CHAP. XV] LINEAR FUNCTIONS MADE SQUARES. 457 



To find 75 three integers the difference of every two of which is a square. 

 Likewise 76 for four integers. To make 77 x+y+z, x+y, y+z, z+x all 

 squares. 



Several 78 solved 3x+ 1 = D , 7x+ 1 = D . 



A. Cunningham 79 found integers x t , , x r such that, if a given number 

 N be added to their sum s or to the sum of any r 1 of them, the results 

 are squares. Froms+A r = o- 2 , s x i +N = a- 2 i) we get x , -. = a 2 a\ (i = l, -,r). 

 Then the initial condition can be written 



(r-l^+N-al ----- al = a\ + +<7 2 4 . 



We may assign any values to <r, cr 5 , , a r such that the left member is 

 positive and hence a sum of four squares. 



A. Gerardin 80 treated the problem to find a number N which can be 

 separated into four parts such that the sum of any two parts is a square. 

 We need only use a number N which is a sum of two squares in three ways. 

 Or we may employ the formula for JV = (a 2 +6 2 )(m 2 +p 2 ) as a sum of two 

 squares and take m=f z g 2 , n = 2fg, whence 



P. von Schaewen 81 remarked that the triple equality (1) is not solvable 

 by the method of Fermat or by any known method and proved that there 

 is a solution x + if and only if a 2 (z 2 1) 2 +46 2 2 2 = D has a solution other than 

 2 = 0,2 = 1. For de Billy's case a = 2, b = 3, the condition is (z 2 l) 2 +9z 2 = D, 

 which has no rational solutions other than 2 = 0, 2 = 1, as proved by Euler 144 

 of Ch. XXII. Thus the triple equation has only the solution z = 0. 



E. Haentzschel 82 treated the following problem. Given e i} e 2 , e 3 , find 

 a rational number s such that s ei, s e 2 , s e 3 shall be rational squares. 

 Their product y 2 /4 must be a square. The relation 



y 2 = 4(s ei) (s 62) (s e 3 } 



is satisfied if s is Weierstrass' function $>(u) and v = g>'(u). Hence the 

 problem is to find a rational value of @(u) such that also &'(u) is rational. 

 The solution is effected by means of the relation between $>(2u) and &(u), 

 and shown to be equivalent to that by Euler 27 for his case of rational d, 

 e z , e 3 [cf. Haentzschel 156 of Ch. V]. Here is treated at length the case 

 e 1 =S; 62, e 3 = 43AP3. 



H. C. Pocklington 83 noted that the first, second, fifth and tenth terms 

 of an arithmetical progression are not all squares, unless the first is zero or 

 all are equal. 



75 Amer. Math. Monthly, 9, 1902, 113, 230. 



76 Ibid., 10, 1903, 206-7. 



Ibid., 141-3. 



78 Math. Quest. Educ. Times, 8, 1905, 79-80. 



"Ibid., (2), 9, 1906,30-1. 



80 Sphinx-Oedipe, 1907-8, 10-12. 



81 Bibliotheca Math., (3), 9, 1908-9, 289-300. 



82 Jahresbericht d. Deutschen Math.-Vereinigung, 22, 1913, 278-284. 



83 Proc. Cambridge Phil. Soc., 17, 1914, 117. 



