Relative primality radius

We defined r to be a primality radius of n iff both n-r and n+r are primes, which requires r<n. If we slightly soften this assumption, writing (\vert n-r\vert,n+r)\in\mathbb{P}^{2}, we can say that a classical primality radius is a natural primality radius and that n is a relative primality radius of r iff  r is a natural primality radius of n.

We thus obtain some kind of duality, where every statement of the form P(n,r,\ \ natural \ \ primality \ \ radius) where P is a ternary predicate, is equivalent to P(r,n, \ \ relative \ \ primality \ \ radius).

As a consequence, we get that the reformulation of Goldbach’s conjecture as « every positive integer n>1 admits a natural primality radius r » is equivalent to this weak form of de Polignac’s conjecture: « every positive integer r>1 admits a relative primality radius n« , which is obviously equivalent to saying that every even integer is the difference of two (non necessarily consecutive) primes.

The strong de Polignac conjecture is equivalent to saying that « every positive even integer is the difference of two consecutive primes in infinitely many ways » or that « every positive integer r is the fundamental (natural) primality radius of infinitely many 1-central numbers n « .