252 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI 



H. Schubert 146 noted that, if in x 2 + y 2 = V? + z 2 the unknowns have 

 no common factor, either all four are odd or in each member one number 

 is odd and one even. In the first case, 



%(x + z) -%(x - z) = %(u + y)-%(u - y), 



whence we must factor an arbitrary number g in two ways with always 

 one factor even and the other odd. In the second case, g must be a product 

 of two even factors and also a product of an even and an odd factor. 



R. E. Moritz 146a proved that every rational number not a square can 

 be expressed in an infinitude of ways as a quotient of two sums or two 

 differences of two squares, and gave one such expression for each such 

 number < 100. 



A. Palmstrom 147 noted that x* = y 2 + z 2 implies x = a 2 + b 2 , whence 

 y = a 3 + ab 2 or a 3 3ab 2 [provided y and z are relatively prime]. P. F. 

 Teilhet 148 obtained all the solutions. 



A. Thue 149 proved that a prime divisor of a E] is a E] . 



Several 149 " found three consecutive integers each a El, including 

 (2n) 2 + (2n) 2 , Sn 2 + 1, (2n - I) 2 + (2n + I) 2 , provided the second be a El, 

 i. e., n be triangular, n = (m 2 + m)/2. 



L. E. Dickson 150 proved that all factors of a sum of two relatively prime 

 squares are sums of two squares by use of the theorem that if a and b are 

 relatively prime every prime divisor of a 2 + b 2 is of the form 4n + 1 and 

 the theorem that every prime 4n + 1 is a sum of squares of two relatively 

 prime integers. 



G. Fontene* 151 proved Gauss' 37 theorem by showing that, it A, B, 

 are primes 4h + 1, there is a (1, 1) correspondence between the decomposi- 

 tions of A a B ft as a product of two factors and its decomposition into a 

 sum of two squares, provided we fix the order of the two squares whose sum 

 is A, or B, etc. 



A. Cunningham 152 expressed each prime 4n + 1 < 100000 as a El . 



P. Pasternak 153 proved that all solutions of x 2 + y 2 = v 2 + w 2 are 



x = raco + np, v = mu np, y = nu mp, w = nu + mp, 

 whence 



x 2 + y 2 = (m 2 + n 2 )(w 2 + p 2 ). 



Thus every integer which can be expressed as a E] in more than one way is 

 itself a product of two sums of two squares. From known theorems it is 

 said to now follow that no prime 4n + 1 is a El in more than one way. 



""Niedere Anal., 1, 1902, 167-171; ed. 2, 1908. 



1460 Ueber Continuanten . . ., Diss. Strassburg, Gottingen, 1902. Cf. Moritz 40 of Ch. IX. 



147 L'interm6diaire des math., 8, 1901, 302. 



148 Ibid., 10, 1903, 210-1. 



149 Oversigt D. Viden. Selsk. Forh., Kristiania, 1902, No. 7. 

 149a Math. Quest. Educ. Times, (2), 3, 1903, 41-3. 



160 Amer. Math. Monthly, 10, 1903, 23. 



U1 Nouv. Ann. Math., (4), 3, 1903, 108-115. 



162 Quadratic Partitions, London, 1904. Errata, Mess. Math., 34, 1904-5, 132. 



163 Zeitschr. Math. Naturw. Unterricht, 37, 1906, 33-35. 



