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The standard appproach is via Baker's method of linear forms in logarithms. We have $x_n+\sqrt{3}y_n=(2+\sqrt{3})^n$, thus $2x_n=(2+\sqrt{3})^n+(2-\sqrt{3})^n$. Now assume that $x_n=7^m$, and consider …

Define the lattice $(\mathcal{L},\prec)$ as the set of all function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ …

A non-trivial lower bound can be obtained using linear forms in $p$-adic logarithms. Suppose that $\left\{\left(\frac{4}{3}\right)^n\right\}$ is small. Clearly it is a rational number with denominator …

For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following: There exists some $c>0$, such that for all $x$ sufficiently large the number of integers $n\in[x, …

You are asking for which functions $f$ the sequence $f(n)$ is equidistributed modulo 1. This is a whole area of mathematics, which began with the work of Weyl in 1916, who discovered the connection be …

The if-part is a special case of Pocklingtons theorem (see https://en.wikipedia.org/wiki/Pocklington_primality_test ). The only if part requires some computation, as you need a quadratic non-reside mo …

I once saw an application of a solved case of the inverse Galois problem. It is well known, that the Dedekind $\zeta$-function of a number field does not determine the number field up to isomorphy. I …

Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed: Suppose $P, …

In the 1960's Turán wrote several papers on a function-theoretic sieve. He managed to express the number of prime twins in terms of roots of $L$-series. He began like you did by expressing $\Lambda$ a …

The integers in the interval $[n, n+D]$ have level of distribution close to $D$, hence you can apply a lower bound sieve (e.g. http://www.math.uiuc.edu/~ford/sieve_notes_intro_brun_hooley.pdf, Theorem …

The inequality you state is not a known consequence of GRH, not even in the case $q=1$. In this case von Koch proved 1901 the error term $\mathcal{O}(X^{1/2}\log^2 X)$. Gallagher and Mueller showed th …

The content of chapter II.9 is contained in many textbooks on analytic number theory. A favourite of mine is Davenport's multiplicative number theory. For binary quadratic forms things are more diffic …

Bruedern, Granville, Perelli, Vaughan and Wooley, (Philos. Trans. Roy. Soc. London Ser. A, 356 (1998) 739 - 761) dealt with the sequence of $k$-free integers. Bruedern (in: Analytic Number Theory …

Such a sieve could only exist under rather special conditions. The easiest case would be $\Omega_p=\{0\}$, $z=\sqrt{x}$. In this case the sifted set consists of all integers of the form $pn\leq x$, wh …

Converging sieves: Let $q_i$ be a sequence of integers with $\sum\frac{1}{q_i}<\infty$, and pick for each $i$ an integer $a_i$ within a finite set $\mathcal{A}$. Then the set of integers $n$ such that …