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 ?


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

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

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