We defined to be a primality radius of iff both and are primes, which requires . If we slightly soften this assumption, writing , we can say that a classical primality radius is a natural primality radius and that is a relative primality radius of iff is a natural primality radius of .
We thus obtain some kind of duality, where every statement of the form where is a ternary predicate, is equivalent to .
As a consequence, we get that the reformulation of Goldbach’s conjecture as « every positive integer admits a natural primality radius » is equivalent to this weak form of de Polignac’s conjecture: « every positive integer admits a relative primality radius « , 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 is the fundamental (natural) primality radius of infinitely many -central numbers « .