1.Introduction

If there is the solution of the simultaneous equation {x^2+y^2=u^2 , x^2+ny^2=v^2}, 
n is called as concordant number.
According to Dickson's book[1], Diophantus treated the simultaneous equation {x^2+m=u^2 , x^2+n=v^2} given the sum m+n.
Euler treated the the simultaneous equation {x^2+y^2=u^2 , x^2+ny^2=v^2}.
We show when n=s(9s+8), simultaneous equation {x^2+y^2=u^2 , x^2+ny^2=v^2} has infinitely many integer solutions.
     
2.Theorem
     
If n=s(9s+8) then, system of equation x^2+y^2=u^2 and x^2+ny^2=v^2 has infinitely many integer solutions.
s is arbitrary.
 
Proof.

x^2+y^2=u^2....................................................................................(1)
x^2+ny^2=v^2...................................................................................(2)

From equation (1), substitute x=2pq and y=p^2-q^2 to equation (2).

Let U=p/q and V=v/q, we obain

V^2 = nU^4+(4-2n)U^2+n.........................................................................(3)

Let n = -16/9+1/9t^2, then equation (3) has a solution (U,V)=Q(2,t).

Hence this quartic equation (3) is birationally equivalent to an elliptic curve below.

Y^2+8/3(-10+t^2)YX/t+(-256/9t+16/9t^3)Y 
= X^3+2/9(2t^2+3t^4-800)X^2/(t^2)+(64/9t^2-4/9t^4)X+6656/81t^2+368/81t^4-8/27t^6-102400/81.....(4)

Transformation is given, 
U = 1/9(18t^2X+8t^2+12t^4-3200+18Yt)/(Yt)
V = 1/81(108t^8X-384000Yt-38368t^4X+102400t^4+5120000X+39360Yt^3+24t^10-25600t^2X-6656t^6-368t^8-960t^5Y+81t^4X^3+108t^4X^2+162t^6X^2-43200X^2t^2-480t^6X)/(Y^2t^3)
X = (6tV-10t^2-80U+160+8t^2U)/(3U^2-12U+12)
Y = (36t^3V-12t^4-512t^2U+992t^2+8t^2U^2+12t^4U^2-3200U^2+12800U-12800)/(9tU^3-54tU^2+108tU-72t).

The point corresponding to point Q is P(X,Y)=(-2/9(2t^2+3t^4-800)/(t^2), 320/27(-41t^2+t^4+400)/(t^3)).

This point P is of infinite order, and the multiples mP, m = 2, 3, ...give infinitely many points.

Hence we can obtain infinitely many integer solutions for equation (3).

Substitute t=9s+4 to n = -16/9+1/9t^2, then we obtain n=s(9s+8).

When m=2 we obtain a parametric solution
(x,y,u,v)=(-4*(1701*s^2+1512*s-64)*(3159*s^2+2808*s+1024),
           9*(3*s+8)*(9*s-16)*(81*s+16)*(81*s+56),
           21552885*s^4+38316240*s^3+22628160*s^2+4976640*s+1064960,
           (9*s+4)*(531441*s^4+944784*s^3+3286656*s^2+2548224*s+65536)).
           
Similarly, when m=3 we obtain 
(x,y,u,v)=(4*(2625849981*s^6+7002266616*s^5+3048503040*s^4-3801530880*s^3-3254722560*s^2-662667264*s-67108864)
            *(23613301440*s^4+23688184320*s^3+11227299840*s^2+2066743296*s-4194304+5208653241*s^6+13889741976*s^5),
           3*(59049*s^3+77112*s^2+29952*s+11264)*(59049*s^3+80352*s^2+32832*s-4096)*(6561*s^3-122472*s^2-145152*s-31744)
           *(6561*s^3+139968*s^2+88128*s+4096),
           54710421075850605525*s^12+291788912404536562800*s^11+699078174555680222400*s^10+993645854683746048000*s^9
           +1088439280059483648000*s^8+1163175140747324620800*s^7+1088625846032931225600*s^6+708460413272771788800*s^5
           +284658394944503808000*s^4+65504498562367488000*s^3+7681123914507878400*s^2+338429679029452800*s+18031990695526400,
           (9*s+4)*(150094635296999121*s^12+800504721583995312*s^11+63495473995087226496*s^10+276404212365046801920*s^9
           +523605733954561105920*s^8+556845828536088133632*s^7+363893003470253850624*s^6+150861212002918858752*s^5
           +40873930132889272320*s^4+8607298586431979520*s^3+1711260209340481536*s^2+151117015560486912*s+281474976710656)).
           
Q.E.D.

3.Example

n=17:
(x,y,u,v)= (88058636, 9209277, 88538885, 95896333)
(x,y,u,v)= (1558919204820754667284, 6254781552139417477437, 6446124521923428875525, 25836199364300466735373)
      

4.Reference

[1].L.E. Dickson:History of theory of numbers, vol 2.