On 1/28/07, maartenvanthiel <

m.v.thiel@...> wrote:

> Anyway, in about 1986, 3000 feet above Afghanistan, working on a palm

> held (of those days), I found out that the "distance" between prime

> twins is a product of 6. 11 13 / 29 31; 29 -/- 11 = 18 = 3 x 6.

> Back home I faxed this to prof. Anthes, Darmstadt university prof in

> history of mathematics and the Scryption museum (history of writing

> and of office material; I was then the director) expert on the history

> of calculators.

> He was very surprised.

> Question: was this not known in 1986???

It's been known for probably thousands of years that prime twins must

be of the form 6n-1 and 6n+1 (since 6n+3 can't be prime). Well,

except 3 and 5 of course.

> Now some years later I was half awake. "Distance between prime twins =

> x 6, 6 is a perfect number (dont know how this is called in English;

> what I mean is that 6 can be divided by 1, 2 and 3, while 1 + 2 + 3 =

> 6). Next perfect number = 28. Would 28 be the distance between prime

> triples?". You can see that I wasnt completely awake..

Depends on what you mean by prime triples: sets of primes of the form

p, p+2, p+6 or p, p+4, p+6 maybe? Then how far apart are they?

> Anyway again. I continued thinking.

> If 3 5 7 is a prime triple, another one would be N N+2 N+4.

> So then the "distance" between 3 5 and N N+2 would have to be a

> product of 6. BUT ALSO the "distance" between 5 7 and N+2 N+4. Which

> cant be..

Yup, p, p+2, and p+4 -- one of them must be divisible by 3, so 3 5 7

is the only possibility.

--Joshua Zucker