Search Results

The Riemann hypothesis is true, if primes are random in certain ways.

In January, Vesselin Dimitrov posted to the arXiv a preprint showing that Mochizuki's work, if correct, would be effective. While this doesn't validate Mochizuki's work it does do a few things: It …

There are two main sources of repeated logs. (These sources can be further refined into natural subcategories, but I'll only mention a couple of those subcategories.) Those two main sources are: Typ …

My question is fairly simple, and may at first glance seem a bit silly, but stick with me. If we are given the rationals, and we pick an element, how do we recognize whether or not what we picked is …

I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard …

This can be done in "essentially linear time." Check out Daniel Bernstein's website: http://cr.yp.to/arith.html Especially note his papers labeled [powers] and [powers2].

This is a problem I have thought alot about. I have not seen any of the modern techniques in your list applied to the problem. Part of the issue is that if you represent $\sigma(n)=2n$ as a Diophant …

I think that there is indeed some possibility to lower the bound, and this is something I've looked at seriously a few times. I spent a semester (in 2019) with the Computational Number Theory Group h …

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_{ …

There are apparently lots of examples. The smallest is $a=1$, $b=2$, and $p=7$.

(This is an extended comment.) There couldn't be anything special about base 10, could there? Notation: Given two positive integers $m,n$, let $m\oplus n$ be the integer that results from prepending …

Doing an internet search I found the paper Groups of small order as Galois groups over $\mathbb{Q}$ by Jack Sonn, from 1989. Theorem 1 of that paper asserts that every group of order less than 672 is …

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 …

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 …

At the present time, the best (general) result is due to James Maynard, who has proven that any admissible 3-tuple takes at most 7 prime factors infinitely often (see this paper). In particular, (usi …