260 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vn 



tion 3, 11. In his reply, Descartes 7 gave a 2 + 2, b z + 2 (a and 6 odd); 

 he 8 took the interpretation that each required number and their sum shall 

 be the sum of three squares in one and but one way, and gave nine examples 

 including 



22 = 9 + 9 + 4, 35 = 25 + 9 + 1, 57 = 49 + 4 + 4. 



But Sainte-Croix desired that each be the sum of 3, but not of 4, squares. 



Fermat 9 asserted that the double of any prime Sn 1 is the sum of 

 three squares; he desired that Brouncker and Wallis seek a proof. Refer- 

 ence will be made under the subject of binary quadratic forms to the 

 assertion of Fermat and proof by Lagrange that any prime Sh + 1 or Sh + 3 

 is expressible in one and but one way as the sum of a square and double of a 

 square. 



The Japanese Matsunago 10 in the first half of the eighteenth century 

 solved x 2 + y 2 + z 2 = u 2 by taking x and y at pleasure, expressing x 2 + y z 

 as a product of two factors and equating the latter to u z and u + z. 

 He noted that x 2 + y 2 + z 2 = u* has the solutions 



x = m 4 n*, y &m 2 n 2 , z = 2(m 2 n 2 )mn, u = m 2 + n 2 . 



L. Euler 11 noted that if Fermat's theorem that every number x is a sum 

 of three triangular numbers (a 2 -f a)/2 is true, then every number Sx + 3 

 is a sum of three squares (2a + I) 2 . 



Euler 12 noted that, to prove that a prime 8m + 3 is of the form 2a 2 + 6 2 , 

 one needs the theorems (of which he had no proofs) : If the integer n is not 

 a sum of two integral squares, then no integer np 2 is a sum of two integral 

 squares; if n is not a sum of three integral squares, it is not a sum of three 

 fractional squares. 



May 6, 1747 (p. 414), Euler wrote that he had verified for small integers 

 m that there always exists a triangular number A = (x 2 + z)/2 such that 

 4(m - - A) + 1 is a prime. If this be true, set n = m A; then 4n + 1 is 

 a E] and 2(4n + 1) is a 121. Set a = 2x + 1. Then n = m A gives 

 8m + 1 = Sn + a 2 . Hence Sm + 3 = 2(4n + 1) + a 2 is a SJ. On pp. 

 442-5, Euler and Chr. Goldbach discussed without result the problem to 

 express 8m + 3 as a EG. June 25, 1748 (pp. 458-460), Euler expressed his 

 belief that any number 4n + 1 or 4n + 2 is a SL The latter would give 



4n + 2 = (2a) 2 + (26 + I) 2 + (2c + I) 2 , 2n + 1 = 2a 2 + (2e) 2 + (2d+l) 2 , 



for b = d + e, c = d e, whence any odd number is of the form 

 2z 2 + ?/ + z\ 



March 24, 1750 (p. 512), Goldbach gave the identity 



/3 2 + 7 2 + (35 - ft - 7 ) 2 = (25 - 0) 2 + (25 - T ) 2 + (5 - ~ T) 2 - 



7 Oeuvres, II, 167; letter to Mersenne, June 3, 1638; Oeuvres de Fermat, 4, 1912, 57. 



8 Oeuvres de Descartes, II, 180-2. 



9 Oeuvres, II, 405; III, 318; letter to K. Digby, June, 1658. 



10 Y. Mikami, Abh. Geschichte Math. Wiss., 30, 1912, 233. 



"Corresp. Math. Phys. (ed., Fuss), 1, 1843, 45; letter to Goldbach, Oct. 17, 1730. 

 /Wd., 263; Oct. 15, 1743. 



