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According to the Wikipedia page on Catalan's conjecture, the problem you pose is open. (Look under the "generalization" heading.) A more general problem is Pillai's conjecture.

It is currently believed that the second conjecture is likely false, but it hasn't been proven quite yet. There is an interval of size 3159 which is not prevented from having more primes than the ini …

Let's use some standard bounds on the $n$th prime, as found in this paper by Pierre Dusart. We have $$ p_{2n} \leq 2n[\ln(2n)+\ln(\ln(2n))-0.9484] = 2n\left[\ln(n)+\ln\left(\frac{2}{e^{0.9484}}\ln(2n) …

[Thanks to Gerhard Paseman for helping me reformulate my original question.] The equation $$ \frac{a^m-1}{a-1}=b^2 $$ was solved by Ljunggren, building on work of Nagell, who showed that if $a>1$, $b …

The best source for explicit, unconditional bounds on such products that I'm aware of is in the work of Pierre Dusart. See his paper Explicit estimates of some functions over primes. In Section 5.4, …

The answer to this question is almost certainly no. It is well known that the ABC Conjecture implies that there are only finitely many triples $(n,n+1,n+2)$ which are all powerful. A similar argumen …

An update on this problem: I found out how to compute effective (and asymptotically accurate) bounds for $\sum_{p\leq x,\, p\equiv a\pmod{k}}\log(p)/p$. Basically it boils down to the usual analytic …

Edited due to mistakes pointed out in the comments: I think the answer to the problem might be yes. Here are some preliminary thoughts. First, you know $f_{\nu}(x)$ divides $f(x)$, since $f(x)$ is …

I just ran across the following paper: Defining Z in Q It uses another characterization of the integers inside the rationals that none of us listed, perhaps because it is so trivial. Namely, the int …

This is impossible from standard results about cyclotomic polynomials and their factors. Note that $\frac{q^{p-1}-1}{p-1}$ is just $$ \prod_{d|(p-1), d>1}\Phi_d(q), $$ where $\Phi_d(x)$ is the $d$th …

Let $\zeta$ be the Riemann zeta-function, and let $t> 0$. I'm interested in the growth rate of $$ \left|\frac{\zeta'}{\zeta}\left(-\frac{1}{2}+it\right)\right| $$ as $t\to\infty$. It is easy to find …

Let $f$ be a monotone decreasing, continuously differentiable function with $\lim_{x\rightarrow \infty}f(x)=0$. Let $\chi$ be a non-principal Dirichlet character. It is standard to show that $\sum_{ …

Color the positive integers using just two colors. By van der Waerden's theorem, we can find a $k$-term arithmetic progression as long as we consider a long interval. I imagine it is possible to fin …

cLet $a,b,c,d\in \mathbb{Z}$ and suppose we have the equation $ac+bd=1$. One way of thinking about this equation is it expresses the fact $\gcd(c,d)=1$. It is well-known that all other similar equat …

This is just a long comment that might be helpful: Treat $a$ as a parameter, and treat $x$ as a variable. The Diophantine expression $$ \exists x\ ((x^2<a) \land (a<(x+1)^2)) $$ defines $a$ as a nons …