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It's elementary: let $d=(a,m)$, $a=a_1d$ and $m=m_1d$. Then dividing both sides by $d^k$ gives $a_1^k+\frac{ka_1^k}{xy^n}=m_1^k$. Let $p$ be any prime divisor of $a_1$. Since $(a_1,m_1)=1$, $p\nmid m_ …

There is a curious Diphantine equation showing up in my research: $$ \frac{1}{a^2-1}+\frac{1}{b^2-1}=\frac{1}{c^2-1}+\frac{1}{d^2-1}. $$ I am trying to find its integer solutions where $a$, $b$, $c$ …

It's well known that if $\alpha $ is a rational root to an integer coefficient polynomial, then its denominator divides the leading coefficient and its numerator divides the constant term. I'm asking …

Yes. Just replace arithmetic progressions by translations of a nonzero (integral) ideal. The same argument shows that the union on the RHS is closed, so the set of units $K^\times$ has to be open. Sin …

Unless I'm mistaken, this is quite easy and only involves the multiplicative property of the quadratic residue. If $n_p=n_q$ then we are done, because this very number is a simultaneous quadratic non …

It may also be helpful to search for "Waring's problem in finite fields". For a bound derived from discrete Fourier analysis, see https://dl.dropboxusercontent.com/u/27883775/math%20notes/analytic-nt. …

In fact the previously deleted sawtooth-like answer is almost right, but the periods of each successive sawtooth must grow much faster than exponentially. Let $d_n(x)=d(x,2^{n^2}\mathbb{Z})$ be the d …

Forgive me for my ignorance, but I'm very surprised to learn that there are two Vinogradovs, both famous in the field of analytic number theory. Guessing from their names and the Russian naming conven …

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the num …

If $\eta_i$ are nonnegative, then there is an algorithm (though not very efficient) finding all the solutions. First note that there is one $\eta_i$ such that $v_2(\eta_i)\le v_2(z)$. where $v_2$ deno …

Although this is a good heuristic, I have to add it doesn't capture the number-theoretic nature of the problem. One way to see this is that the same heuristic would imply every odd number is the sum o …

Let $N_{\bf n}$ be the number of solutions to $\sum x_i^s=n_i$. Them you only need to show $\sum N_{\bf m}N_{\bf m+n}\le\sum N_{\bf n}^2$, which is just Cauchy-Schwarz.

Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance o …

Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that i …

By the Pólya-Vinogradov inequality, $|S(\chi_N,x)|\ll (p_1\cdots p_N)^{1/2}$, whose order you can estimate using the prime number theorem.