In a message dated 05/04/2001 05:19:38 GMT Daylight Time, "Gary Untermeyer"
<[EMAIL PROTECTED]> writes:
> Greetings,
> Although this is not a question regarding mersenne primes, I thought I'd
> throw this out to the readers here.
You will find [EMAIL PROTECTED] is also good for this kind of
question.
> Let x be a prime number. Consider the series of numbers that take the
> following form:
>
> x, x + n, x + 2n, x + 3n, x + 4n, x + 5n, x + 6n, where n is an
> even positive whole number.
>
> In this series of seven numbers, can anyone tell me why, if ALL of these
> numbers are prime, that the minimum value of n is 210 if all the terms
> are _consecutive_ prime numbers?
If n is not a multiple of 2 (i.e. even), then one of the first 2 terms
x, x+n
is (obviously) a multiple of 2 and so is not prime.
Similarly, if n is not a multiple of 3, then one of first 3 terms
x, x+n, x+2n
is a multiple of 3 and so is not prime.
And so on, for primes 2, 3, 5, ...
Inverting the argument, if the sequence has N terms, then n must be a
multiple of all the primes <= N. So in your example (N = 7), n must be a
multiple of 2, 3, 5 and 7, i.e. of 210. This is still the case for N = 8
(your next case), 9, and 10. For N=11, n must be a multiple of 210*11=2310.
Note that this is true for _all_ arithmetic progressions of primes, not just
for progressions of _consecutive_ primes.
Mike Oakes
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