On the Analysis of Square Kumheri. 89 



S. If A2=B=+CS then A*=D2+E=^=F2+G='; that is, if 

 any square number contain the sum of two squares, the square of 

 that number will contain the sum of two squares in two different 

 ways. 

 Demonstration a^ = {h^ -\'C''Y =[h^ - c^Y -\-{2bcY =a^h'' -\-a'' c^ . 



3. If A* and B* contain each, the sum of two squares in two 



different ways, then A*- B'* will contain the sum of two squares in 



twelve different ways. 



Demonstration. Fnl Pi.'^=a^-\-h^=c^-\-d^ 

 and B*=m2-|-w2=?/3_{_^2. 



then A*-B* = (m2 4-w2)-(a='+6^)=a2+g2^72_j_5i 



» =(3/^+2;"Hc2 4-<Z')=v2+p=o2+7r2 jmakingin 

 all eight sets of squares, deducible from the general expressions [A] 

 and [B]. In addition to these, we may derive four sets more from 

 the simple multiplication of the expressions representing the value 

 of A*-B^ 



F0r(m2 4-n2)-(a2_f.^,2)_^2(„2_^j2)_]_^2(^3^52J=^2A4_}.^2A4^ 



(m^ -f W2)-(a2 +62)_gj2(^24-;j2^_^52(;;^3_|_^2)=a2B* + ^2B«, 

 (?/2+2r2 )-(ca+j2 )=y2(c2_j_^2)^_2.2(c2^^2)=2^2A4+^2A4j 



(V+'2'')'(c'+^')=c2(2/'+2;')+t?^(y2+«=')=c='B^+fZ2B*. 



Q. E. D. 



Remark.— The product A * -B * = ( A-B) * . It may be shown that 

 A and B each equal the sum of two squares when A^ and B^ each 

 equal the sum of two squares. Therefore the fourth power of the 

 product of any two numbers that are prime to each other, and that 

 consist each of the sum of two squares, may be resolved, into the 

 sum of two squares in twelve different ways. The smallest number 

 that can be thus resolved is (5.13)* =65*. In the same way it may 

 be shown that the expression (A'B)% or the number 65® may bo 

 resolved into the sum of two squares in eighty four different ways. 



4. A convenient method for finding two squares whose sum shall 

 be a square, is the following. Let a^ — b^=c^. For c^ put any 

 square number whatever; then, by the common rule, representing by 



m and n, any unequal factors of c^, we have a= -^ — » and 6= ~"o~« 



Putting for c^ any square number a^^^y^ where a,!?,/, represent any 

 prime factors, we have 2a=a2§3y2_|.i_a2g2j/_|.j/=:cc2g2_j_y2_j. 



Vol. XXV.— No. 1. 12 



