What is the lowest arithmetic sequence of positive prime integers that has 3 terms? 5 terms? 8 terms?

What is the constant difference for the lowest N positive prime integers in arithmetic sequence?

What would the first term be for such a sequence?

(A prime sequence is "lowest" if the average of its terms is the lowest. If any are tied then it is the one with the smallest starting term.)

I found a solution to the next 2 parts:

first, let Pn be the nth prime number, hence, P1=2, P2=3, P3=5, P4=7 etc...

Hypothesis: Pn + d * (P1*P2*P3...*Pn-1) is prime for each d
Proof:

Define M= P1*P2*...*Pn-1
first, note that
M= gcm(P1,P2,...Pn-1)
(Trivial).

now let 1 < k < n-1

then Pn(mod Pk) = Pn + d * M (mod Pk)
because M (mod Pk) = 0

then Pn+d cannot be devided by any prime number smaller than P.

Now, let P' be a prime number equal or greater than P. Assume (d * M) / P' is an integer. then d / P is an integer (Because P1,P2... are primes). But d < P <= P' - contradiction.

so we have proven the theorem - hence, for a squence of n numbers, take the nth prime number as first number, and chose the difference as multiplation of all primes before it...

Please let me know if I have a mistake...

Posted by ronen on 2003-12-08 15:05:05