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From: raycb on 16 Jul 2010 10:40
I wonder if the even number between primes that differ by 4x + 2 can
be written as the sum of two primes from sets that differ by 4x +
2 ...

A quick check seems to support it, but it may fail for the first
"small" few numbers. For example, 67 and 97 differ by 30, and 67 + 15
= 82 = 41 + 41, where 41 and 71 differ by 30. OTOH, 7 and 13 differ by
six, but there aren't appropriate primes to add up to 7 + 3. With 41
and 47, 41 + 3 = 44 = 13 + 31, where 13 and 19, and 41 and 47 differ
by 6.
From: raycb on 16 Jul 2010 10:52
> where 13 and 19, and 41 and 47 differ
> by 6.

should be 31 and 37
From: Ludovicus on 16 Jul 2010 11:18
On 12 jul, 09:08, raycb <ra...(a)live.com> wrote:
> Let 2n be an even number larger than 4 such that 2n - 1 and 2n + 1 are
> prime.
>
> Conjecture: 2n can be written as the sum of two primes in a way that
> both primes are members of twin prime pairs.
> I've gone up to 2n = 1290 without finding a counterexample.

If 2n - 1 and 2n + 1 are prime then, naturally,
2n must be the sum of members of twin prime pairs.
But even the isolated conjecture is false.
94 is not the sum of members of twin prime pairs.
Ludovicus
From: raycb on 16 Jul 2010 11:21
On Jul 16, 12:18 pm, Ludovicus <luir...(a)yahoo.com> wrote:
> On 12 jul, 09:08, raycb <ra...(a)live.com> wrote:
>
> > Let 2n be an even number larger than 4 such that 2n - 1 and 2n + 1 are
> > prime.
>
> > Conjecture: 2n can be written as the sum of two primes in a way that
> > both primes are members of twin prime pairs.
> > I've gone up to 2n = 1290 without finding a counterexample.
>
> If 2n - 1 and 2n + 1 are prime then, naturally,
> 2n must be the sum of members of twin prime pairs.
> But even the isolated conjecture is false.
> 94 is not the sum of members of twin prime pairs.
> Ludovicus

94 would only come under consideration if 93 and 95 were prime.
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