Existence of solution to $3a^2 = b^2 + c^2 + d^2$ with constraints.

Question

I am searching for solutions to the following equation: $$ 3a^2 = b^2 + c^2 + d^2 \tag{1} $$ where $a,b,c,d$ are distinct positive integers, satisfying $$ ac = bd.\tag{2} $$

I have found solutions to $(1)$ via a search (over odd $a$) under the constraint $abcd$ is a square, which is weaker than $(2)$. None of these has satisfied $(2)$ however. As an example the smallest solution I found is $$ (a,b,c,d) = (637, 361, 481, 925) = (7^2\cdot13, 19^2, 13\cdot 37, 5^2\cdot 37) $$ which doesn't satisfy $(2)$.

My apologies for the vagueness of this question, but I just wondered if anyone could offer some insight on this problem? Perhaps there is a simple proof of impossibility, or a restriction on the variables that means the magnitude of any possible solution must be beyond the scope of my laptop.

I welcome anybody's thoughts on this!

A few of my own:

• By Legendre's three square theorem, solutions to $(1)$ usually exist (always when $a$ is odd), and there are often many solutions, so the problem doesn't feel too restrictive.
• Clearly if $(a,b,c,d)$ is a solution, then so is $(ka, kb, kc, kd)$. The solutions I found went beyond this extension.

Answer 1

There are no integer solutions to $$ 3\,a^2 = b^2 + c^2 + d^2 \tag{1} $$ and $$ a\,c = b\,d \tag{2} $$ except $\,|a|\!=\!|b|\!=\!|c|\!=\!|d|.\,$ For equation $(2)$ we have a general solution $$ a = y\,w,\;\; b = x\,y,\;\; c = x\,z,\;\; d = w\,z. \tag{3} $$ Substituting $(3)$ into $(1)$ we simplify to get $$ (w^2+x^2)\,z^2 = (3\,w^2-x^2)\,y^2. \tag{4} $$ The solution to $(4)$ is $$ w^2+x^2 = t\,y^2, \quad 3\,w^2-x^2 = t\,z^2. \tag{5} $$ This corresponds to the LMFDB elliptic curve 24.a4 $$ y^2 = x^3 - x^2 - 4x + 4 = (x-1)(x-2)(x+2) \tag{6} $$ with rank $0$ and hence has no rational generator. The only rational solutions are $$ (-2,0),\; (0,\pm2),\; (1,0),\; (2,0),\; (4,\pm6). \tag{7} $$ The point $\,(0,2)\,$ of order $4$ and $\,(1,0)\,$ of order 2 generate a subgroup of order $8$.

NOTE: The connection between equaton $(5)$ and equation $(6)$ is contained in my WXYZ Project. Briefly, define the constants $$ w_0 = 0, x_0 = y_0 = z_0 = w_1 = 1, \\ x_1 = 2, y_1 = \sqrt{5}, z_1 = \sqrt{-1}. \tag{8} $$ This determines the four sequences $\,\{w_n, x_n, y_n, z_n\} \,$ which satisfy $$ w_n^2+x_n^2=y_n^2,\quad 3\,w_n^2-x_n^2=z_n^2 \tag{9} $$ corresponding to equation $(5)$. The sequence $\,\{w_n\}\,$ is an elliptic divisibility sequence with $j$-invariant $35152/9$ which can be looked up in LMFDB and the first curve with that $j$-invariant is equation $(6)$.

NOTE: Both equation $(1)$ and equation $(2)$ are homogeneous quadrics. The general solution to them involves an elliptic curve. There should be references to such a transformation but I don't have one at hand now.