2
$\begingroup$

I am wondering about the following question.

Do there exist infinitely many prime numbers $p$ such that there exist integers $m,n$ with $5(m+1) \geq 7n$ and $$5m < p < 7n?$$

If not, what other numbers would we have to replace $5$ and $7$ by for there to be? If so, what is the minimal set of numbers we would have to replace $5$ and $7$ by? This leads onto my second question below.

Find numbers $a,b$ with $|a-b|$ minimal such that there exist infinitely many primes $p$ with the property that there exist integers $m,n$ with $a(m+1) \geq bn$ and $$am < p < bn.$$

$\endgroup$
5
  • 1
    $\begingroup$ Where is $n$ involved? $\endgroup$
    – Bernard
    Jun 18 '16 at 14:04
  • $\begingroup$ @Bernard Thanks, that was a typo. $\endgroup$ Jun 18 '16 at 14:07
  • 1
    $\begingroup$ The standard Dirichlet result gives infinitely many primes $p$ of the form $35n-1$. For such primes we have $5(7n-1)<p<7\cdot5n=5\cdot7n$ $\endgroup$
    – almagest
    Jun 18 '16 at 14:07
  • $\begingroup$ @almagest: that should be an answer. I was thinking down that line. $\endgroup$ Jun 18 '16 at 14:08
  • $\begingroup$ @RossMillikan then your welcome to put it up. I am too busy on something else to think about the second part of the question. $\endgroup$
    – almagest
    Jun 18 '16 at 14:09
4
$\begingroup$

Following almagest: The standard Dirichlet result gives infinitely many primes pp of the form $35n−1$. For such primes we have $5(7n−1)\lt p\lt 7⋅5n$

The same argument works with $a=6, b=7,$ primes of the form $42n-1$

$\endgroup$
2
  • $\begingroup$ What is the standard Dirichlet result? $\endgroup$ Jun 18 '16 at 14:13
  • $\begingroup$ Any arithmetic progression $a+kd$ where $a,d$ are coprime and $k \in \Bbb N$ contains an infinite number of primes. Here in Wikipedia @SubhadeepDey because I stole almagest's comment for the answer by his request $\endgroup$ Jun 18 '16 at 14:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.