INDETERMINATE PROBLEMS. 197 



Suppose a = 22, b zz 3, and m = 2, then a? rz +6+I X 22 



== 14, and y = -~ ^~-X2 2 = 6. But 196+36+252 = 484 



Cor. If b zz j, the hypothecs will be x*-\-y 2 -{-xy rz a* ; and 



* = w »+ M +i X "> and ^ = ;<4~?I X *" Thus > lf * = J 3» and OT 



= 3, then x = -j^rX 13 =7, and y = ~^T X 13 = 8. 

 But 49+64+56 = 169 = (13) 2 . 



PROBLEM IV. 



To find two numbers •, y#c#, ^><atf #wA, increafed by unit, jhall be a 

 fquare, and their fum, increafed by unit, a given fquare. 



Let the numbers be denoted by x* — 1 and ^ y % — 1, and 

 the firfl: condition will be obferved. The laft requires, that 

 x* — i+j 2 — 1 + 1, or ff*+y* — 1 — #*• By tranfpofition, x % — 1 

 zz a 1 — j/% and by refolution, (x-\-i)(x — 1) zz (a-\-y){a— y) ; 

 whence x+i zz ma — my t and mx — m zz a-\-y. Tranfpoling the 

 firfl equation, x zz ma — my — 1 ; and tranfpoling the fecond, mx zz. 



a-\-y-\-m, and dividing, x — ; whence zz 



ma — my — 1, and reducing, a-\-y-\-m 1= itfa — m % y — m, or 



a — 2m- 



y+7 



my-{-y zz m*a — 2m — a, and therefore y zz tt . But 



a+y+m m* +2ma — I 



x = m. > whence * = — q^— 



Suppose 



