2
$\begingroup$

Under Goldbach conjecture, say a positive integer $ r $ is a primality radius of a large enough composite integer $ n $ if and only if both $ n-r $ and $ n+r $ are prime. Let for given $ n $ the quantity $ N_{2} $ to be the number of primality radii of $ n $ and $ k $ the greatest positive integer not exceeding $ \sqrt{N_2} $ (the assumption of GC entails this number always exists).

Can one always find a $ k $-term sequence of primality radii of $ n $ in arithmetic progression with minimal positive common difference ?

$\endgroup$
1
$\begingroup$

This is not true, even without the minimal positive common difference condition.

The first counterexample is $n=81$, which has $10$ primality radii that contain no arithmetic progression of length $3$: $$\{2,8,20,22,28,50,58,68,70,76\}$$

More generally, we can note that if the radii are in AP, then so are the equivalent $n-r$ values. That is, we are also finding primes in AP. The number of radii grows relatively quickly and the required $k$ soon exceeds the longest known prime APs. The current record prime AP has length $26$, but:

$n=12285, N_2=690, k=26$
$n=13650, N_2=738, k=27$

OEIS, number of radii: http://oeis.org/A002375
Reference for record prime APs: http://primerecords.dk/aprecords.htm

$\endgroup$
  • $\begingroup$ Thank you very much for this insightful answer. $\endgroup$ – Sylvain Julien Oct 23 '18 at 10:16

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.