i am wondering if there is a complete solution for the equation $a^2+pb^2-2c^2-2kcd+(p+k^2)d^2=0$ in which $a,b,c,d,k$ are integer(not all zero) and $p$ is odd prime.
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I'm not getting much 2-adic information for this one, but it should be easy enough to check all solutions mod 8 and mod 16 and see what happens. To restrict anything, one property requires $p \equiv \pm 3 \pmod 8$ and the other requires $p \equiv 3 \pmod 4.$ Put them together, when $$p \equiv 3 \pmod 8 $$ and $$ p | k, $$ then all four of your letters $$ a,b,c,d = 0.$$ The proof uses two flavors of anisotropy for binaries. The assumption is that at least one of $ a,b,c,d $ is nonzero and $\gcd(a,b,c,d) = 1.$ First we have forced $a^2 - 2 c^2 \equiv 0 \pmod p,$ so $a,c \equiv 0 \pmod p$ as $(2 | p) = -1.$ But then $ p b^2 + p d^2 \equiv 0 \pmod {p^2},$ or $ b^2 + d^2 \equiv 0 \pmod p,$ so $b,d \equiv 0 \pmod p$ as $(-1 | p) = -1.$ So $ p | \gcd(a,b,c,d)$ contrary to assumption. Otherwise, given a fixed $(p,k)$ once you have a nontrivial solution you get infinitely many using automorphs of the indefinite part in variables $(c,d).$ That is, there may be many parametrized families of solutions of one type or another. But you can figure some of those out with a computer algebra system more easily than I can by hand. The next interesting case is when $12 k^2 + 8p$ is a square, which means that the binary form $T(c,d)=2c^2+2kcd-(p+k^2)d^2$ factors. So $3 k^2 + 2p$ is a square, which is not possible for even $k,$ so $k$ is odd and $2p \equiv 6 \pmod 8,$ or $p \equiv -1 \pmod 4.$ Unless $p=3$ we also need $p \equiv -1 \pmod 3,$ or $p \equiv -1 \equiv 11 \pmod {12}.$ For example, with $p=11, k=1,3 k^2 + 2p = 25, p + k^2 = 12,$ we have $$ a^2+11b^2-2c^2-2cd+12d^2 = a^2+11b^2-2(c-2d)(c+3d).$$ The value of the factorization is that we can take, for instance, $c = 2 d + 1, c + 3 d = 5 d + 1,$ and $$ a^2+11b^2-2(5d+1) = 0.$$ Now $a^2 + 11 b^2$ is not even unless it is also divisible by $4.$ We also need $ a^2 \equiv b^2 \equiv 1 \pmod 5.$ Put them together, we have a parametrized solution of sorts, with $$ a \equiv 1,4 \pmod 5, \; \; b \equiv 1,4 \pmod 5, \; \; a \equiv b \pmod 2$$ take $c = 2 d + 1$ and $$ d = \frac{ a^2+11b^2-2}{10}.$$ |
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Perhaps it would help if we knew where the question is coming from. For what it's worth, you can write the equation in the form $$ a^2 + (c-kd)^2 + pb^2 + pd^2 = 3c^2, $$ so you are looking at a parametrized subset of the equation $$ A^2 + B^2 + pC^2 + pD^2 = 3c^2. $$ If $p \equiv 1 \bmod 4$, then $p(B^2+D^2) = R^2 + S^2$ is a sum of two squares, and your solutions must occur among those of $$ R^2 + S^2 + T^2 + U^2 = 3c^2. $$ Both quadrics can be parametrized by the standard method of sweeping lines if you know one solution. For arbitrary primes $p$ such a solution seems to be difficult to find. And even armed with such a parametrization you then would have to figure out which of them satisfy the additional conditions coming from the original equation. |
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There are lots! Here is a selection: $(a,b,c,d,k,p)=(3,3,4,1,1,3)$ $(a,b,c,d,k,p)=(3,3,4,1,7,3)$ $(a,b,c,d,k,p)=(3,3,4,2,2,3)$ |
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