Re: Goldbach minus


Tim Peters wrote:
[vernonner3voltazim]
We are most of us here probably familiar with Goldbach's Conjecture,
about any even number being describable as the sum of two primes.

It occurred to me to wonder if any even number might also be
describable as the difference between two primes.
For example:
2=5-3
4=11-7
6=13-7
8=19-11
10=13-3
etc.
Perhaps this is already known, if not so widely as the original
Conjecture.
Perhaps a counterexample is known. Just wondering....

It's still an open question:

http://primes.utm.edu/notes/conjectures/

Not only isn't there a counterexample known to that, but there isn't a
counterexample known to the stronger conjecture that there are an infinite
number of such pairs for every even difference, or to the stronger-still
conjecture that there are an infinite of such adjacent-prime pairs for every
even differece. For an example of the last, you have 10=13-3, but 10 is
also the difference of /adjacent/ primes in "many" ways, like:

149-139
191-181
251-241
293-283
347-337
419-409
431-421
557-547
587-577
641-631
701-691
719-709
797-787
821-811
839-829
929-919
1031-1021
1049-1039
1061-1051
1163-1153
1181-1171
...

As far as anyone knows, that list extends without bound, and likewise for
all even differences. But it hasn't yet been proved for any specific
difference, not even for 2.

There is another conjecture which is related, and quite strong. It
says, given a finite set {a_1, ..., a_n} of integers, such that for any
prime p, the collection of residue classes {a_1 mod p, ..., a_n mod p}
is not all residue classes mod p, then there are infinitely many
integers q such that q + a_1, ... ,q + a_n are all prime. The
conjecture even states the (believed) asymptotic density of such q.
The specific case with {a_1, a_2} = {0, 2n} reduces to the conjecture
that every even number can be written as the difference of two primes
in infinitely-many ways.

.

Relevant Pages

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