Consider the sequence of primes at prime positions in the sequence of primes.

3, 5, 11, 17, 31, 41, 59, 67, 83, 109...

These are the 2nd prime, the 3rd prime, the 5th prime, and so on. Call these numbers superprimes.

W, X, Y, and Z are 4 superprimes in arithmetic progression, and W<X<Y<Z. Find the smallest possible value of Z.

Method using Excel.

1. Generate the first 50 primes with prime indices (mathematica does this and there is also a list in Sloane).

2. Take successive differences {2-1,3-2..}{3-1,4-2...}{4-1,5-2...} etc.

3. Sort {value, index, number, smaller index} by number. We are looking for a sequence of 3 where the indices and smaller indices match up, e.g.

431 23 78 20
509 25 78 23
587 28 78 25

4. Checking back with the original list gives 353 as the remaining value.

Posted by broll on 2011-07-19 02:38:59