6 are given numbers, p and q are unknown primes, while z is unknown but relatively prime to a, 6, p, q. Set J a :J 6 = m : n, where m and n are relatively prime. Since (p + 1) Ja = (q + !)/&, we may set p+I~nx, q+I~mx. The usual second condition gives //* cM2 = za(nz— l)+zb(mrc — 1), •^ # I o ^ — a — o Let the latter fraction in its lowest terms be r/s. Then z=kr, fz = ks. Since J(kr)^kfr, we have s^Jr. Hence we have the useful theorem: if 2:j2=r':s', a'<jV, then r' and s' have a common factor >1. (50 The unfruitful case a = 3, 6 = 1, was treated like the next. (52) Let a = 5, 6=1, whence m = 6, n = l, 2:j2=6x:llx-6. By the theorem in (5), x must be divisible by 2 or 3. Euler treated the cases 2(2$+l). But this classification is both incomplete and