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April 17

Intersecting two Pell equations

Does the triplet (12, 7, 10) constitute the only positive integer solution to the system of Diophantine equations $\frac{x^2}{3}+1=y^2=\frac{z^2}{2}-1?$

• Observation: This is equivalent to locating the rational points found on the conical surface $2x^2-12y^2+3z^2=0.$
• Attempt: It is clear that x is a multiple of 3, and z is even, yielding $3a^2-2b^2+2c^2=0,$ from where it follows that a is also even, giving $6{\alpha }^2-{\beta }^2+{\gamma }^2=0,$ at which point I am all fresh out of ideas.

— 79.113.238.17 (talk) 12:09, 17 April 2018 (UTC)

This boils down to finding n so that 24 n² + 1 and 48 n² + 1 are both perfect squares. Letting p² = 48 n² + 1 and q² = 24 n² + 1, we have the ratio Template:Frac → Template:Sqrt as n → Template:Math, implying that p and q are to be found among the continued fraction approximations of Template:Sqrt. A computer aided search for continued fraction depth below 10⁴ (by which point p and q are both 3,828 digits long) reveals no other solutions, apart from the one already mentioned above. — 79.113.238.17 (talk) 19:02, 18 April 2018 (UTC)

• I fail to understand how you deduced your "observation", since you go from two equations to one (even if you allow rational rather than integer solutions). But assuming it holds...

Your last equation can be rewritten as $6{\alpha }^2=(\beta -\gamma )(\beta +\gamma )$, and we can solve it in integers (any solution in rationals is a solution in integers divided by some number). Both terms in the RHS product have the same parity. They cannot be odd because the LHS is divisible by 2. Hence both RHS terms are divisible by 2.

Define $2m=(\beta -\gamma ),2n=(\beta +\gamma )$. For a given α, integers m,n satisfy $3{\alpha }^2=2mn$ iff. integers $\beta =m+n,\gamma =m-n$ satisfy the original equation. There are infinitely many such triplets; there is no solution for odd α, and for even α, there is a finite number of solutions easy enough to enumerate depending on α's prime factorization. Tigraan^{Click here to contact me} 08:50, 18 April 2018 (UTC)

U = V = W is indeed different than 2 V = U + W, but it is clear that the solutions to the former are among those of the latter. — 79.113.238.17 (talk) 18:04, 18 April 2018 (UTC)

• Here's what I've been able to deduce so far. As x is divisible by 3 and z is divisible by 2, then x^2 = 9a^2, and z^2 = 4b^2, putting these in to the original equation gives us $3a^2+1=y^2$ and $2b^2-1=y^2$, which are much easier to deal with as there are no fractional terms.

From the 2nd equation, we know that y is odd, then we can rewrite y^2 using y^2=8c+1, so the 2 equations can be simplified to $3a^2=8c$ and $2b^2=8c+2$. The 1st equation now tells us that a is divisible by 4 so we can set a^2 = 16d^2. Putting this back in to our original equation gives us $48d^2+1=y^2$. From the 2nd equation, we get $b^2=4c+1$ and thus b must be odd, and as c is divisible by 6 (from simplifying $3a^2=8c$ to $6d^2=c$), b cannot be divisible by 3. We can rewrite this as $b^2=24d^2+1$. I'm unable to get any further to proving that there is only one solution (I couldn't find any others with a computer search), but maybe this can help someone else get there. Iffy★Chat -- 10:45, 18 April 2018 (UTC)