Project Euler Forum

I moved this from "Suggestions etc" to this Forum.
Our interpretation is that it is a dream a lot of engineers share.

"A pair of consecutive prime numbers with a difference of six is called a sexy pair (since "sex" is the Latin word for "six"). The first sexy pair is (23, 29)."

I think this is an error. There are many sexy pairs less than (23, 29): (5,11), (7,13), (11,17), (13,19), (17,23).

Your examples are not consecutive.

Ah, you're right.

Ah. That's the first time I post on this forum, I'm totally confused.
I've "proved" that there are no "engineers' paradises". I may assume that my proof is flawed, but I fail to find the bug. I would appreciate any help.
Now, here we have a number n > (say) 30, which is EP. It's practical, so we have to represent 1, 2, 3, 4, ..., 8 as sums of its distinct divisors. 2 cannot be represented other way than itself, so n is even. 3 = 1 + 2, everything's good. 4 = 1 + 3, but if n is divisible by 3, then n + 3, n + 9 are divisible by 3 and therefore not prime, fail. So n has to be divisible by 4.
Next, 5 = 1 + 4, 6 = 2 + 4, 7 = 1 + 2 + 4. Next, 8. If n is not divisible by 5, 7 or 8 (it cannot be divisible by 6, as earlier), we can't represent 8, and that's a fail.
Assume n is divisible by 8. Then n + 4 and n - 4 are not, but as they're practical, they must be divisible by 5 or 7. They cannot be both divisible by 5 or 7, since their difference doesn't, so one of them is divisible by 7, say, n + 4. But then n - 3 is divisible by 7 and is not equal to 7, therefore not prime. Same for n - 4 and n + 3. Fail, n is not divisible by 8.
If not, n + 8 and n - 8 are also not divisible by 8, similarily one of them is divisible by 5, say, n + 8. But then n + 3 is also divisible by 5, therefore not prime, same for n - 8 and n - 3. We've come to a contradiction.
Where am I wrong? %)
Sorry if I give away something important.

n may not be divisible by 3, but that doesn't mean other practical numbers can't be.

Oh. That was silly of me. =)
Thanks for your help!

Can I safely assume there are 28,388 sexy triple-pairs (n-9,n-3)-(n-3,n+3)-(n+3,n+9) below 10⁹?