décembre 2015 – ideasfornumbertheory

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 $n$ « .

ideasfornumbertheory

a blog to give new ideas in number theory

Mois : décembre 2015

Relative primality radius