56 History of the Theory of Numbers. [Chap, ii 
Wallis^'^ for use in problem (ii) gave a table showing the sum of the 
divisors of the square of each number < 500. Excluding numbers in whose 
divisor sum occurs a prime entering the table only once or twice, there are 
left the squares of 2, 4, 8, 3, 5, 7, 11, 19, 29, 37, 67, 107, 163, 191, 263, 439, 499. 
By a very long process of exclusion he found only two solutions within the 
limits of the table, viz., Frenicle's" and 
(rj(7.11-29.163-191439)2[ = ]3.7-13-19-31-67(^ 
Jacques Ozanam" stated that Fermat had proposed the problem to find 
a square which with its aliquot parts makes a square (giving 81 as the answer) 
and the problem to find a square whose aliquot parts make a square. For 
the latter, Ozanam found 9 and 2401, whose aliquot parts make 4 and 
400, and remarked that he did not believe that Fermat ever solved these 
questions, although he proposed them as if he knew how. 
Ozanam^ noted that the sum of 961 = 31^ and its aliquot parts 1 and 31 
is 993, which equals the sum of the aliquot parts of 1 156 = 34". As examples 
of two squares with equal total sums of divisors [WaUis' problem (m)], he 
cited 16 and 25, 326^ and 407^, while others may be derived by multiplying 
these by an odd square not divisible by 5. The sum of all the divisors of 
9^ is 11^ that of 20^ is 31l The numbers 99 and 63 have the property that 
the sum 57 of the aliquot parts of 99 exceeds the sum 41 of the aliquot 
parts of 63 by the square 16; similarly for 325 and 175. 
E. Lucas^^ noted that the problem to find all integral solutions of 
(1) l-\-x+z^-\-x^ = y^ 
is equivalent to the solution of the system 
(2) l+x = 2w2, l+x2 = 2t;2, y = 2uv, 
and stated that the complete solution is given by that of 2y^— x^ = l. 
E. Gerono^^ proved that the only solutions of (1) are 
(x, 2/) = (-l, 0), (0, ±1), (1, ±2), (7, ±20). 
E. Lucas^^ stated that there is an infinitude of solutions of Fermat's 
problem (i); the least composite solution is the cube of 2-3-5-13'41-47, the 
sum of whose divisors is the square of 2^3^5^7- 13- 17-29. [This solution was 
given by Frenicle.^®] For the case of a prime, the problem becomes (1). 
A. S. Bang^^ gave for problem (i) the first of the three answers by Wallis;*' 
for (it), (7(43098^) = 1729^ for {in), 29-67, 2-3-5-37 of Wallis^* and the first 
two by Frenicle;^^ all without references. 
"A Treatise of Algebra, 1685, additional treatises, Ch. IV. 
"Letter to De Billy, Nov. 1, 1677, published by C. Henry, Bull. Bibl. Storia Sc. Mat. e Fia., 12, 
1879, 519. Reprinted in Oeuvres de Fermat, 4, 1912, p. 140. 
"Recreations Math6matiques et Phys., new ed., 1723, 1724, 1735, etc., Paris, I, 41-43. 
"Nouv. Corresp. Math., 2, 1876, 87-8. 
••Nouv. Ann. Math., (2), 16, 1877, 230-4. 
"Bull. Bibl. Storia Sc. Mat. e Fis., 10, 1877, 287. 
"Nyt Tidsskrift for Mat., 1878, 107-8; on problems in 1877, 180. 
