CHAPTER VIII. 



SUM OF FOUR SQUARES. 



Diophantus, IV, 31 [32], desired four numbers x { such that the sum of 

 their squares increased [diminished] by the sum of the numbers is a given 

 number n. He took n = 12 [n = 4]. Since x 2 x + \ is a square, 

 2x] #; + 1 is the sum of four squares, here 13 [5]. Hence we have to 

 divide 13 [5] into four squares and subtract \ from [add \ to] each of their 

 sides to obtain the sides of the required squares. Since 



rn is 2i 17] 



.10' 10' 10' 10J* 



the sides of the required squares are 

 11 _7_ 19 13 

 10' 10' 10' 10' 



G. Xylander 1 noted that if we take 1430 in place of 4 in the second prob- 

 lem, we get the solution 6 2 , IP, 21 2 , 30 2 . 



C. G. Bachet la remarked that Diophantus apparently assumed here 

 and occasionally in Book V that any number is either a square or the sum 

 of 2, 3 or 4 squares [Bachet's theorem], and added that he himself had 

 verified this proposition for all numbers up to 325 and would welcome a 

 proof; he gave decompositions into 4 or fewer squares of each number up 

 to 120. He mentioned the generalization of Diophantus IV, 31 to the 

 problem to find k numbers such that the sum of then- squares increased by 

 the sum of the numbers is a given number n. Thus n + k/4 is to be the sum 

 of k squares. Bachet stated that if k ^ 4 there is no condition. 



Fermat, in his comment quoted in Ch. I 36 , stated that he possessed a 

 proof that every number is a sum of four squares. In stating the theorem 

 elsewhere, Fermat 2 remarked that Diophantus seems to have known the 

 theorem. 



The reason for ascribing a knowledge of this theorem to Diophantus 

 lies in the fact that he made no mention of a condition on a number in order 

 that it be a sum of four squares, in the three cases IV, 31, 32 and V, 17, in 

 which he mentioned the subject, but that he gave necessary conditions for 

 representation as a sum of two or three squares (Chs. VI, VII). 



Diophantus, V, 17, sought to divide a given number into four parts 

 such that the sum of any three of the parts is a square. Thus three tunes 

 the sum of the four parts is the sum of four squares. Let the given number 



1 Diophanti Alexandria! Rerum Arith., . . . , G. Xylandro, Basileae, 1575, 104. 



10 Diophanti Alex. Arith., 1621, 241-2. 



2 Oeuvres, II, 65; III, 287; letter to Mersenne, Sept. or Oct., 1636; to be proposed for 



solution to Sainte-Croix. Mersenne communicated it to Descartes, March 22, 1638. 



The latter ascribed the theorem to St. Croix (Oeuvres de Descartes, II, 256). Fermat, 



Oeuvres, II, 403-4; III, 315, letter to Digby, June, 1658, proposed that Brouncker and 



Wallis seek a proof of the theorem. 



275 



