42 History of the Theory of Numbers. [Chap, i 
then A:a = S:5; he took A = 3B, S^B, 3^5, but found no solution. For 
p = 5, Q = 41, we have 72 = 251, 38A = 21a; set A = 495, whence 3-576 = 
38-75, where b is the sum of the divisors of B; set B = 9C, whence C:c = 
13:14, C=13, yielding the amicable numbers 5-41A., 251A, where 
A = 3^-7213 = 5733 [the pair VII in Euler's^'^^ ijgt and (7) in the table below]. 
Again, to make A/a = 3/8, set A = 35, whence a = 46 and the condition is 
6 = 25, whence 5 is a perfect number prime to 3. Using 5 = 28, we get 
A = 84. For use in such questions, Kraft gave a table of the sum of the 
divisors of each number^ 150. He quoted the rule of Descartes. 
L. Euler^*^^ obtained, in addition to two special pairs, 62 pairs [including 
two false pairs] of amicable numbers of the type am, an, in which the 
common factor a is relatively prime to both vi and n. He wrote jm for 
the sum of all the divisors of m. The conditions are therefore 
jm=jn, fa-jm = a{m-{-n). 
If m and n are both primes, then 7n = n and we have a repeated perfect 
number. Euler treated five problems. 
(1) Euler's problem 1 is to find amicable numbers apq, ar, where p, q, r, 
are distinct primes not dividing the given number a. From the first con- 
dition we have r = xy — \, where x — p-\-\, y = q-\-\. From the second, 
xyi a = a{2xy —x — y). 
Let a/{2a— \a) equal 6/c, a fraction in its lowest terms. Then 
y = bx/{cx — b), {cx — b){cy — b)=b^. 
Thus x and y are to be found by expressing 6^ as a product of two factors, 
increasing each by 6, and dividing the results by c. 
(li) Fu-st, takea = 2". Then6 = 2^ c=l, x, 2/ = 2"**+2^ Letri-A; = m. 
Then 
p = 2"»(22*=+2*)-l, g = 2'"(l+2*)-l, r = 2^"'(2^''+^+2^''+2'')-l. 
When these three are primes, 2"*"^^'^^ and 2'"''"^ are amicable. Euler noted 
that the rule communicated by Descartes to van Schooten is obtained by 
taking A:= 1, and stated that 1, 3, 6 are the only values ^ 8 of m which yield 
amicable numbers (above^^^). For k = 2 or 4, Euler remarked that r is 
divisible by 3; for k = 3, vi<Q, and for k = 5, mS2, p, q, or r is composite. 
(I2) Take a = 2J, where /=2"+^+e is a prune. Then 2a-fa = e+l. 
If e+1 divides a, we have c = l. Set e+ 1 = 2^, A;^?n, n = m+A:. Then 
/=2*(2'"+^ + l)-l, a = 2"'+i, 6 = 27, b""={x-b){y-b). 
For k = l, /=2'"+2+l is to be a prime, whence m+2 is a power of 2. 
If w = 0, 6=/=5, and either x = y, p = q; or x, y = Q,30; p, q = 5, 29, whereas 
p and q are to be distinct and prime to 10. If m = 2, /=17, 68^ is to be 
resolved into distinct even factors; in the four resulting cases, p, q, r are 
'"De numeris amicabilibus, Opuscula varii argumenti, 2, 1750, 23-107, Berlin; Comm. Arith., 1, 
1849, 102-145. French transl. in Sphinx-Oedipe, Nancy, 1, 1906-7, Supplement 
I-LXXVI. 
