History Of The Theory Of Numbers - I

56 HISTORY OF THE THEORY OF NUMBERS. [CHAP. 11
Wallis62 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's51 and
<rj(7.11.29.163-191439)2} = I3.7-13-19-31-67}3.
Jacques Ozanam53 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.
Ozanam6* noted that the sum of 961 = 312 and its aliquot parts 1 and 31 is 993, which equals the sum of the aliquot parts of 1156 == 342. As examples of two squares with equal total sums of divisors [Wallis7 problem (Hi)], he cited 16 and 25, 3262 and 4072, while others may be derived by multiplying these by an odd square not divisible by 5. The sum of all the divisors of 92 is II2, that of 202 is 312. 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. Lucas65 noted that the problem to find all integral solutions of
(1)
is equivalent to the solution of the system
(2) l-fz=2w2, l+x2 = 2y2, y = 2uv,
and stated that the complete solution is given by that of 2t^— aj2= 1. E. Gerono66 proved that the only solutions of (1) are
(x, y)-(-l, 0), (0, ±1), (i, *2), (7, *20).
E. Lucas67 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 2732527- 13- 17-29. [This solution was given by Frenicle.46] For the case of a prime, the problem becomes (1).
A. S. Bang68 gave for problem (i) the first of the three answers by Wallis;47 for (w), o-(430982) = 17293; for (m), 29-67, 2.3-5-37 of Wallis48 and the first two by Frenicle;49 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 Fis., 12,
1879, 519. Reprinted in Oeuvrea de Fermat, 4, 1912, p. 140.
"Recreations Mathematiques 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. BiU Storia Sc. Mat. e Fis., 10, 1877, 287. "Nyt Tidsskrift for Mat., 1878, 107-8; on problems in 1877, 180.