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April 17

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Intersecting two Pell equations

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Does the triplet (12, 7, 10) constitute the only positive integer solution to the system of Diophantine equations

  • Observation: This is equivalent to locating the rational points found on the conical surface
  • Attempt: It is clear that x is a multiple of 3, and z is even, yielding from where it follows that a is also even, giving at which point I am all fresh out of ideas.

79.113.238.17 (talk) 12:09, 17 April 2018 (UTC)[reply]

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 pq2 as n → , implying that p and q are to be found among the continued fraction approximations of 2. 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)[reply]
  • 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 , 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 . For a given α, integers m,n satisfy iff. integers 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. TigraanClick here to contact me 08:50, 18 April 2018 (UTC)[reply]
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)[reply]
  • 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 and , 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 and . 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 . From the 2nd equation, we get and thus b must be odd, and as c is divisible by 6 (from simplifying to ), b cannot be divisible by 3. We can rewrite this as . 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. IffyChat -- 10:45, 18 April 2018 (UTC)[reply]