4 REPORT — 1855. 



X- + Ay"=w" we can obtain another solution of the same equations, in larger 

 integers, by only taking new X=Ay'^ — x* and new Y^2xyzw." We shall 

 give here only a few instances of the use of this most important theorem. 



When K-=7 , then the proposed equations ■T^+y-=n=z'" and x'' +1 y^=U-=-w^ 

 are obviously fulfilled by a7=3, y=4, 2'=5, and w=ll ; whence for a second 

 solution we have only to take new x= 7 X 4^— 3'*=1711andnewy=2a:y£rz/;=1320, 

 giving .*. new 2'=2161 and new «;=38S9 ; and thence again a third set of 

 answers are new x=1 x 1320''— lyil'* and 



weM;y = 2x 1711x1320x2161x3889. 



When A=10, one solution is obviously a;'=3 and y = 4, from which new 

 solutions can be obtained as above. When A= 11, then taking » = 5, a pos- 

 sible remainder of squares to modulus 11, the aM,rj7iary equations a;"+y-=5z^ 

 and a;^+ ll3/^=:5w" are obviously fulfilled bya;=l, 3/ = 2, z=i\, and j<;=3 ; 

 whence by our general theorem we have X = a?V — yW=35 and Y = 2xyzw 

 =:I2, which are the least values of x and y to answer the proposed equations 

 x" + y^=n=z- and a:'"+lly"^=n = w^, giving z=37 and w = 53 ; and thence 

 again another set of answers are r\Q-w x=-\\y*—x*, .•.==1272529 and new 

 y=2xyzio—10-K 12 X 37 x 53 = 1647240, and thence again new X=lly''—a;'' 

 = 11 X 1647240^- 1272529\&c. 



When A = 4, tlie proposed equations {a^-\-y^^=0 and a?* + 4y^=n) are 

 proved to be impossible, whence by taking a=:6=— 2 and n= — 1, it follows 

 from the foregoing general theorem that the auxiliary equations ly" — x~-=z" 

 and 2cr — y- = ?y^ must be also impossible, »'. e. there cannot be four square 

 numbers, lo", x?, y^, z^, in arithmetical progression. 



Chapter III. treats of the possible and impossible cases of the two simul- 

 taneous equations .r— y^=D and .r^ — Ay" = n. In the paper, of which we 

 here present a very short abstract, this is rigorously demonstrated to be im- 

 possihle when A is any integer < 13, except 7 or 11 ; the solutions of the 

 possible cases in integers x, y prime to each other are obtained with great 

 facility and generality from the following new and important — 



General Theorem. — The values of X and Y to fulfil X" — Y'^ = n:=Z^ and 

 X^ — a6Y-=n=:W- can be got from the solution of the auxiliary equations 

 x^ — ay'^^nz" and bx" — y'=^nv?, since in fact y>.-=^x'vr-\-y'z" and Y ■=^1xyzvi) 

 will answer the purpose, as is easily demonstrated. 



By taking S= 1, and interchanging z and lo in this general theorem, we see 

 that the solution of X' — Y"=^7i^ and X^— aY'=W" can be obtained from the 

 solution of x^ — y'^-=znz''' and x^ — ay'^^mv^ merely by taking X = a;"z" + yV 

 and Y=^2xyzw. And then again, by taking n=l, this general theorem shows 

 how to find a solution in great integers from a known solution in smaller 

 integers of x^—y'^^=s" and x'—ay-=.u;^ ; for then new X=:x'z- + yV=^x* — ay* 

 and new Y = 2xyzv> in all cases. 



Ex.gr. Let a:='l , so that the two equations to be solved are a?^ — y-=n=:2* 

 and .r^ — 7y^=n = w'''; then taking n = 2, a possible remainder of square 

 numbers to divisor 7, we see that one obvious solution of the two auxiliary 

 equations ar — y"='2z' and x' — ly^=.2vr is ^=3, y=l, r = 2, and m;=1 ; and 

 .•. by the foregoing X=:(;'V-f-y-w" = 37 and y^=^2xyzw^\2, which are the 

 least integers to answer the two proposed equations; they give £r=35 and 

 w=\Q ; and from this solution we find another, as indicated above, viz. new 

 X=^''— 0^=37' — 7. 12^=172900!) and new 



Y=2xyzw=Z1 X 24 X 35 X 19 = 590520. 

 And now using these values of X and Y for x and y, we thence get another 

 solution by the same formulae, viz. 



new X=a:' — a/= 1729009'' — 7 . 590520^=&c. 



