CHAPTER VII. 



SUM OF THREE SQUARES. 



Diophantus V, 14 relates to the division of unity into three parts such 

 that if the same given number a be added to each part the sums will be 

 squares. This problem is equivalent to the determination of three squares, 

 each > a, whose sum is 3a + 1. Diophantus stated that a must not be 

 of the form SI + 2. 



C. G. Bachet 1 stated that this condition is not sufficient and gave as a 

 sufficient condition that a must not be of the form 8k + 2 or 32k + 9, 

 stating that he had tested the numbers a < 325. He also divided 5 into 

 three parts such that each increased by 3 is a square; since 



3-3 + 5 = 1 + 2 2 + 3 2 , 



he took the sides of the squares to be 1 + 7N, 2 + N, 3 5N, whence 

 N = 4/25. 



Fermat 2 remarked that Bachet's condition fails to exclude a = 37, 149, 

 etc., and himself gave the correct sufficient condition that a must not be of 

 one of the forms 



8& + 2, 4-S& + 2-4 + 1, 4 2 -8fc + 2-4 2 + 4 + 1, 



4 3 -8fc + 2-4 3 + 4 2 + 4 + 1, 

 [Thus a must not equal 



4*-8& + 2-4" + (4" - l)/3 = [(24& + 7)4" - l]/3, 



so that 3a + 1 must not be of the form (24fc + 7)4" and hence not 

 (8m + 7)4 n , since m is a multiple of 3 if 3a + 1 is of the latter form.] 



Regiomontanus 3 (Johannes Miiller, 1436-1476) proposed in a letter the 

 problem of solving the pair of equations 



x + y + z = 116, x* + y 2 + z 2 = 4624 = 68 2 . 



Fermat 4 stated that no integer 8k + 7 is the sum of three rational 

 squares. Descartes 5 proved this for integral squares by noting that a 

 square is of one of the forms 4k or 8k + 1. 



Fermat 6 treated the problem to find two numbers each of which, as 

 well as their sum, is composed of three squares only [not composed of one 

 or two squares]. He took any such number, as 11, and multiplied it by 

 two squares whose sum is a square, for example, 9 and 16. The problem 

 was proposed by Sainte-Croix to Descartes in April, 1638, with the illustra- 



1 Diophanti Alex. Arith., 1621, 310-3. 



2 Oeuvres, I, 314-5; French transl., III, 257-8. 



3 C. T. de Murr, Memorabilia Bibl., 1, 1786, 145. 



4 Oeuvres, II, 66; III, 287; letter to Mersenne, Sept. or Oct., 1636. The latter communi- 



cated it to Descartes. 



6 Oeuvres, II, 92; letter from Descartes to Mersenne, March 31, 1638. See also p. 195. 

 6 Oeuvres, II, 29, 57; letters to Mersenne, July 15 and Sept. 2, 1636. 



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