458 HISTORY OF THE THEORY OF NUMBERS [CHAP, xv 



E. Haentzschel, A. Korselt, and P. von Schaewen 84 treated the problem 

 to find 3 numbers in arithmetical progression the sum of any two of which 

 is a square (Diophantus III, 9). 



A. Gerardin 85 noted further cases of Euler's relation (2) : 



13920 2 =(7 4 -3 4 )(17 4 -1), 62985 2 =(14 4 -5 4 )(18 4 -1), 



3567 2 =(5 4 -4 4 )(21 4 -20 4 ), 2040 2 =(2 4 -l)(23 4 -7 4 ), 



7800 2 =(9 4 -7 4 )(ll 4 -2 4 ), 2308S0 2 =(17 4 -9 4 )(29 4 -ll 4 ). 

 He and A. Cunningham 86 noted solutions of 



P(x+y)+Qx=H, P(x+y)+Qy=H. 



E. Turriere 87 obtained a second solution from one of a+a'=D, 

 bx+b'=n. 



H. R. Katnick 88 noted that zn can be made squares if n is even. 

 M. Rignaux 89 gave Rolle's 18 solution in factored form, and also 



A = 



In terms of any given solution are expressed two new solutions. 

 On linear functions made squares, see Genocchi 44 of Ch. XIV. 



84 Jahresber. d. Deutschen Math.-Vereinigung, 24, 1915, 467-471; 25, 1916, 138-9, 139-145, 



351-9. 



85 L'intermediaire des math., 22, 1915, 230-1 (50-1). 



86 Ibid., 75, 233-5. 



87 L'enseignement math., 18, 1916, 423-4. 



88 Amer. Math. Monthly, 24, 1917, 339-40. 



89 L'intermediaire des math., 25, 1918, 129. 









