Can anyone either direct me to an relatively elementary proof in the literature--or show me why this (Conjecture 1 stated below) is true--or if I am mistaken and it is not true:


1. For any 4-tuple $\xi = (x\_0,x\_1,x\_2,x\_3) \in \mathbb{Z}^4$, let us define the quadratic form $Q(\xi) \doteq x\_0^2+x\_0x\_3+2x^2\_1 + x\_1x\_3+13x^2\_2+2x^2\_3$.
2. Now let $q \in \mathbb{N}$ be a prime **power** such that $(26,q)=1$, and let $\cal{B}$ be the set $\{B=(b\_0,b\_1,b\_2,b\_3) \in \mathbb{Z}^4$; $Q(B) \equiv\_q 1\}$.
3. Now let $\cal{A}$ be the set $\{A = (a\_0,a\_1,a\_2,a\_3) \in \mathbb{Z}^4; Q(A) = 2^k$ for some nonnegative integer $k\}$.



> 
> **Conjecture 1:** Then using the notation as above, then for any $B =(b\_0,b\_1,b\_2,b\_3) \in \cal{B}$ there is an $A =(a\_0,a\_1,a\_2,a\_3) \in \cal{A}$ that satisfies $a\_i \equiv\_q b\_i$ for each $i \in \{0,1,2,3\}$.
> 
> 
> 


I do need this result for a paper that I am writing. But I must admit to not understanding the Strong Approximation Theorem well at all, nor the associated math around this--which is what I think this looks like. [I am in graph theory]. 


Now I believe that if the above holds for all $q$ as prescribed in 2 then it holds for all integers $q$ such that $(q,26) = 1$.


If it would further help things let $q$ as in 2. be a power of 3, at least to start with.


Many Thanks!



Strong Approximation for solutions to quadratic Diophantine equations
The conjecture follows from Theorem 2.1 in Hsia-Jöchner: Almost strong approximations for definite quadratic spaces, Inventiones 129 (1997), 471-487. The paper is available [here](https://link.springer.com/content/pdf/10.1007/s002220050169.pdf).


The details of this implication are nontrivial. Let $q=\ell^s$ be a prime power (so $\ell$ is the underlying prime), and let $B\in\mathcal{B}$. Let $k$ be a large positive integer divisible by $\varphi(q)$. Then the $2$-power $\alpha:=2^k$ is large, and $Q(B)\equiv 1\equiv\alpha\pmod{q}$. By a standard lifting argument, there exists a primitive vector $v\_\ell\in B+q\mathbb{Z}\_\ell^4$ such that $Q(v\_\ell)=\alpha$. Let us also define $v\_2\in\mathbb{Z}\_2^4$ and $v\_{13}\in\mathbb{Z}\_{13}^4$ as the integer vector $(2^{k/2},0,0,0)\in\mathbb{Z}^4$. 
We observe that $Q(v\_2)=Q(v\_{13})=\alpha$, and $v\_{13}\in\mathbb{Z}\_{13}^4$ is primitive, but $v\_2\in\mathbb{Z}\_2^4$ is not. Now we apply the mentioned theorem of Hsia-Jöchner for the primes $T:=\{2,13,\ell\}$ and the above data. The conditions of the theorem are satisfied, because $Q$ is positive definite and isotropic over $\mathbb{Q}\_2$ (cf. Section IV.2 in Serre: A course in arithmetic). We conclude that there exists $A\in\mathbb{Z}^4$ such that $Q(A)=\alpha$ and $A\equiv v\_p\pmod{p^s}$ for all $p\in T$. In particular, $A\in\mathcal{A}$ and $A\equiv B\pmod{q}$. We are done.


