1.Introduction

A problem of square of each three numbers plus product of remaining two, i.e. is there a solution of simultaneous equation
{x^2+yz=u^2, y^2+xz=v^2, z^2+xy=w^2}. 
According to Dickson's book[1], L.Euler gave parametric solution below.
(x,y,z,u,v,w)=(8ts+5t^2, 8ts+13s^2, 64ts+20t^2+52s^2, 5t^2+24ts+26s^2, 10t^2+24ts+13s^2, 20t^2+65ts+52s^2)

We show simultaneous equation {x^2+yz=u^2, y^2+xz=v^2, z^2+xy=w^2} has infinitely many parametric solutions.
     
2.Theorem
     
A simultaneous equation {x^2+yz=u^2, y^2+xz=v^2, z^2+xy=w^2} has infinitely many parametric solutions.
 
Proof.

Let X=x/z, Y=y/z, U=u/z, V=v/z, W=w/z.

X^2+Y=U^2.......................................................................(1)

Y^2+X=V^2.......................................................................(2)

1+XY=W^2........................................................................(3)

From equation (3), substitute X = (W+1)/t and Y = Wt-t to equation (1) and (2).

From equation (1), we obtain a solution for (U,W) is below.

U = (s^2-2t+st^2)/(2s+t^2), W = (ts^2-2s+t^2)/(2s+t^2)..........................(4)

Substitute (4) to equation (2), then we obtain

V^2 = (t^4s^4-8t^3s^3+17t^2s^2+2s^3+4ts+2t^3)/((2s+t^2)^2)......................(5)

Let f^2=t^4s^4+(2-8t^3)s^3+17t^2s^2+4ts+2t^3....................................(6)

Quartic equation (6) has a solution (s,f)=Q(1/2, 1/4(t^2+8t+2)).

Hence this quartic equation (6) is birationally equivalent to an elliptic curve below.

Y^2+2(8t+34t^2+3-12t^3+t^4)YX/(t^2+8t+2)+(t^2+4t^5+t^6+8t-30t^4+2-8t^3)Y
= X^3+1/2(6+8t^2+t^8+608t^5-378t^4+144t^3+96t-570t^6+72t^7)X^2/((t^2+8t+2)^2)+(-1/4t^8-t^4-4t^7-17t^6-8t^5)X-3/4t^4-t^6-1/8t^12-76t^9+189/4t^8-18t^7-12t^5+285/4t^10-9t^11
Transformation is given, 

S = (6+8t^2+t^8+608t^5-378t^4+144t^3+96t+8X+4Y-570t^6+72t^7+136t^2X+64tX+2t^4X+32t^3X+2Yt^2+16Yt)/(4Yt^2+32Yt+8Y)

T = (64t^7X^3+19667/2t^12+11312t^4X^3+461064t^8X+3136t^2X^3+22368t^3X^2+1728X^2t-164832t^7X+1086888t^10X-18648t^6X^2+8960t^3X^3-81306t^8X^2+15546t^4X^2+1056t^5X^2+126844t^8
  +768t^5-3664t^11+24t^4+10536X^2t^2+52464t^7X^2+446286t^12X+512tX^3-362496t^9X+52120t^13+1950t^10X^2-617280t^11X-10752t^9X^2+4530t^14X-75504t^13X+3/2t^16X+168t^15X-159676t^15
  -420358t^14+2t^8X^3+3t^12X^2+255584t^9Y+264t^11X^2-542316t^8Y+784t^6X^3+4480t^5X^3-18713/2t^16-64t^13Y+2816t^12Y+8032t^11Y+3208t^17+1063/2t^18+26t^19+1/4t^20-244096t^7Y-126712t^6Y
  +72X^2+34808Yt^4+11904Yt^3+36X-8Y+32X^3-94712t^10Y+8528t^6-62208t^5X+132360t^6X+45440t^7+182752t^9+59632t^5Y+115396t^10+9312t^2X+1152tX+23208t^4X+3264t^3X+1004Yt^2-352Yt)
  /(8Y^2t^6+192Y^2t^5+1584Y^2t^4+4864Y^2t^3+3168Y^2t^2+768Y^2t+64Y^2)
  
X = (4t^2T-t^4+40t^3+32tT+16t+8T-2+32tS+136t^2S+12S-48t^3S+4t^4S)/(8S^2-8S+2)

Y = (-1+192t^2+192tT+792t^2T+4882t^4S+2976t^3S+1176t^2S+1656t^5+576t^4+680t^3+960t^6S-1680t^5S+396t^4T+2t^8S^2+12S^2+192tS^2+288t^3S^2-1140t^6S^2-756t^4S^2+144t^7S^2+16T+1216t^5S^2
  -136t^7S+12S+48t+t^6+64tS+16t^2S^2+1216t^3T+48t^7+2t^6T+48t^5T)/(8S^3t^2+64S^3t+16S^3-12t^2S^2-96tS^2-24S^2+6t^2S+48tS+12S-t^2-8t-2).


The point corresponding to point Q is P(X,Y)=(-1/2(6+8t^2+t^8+608t^5-378t^4+144t^3+96t-570t^6+72t^7)/((t^2+8t+2)^2), 2(1-286t^2-11586t^8+3736t^5+16t^11-2752t^4-364t^3+40t+2271t^6+19560t^7+4616t^9-832t^10)/((t^2+8t+2)^3)).

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

Hence we can obtain infinitely many parametric solutions for equation (5).

Consequently, a simultaneous equation {x^2+yz=u^2, y^2+xz=v^2, z^2+xy=w^2} has infinitely many parametric solutions.

Q.E.D.


3.Examples

When k=1 we obtain two parametric solutions below.

(x,y,z,u,v,w) = ( 1+8*t, t*(t-8), 4+4*t^2, 1-8*t+2*t^2, t^2+8*t+2, t-4+4*t^2)

(x,y,z,u,v,w) = ( -4*(t-2)*(t+2), -4*t^2-20+8*t, (t+3)*(3*t-7), -2*t^2-4*t+26, 2*t^2-18*t+8, -5*t^2+2*t+11)
t is arbitrary.

When k=2 we obtain a parametric solution,

x = (t^12+96*t^11+1920*t^10-16568*t^9+87936*t^8-204048*t^7+289022*t^6+91008*t^5+61824*t^4+9152*t^3+960*t^2+240*t+1)
   *(72*t^13-2303*t^12+18720*t^11-5184*t^10+48088*t^9+67968*t^8+70848*t^7+358718*t^6+342720*t^5+147456*t^4+32048*t^3+1728*t^2+216*t+1)
y = (t^12-240*t^11+960*t^10-9152*t^9+61824*t^8-91008*t^7+289022*t^6+204048*t^5+87936*t^4+16568*t^3+1920*t^2-96*t+1)
   *(t^13-216*t^12+1728*t^11-32048*t^10+147456*t^9-342720*t^8+358718*t^7-70848*t^6+67968*t^5-48088*t^4-5184*t^3-18720*t^2-2303*t-72)*t
z = 12*(t^6+48*t^5-189*t^4+536*t^3+189*t^2+48*t-1)*(3*t^8-48*t^7+8*t^6-120*t^5+9*t^4-608*t^3-186*t^2-48*t+1)*(t^2+1)*(t^2-16*t-1)
   *(t^8+48*t^7-186*t^6+608*t^5+9*t^4+120*t^3+8*t^2+48*t+3)
u = -(2*t^2-8*t+1)*(-1+832*t+35056*t^2+1614224*t^3+23033840*t^4+805744262*t^12+13289849536*t^11+18582069040*t^10+15970839728*t^9+8415011376*t^8
   +3253863552*t^7+883526592*t^6+181550208*t^5+3*t^24-576*t^23-23024*t^22+638896*t^21-5444848*t^20+68061440*t^19-311888648*t^18+1181851392*t^17
   -1999094064*t^16+3859948560*t^15-1679424048*t^14+8411036480*t^13)
v = (t^2+8*t+2)*(-3-576*t+23024*t^2+638896*t^3+5444848*t^4-805744262*t^12+8411036480*t^11+1679424048*t^10+3859948560*t^9+1999094064*t^8+1181851392*t^7
   +311888648*t^6+68061440*t^5+t^24+832*t^23-35056*t^22+1614224*t^21-23033840*t^20+181550208*t^19-883526592*t^18+3253863552*t^17-8415011376*t^16+15970839728*t^15
   -18582069040*t^14+13289849536*t^13)
w = -36+2305*t+32292*t^2-1884096*t^3-39658520*t^4+36*t^26+62641580328*t^12+62942511168*t^11+22625475400*t^10+5881305600*t^9-2785501584*t^8-2474757988*t^7
    -1112974704*t^6-260546688*t^5-32292*t^24-1884096*t^23+39658520*t^22-260546688*t^21+1112974704*t^20-2474757988*t^19+2785501584*t^18+5881305600*t^17
    -22625475400*t^16+62942511168*t^15-62641580328*t^14+57274524934*t^13+2305*t^25
t is arbitrary.


      

4.Reference

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