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I've decided to simplify the argument found in notes of Jameson, and at the same time improve the bounds and ranges of applicability. I'm rewriting for the purpose of understanding and the specific g …

Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference. Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, usin …

I will not comment on the soundness of the approach, but I will render a subjective opinion: I don't like it. One of the reasons is that I have found most people do not have a good understanding of …

Although this is really a comment, I decided it was important enough to post as an answer. Having studied the problem posted above, I was suspicious of the given claim because of symmetry reasons. It …

As I understand the claim $\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, it is false for some $n$ and $x$ with $n$ close to $x$. Let us take $x$ to be $P_4=210$, the fourth primorial. Let us take $n$ …

Since I haven't posted this construction on MathOverflow (only referred to an ArXiv posting at Erik Westzynthius's cool upper bound argument: update? ), let me show that one can have intervals of th …

I called it $\pi^{-1}(m)$ in a number theory article I posted on the ArXiv. I did not scour the literature, but I conjecture that Erdos never came up with a name for it (he didn't in the papers I saw …

Let's assume a limited (and unproved) version of Linnik's theorem: There is a prime $q$ of the form $kp + 1$ for $k \leq (p-2)$ and $p$ a prime. Experimentally this is true, and can be proved for ma …

I've decided to collect some basic observations and references for the benefit of future readers. A more challenging problem is to ask for integers $m$ and $p$ such that for all integers $k$, $p_0 = …

EDIT 2015.07.15 I believe there is no such integer $x$. See below for the rest of the edit. END EDIT 2015.07.15 I had some initial thoughts which seemed promising. They do not lead to a proof, but …

For squarefree $d$, we can translate this into a design problem. Given an $r$ by $c$ array (which correspond to your $r$ many $k$-tuples, but I use $c$ instead of $k$), you need to divide the $k$ dis …

Look for a large gap in the distribution of primes. For this conjecture, the gap between $n!+2$ and $n!+n$ will suffice. Set $y = n!+2$ (which is composite) and set $m$ (which will be $\frac{x+y}{2} …

This is the closest I can come to a positive test, but I don't know how well it will work for you. It is essentially taking the intersection of possible solution sets. Let us cut down on symmetry by …

I imagine it took Cole longer than he said. If I were to undertake the project, here is how I would proceed: I would start sifting the set of numbers {134k + 1} for primes. One can modify the Sieve …

Indeed, the suggestion given in the other thread is quite appropriate. Use a lower bound from Dusart for $\pi(cn)$, and an upper bound for $\pi(n)$, and you want the difference between these bounds t …