44 History of the Theory of Numbers. [Chap, i 
Multiply by 6/a and replace bja by 2ab — ac [see case (1)]. Thus 
exy — bhx — hgy = b{f—l), e=bf—bgh-{-cgh. 
Thus L^b'^gh+be{f—l) 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. 
(3i) Let a = 4. Then 6 = 4, c = l, e = 4/— 3^/i. The case/= 3 is excluded 
since it gives e = 0. For/ =5, g = 2, h = S, we again get (a) and also (j3). 
For/=5, ^ = 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 ff into two factors, Af , N, such that 
M+bff N+bff 
are integers and primes, while in r+1 = {p+l){q+l)/jf, 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 = 3^-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-\-\ = kn. For m = \, n = 3, a = 4 or 8, no amicables are found. 
Form = 3, n = l, the cases a =10, A: = 8 and a = 3^-5, ^' = 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 
JoM 6 = m:n, where m and n are relatively prime. Since(p-fl)j a = (54-l)j6, 
we may set p-\-\=^nx, q-{-\=mx. The usual second condition gives 
r r / K 7 / ^ — nxfa 
nx\a'{z = za{nx — l)-j-zb{mx — l), C —-, 1^ r* 
■^ -^ j^ {na-\-mb)x — a — b 
Let the latter fraction in its lowest terms be r/s. Then z = kr, jz = k$. 
Since f{kr)'^kCr, we have s'^ff. Hence we have the useful theorem: 
if z:Cz = r':s', s'<\r', then r' and s' have a common factor > 1. 
(5i) The unfruitful case a = 3, 6 = 1, was treated like the next. 
(52) Let = 5, 6=1, whence w = 6, n = l, 2:J z = 6a::llx — 6. By the 
theorem in (5), x must be divisible by 2 or 3. Euler treated the cases 
x = 3(3^+1), x = 2{2t+\). But this classification is both incomplete and 
