276 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vm 



be 10. Then 30 is to be divided into four squares each < 10. Since 

 30 = 16 + 9 + 4 + 1, we take 9 and 4 as two of the squares and divide 

 17 into two squares each < 10 [the squares of 1016/349 and 1019/349]. 

 If we subtract each of the resulting four squares from 10, we obtain the 

 required parts 1, 6, etc. In V, 16, the number 10 is divided into three 

 such parts. For a generalization to n parts, see Kausler 47 of Ch. XV. 



Regiomontanus 3 (J. Miiller) proposed in a letter the problems to find 

 four squares whose sum is a square and twenty squares whose sum is a 

 square > 300000. 



Jakob von Speyer 30 gave 



1 + 2 2 + 4 2 + 10 2 = II 2 , 2 2 + 4 2 + 7 2 + 10 2 = 13 2 . 



A. Girard, 4 in commenting on Diophantus V, 15, stated that there are 

 numbers, as 7, 15, 23, 28, 31, 39, not a sum of three squares, but that any 

 integer is a sum of four squares. 



R. Descartes 5 announced the theorem (" whose demonstration he judged 

 so difficult that he dared not undertake to find it ") : Any number which 

 is the sum of three squares and exceeds 41 can be expressed also as the sum 

 of four squares, excepting only the products of 6 or 14 by 4, 4 2 , 4 3 , 

 There are no other numbers which are not composed of four squares, except 

 2 -4 n , which is not a square, nor composed of three or four squares, but only 

 of two. 



Fermat 6 stated that he had much trouble in finding the new principles 

 needed to apply his method of infinite descent to show that every number 

 is a square or the sum of 2, 3 or 4 squares; but stated that he had finally 

 proved that if a given number is not of this nature there would exist a 

 smaller which is not. 



L. Euler 7 admitted that he could not prove Bachet's theorem that 

 every integer is a 12, nor give a general rule to express n z + 7 as a SO. 

 Oct. 17, 1730 (p. 45), he noted that, if Fermat's theorem that every integer 

 x is a sum of three triangular numbers (a 2 + a)/2 is true, then 8x + 3 is 

 the sum of the three squares (2a + I) 2 . Hence Sx + 4 and Sx + 7 are HI. 

 [Cf. Beguelin 75 of Ch. L] Since m*(Sx + 4) = W(2x + 1), it remains only 

 to prove that 4x + 2 is a 12. Oct. 15, 1743 (p. 263), Euler noted that, if 

 np 2 is a 3D, n is a sum of four integral squares. Thus if it be true that 

 8m + 3 is a GS, 8m + 4 is a 31 and also 2m + 1, so that every integer is a 12. 

 Ma'y 6, 1747 (p. 419), he stated that Bachet's theorem depends on the 

 unproved fact that every number 4m + 2 is the sum of two numbers 4x + 1 

 and 4y + 1, neither having a factor 4p 1 [and hence each a El]. For, 



C. T. de Murr, Memorabilia Bibl., 1, 1786, 160, 201. 



Ibid., 168. 



4 L'arith. de Simon Stevin . . . annotations par A. Girard, Leide, 1625, p. 626; Oeuvres 



math, de S. Stevin par A. Girard, 1634, p. 157. 

 B Oeuvres, 2, 1898, 256, 337-8, letters to Mersenne, July 27 and Aug. 23, 1638. The limit 



33 given in the first letter was changed to 41 in the second. 

 6 Oeuvres, II, 433, letter to Carcavi, communicated Aug. 14, 1659, to Huygens. 

 'Corresp. Math, et Phys. (ed., P. H. Fuss), St. Petersburg, 1, 1843, 24, 30, 35; letters to 



Goldbach, June 4, June 25, Aug. 30, 1730. 



