## Take any even number! A prime puzzle

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• Clearly, you guys/gals are bright! But it s the simplest of questions that vexes us all. Take any even number, for example. I ll choose 20, for simplicity s
Message 1 of 3 , Mar 31, 2006
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Clearly, you guys/gals are bright!

But it's the simplest of questions that vexes us all.

Take any even number, for example.

I'll choose 20, for simplicity's sake.

Now the high/low prime partition for 20 is 17:3

But 17 - 3 = 14

Now the high/low prime partition for 14 is 11:3

But 11 - 3 = 8

Now the high/low prime partition for 8 is 5:3

And finally 5 - 3 = 2

So the even number 20 will resolve to 2, via the above rendition.

However, some even numbers will resolve to zero, for example 16 or 22.

In fact every even number will resolve to 0 or 2, (which I cannot prove) unless it happens to be the unproven Goldbach exception.

Question: What is the highest even number that will resolve to zero?

Really hope you can help me out here.

Cheers

Bob

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• ... It s 30, but you ll never prove it. :) Proving that it s 30 would seem to be roughly as difficult as proving the Goldbach conjecture. The numbers which
Message 2 of 3 , Mar 31, 2006
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--- Bob Gilson wrote:
> Question: What is the highest even number that will resolve to zero?

It's 30, but you'll never prove it. :)

Proving that it's 30 would seem to be roughly as difficult as proving
the Goldbach conjecture.

The numbers which resolve to 0 are 4, 6, 10, 16, 22, and 30.

You could create other related conjectures. One might be:

Assume that you can choose ANY pair for a number, rather than
always having to choose the largest pair. Which even numbers
cannot resolve to 0? It appears that only 2, 8, and 12 cannot
resolve to 0. Again, proving it is the tough part. You need
to show that every even number > 12 can be expressed as a sum
of primes p+q where abs(p-q) is not 2, 8, or 12, which is
basically Goldbach plus a twist.
• ... It s 30, but you ll never prove it. :) Proving that it s 30 would seem to be roughly as difficult as proving the Goldbach conjecture. The numbers which
Message 3 of 3 , Mar 31, 2006
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jbrennen <jb@...> wrote: --- Bob Gilson wrote:
> Question: What is the highest even number that will resolve to zero?

It's 30, but you'll never prove it. :)

Proving that it's 30 would seem to be roughly as difficult as proving
the Goldbach conjecture.

The numbers which resolve to 0 are 4, 6, 10, 16, 22, and 30.

You could create other related conjectures. One might be:

Assume that you can choose ANY pair for a number, rather than
always having to choose the largest pair. Which even numbers
cannot resolve to 0? It appears that only 2, 8, and 12 cannot
resolve to 0. Again, proving it is the tough part. You need
to show that every even number > 12 can be expressed as a sum
of primes p+q where abs(p-q) is not 2, 8, or 12, which is
basically Goldbach plus a twist.

And there was I, drinking a Brandy & Coke, watching the moon rising over the Waterfront in Cape Town, suddenly having this really amazing truly original thought, only to find (as usual), that someone's thought about it before. Ah well, hey ho, BUT one day.....

Thanks for the enlightenment, anyway.

Bob

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