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```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```