30 semiprimes, each less than 10
8, form an arithmetic sequence with strictly positive common difference.
Determine all of them.
Note: A semiprime is the product of exactly two primes.
*** For an extra challenge only, find a semi-analytic solution (simple calculator + p&p) to this puzzle.
"Prime Puzzles" actually
states the answer, and goes further.
Included is an interesting discussion of fast computer methods: byte vs bit searches, and the use of a HW cache (super-fast) for the first interval check, supplemented by those using SS memory arrays.
Wikipedia shows
work on prime arithmetic sequences (as opposed to semiprime sequences). These are termed CPAP-k, for
Continuous Primes in Arithmetic Progression. CPAP-k of length k.
is the form p = start + n distance, with n = 0, 1, ..., k-1
Examples are known to k=20, but the minimal ones are only known to k=6, where minimal means the lowest start. Beyond k=6 there may be larger d's but with a smaller start.
I haven't found a correspond SP (CSPAP-k) list, for example, in
the OEIS. I set out to search for the minimal ones, and I got a little hung-up wondering if "even SP" sequences were possibly minimal. (They would spawn from CPAPs times the prime 2. E.g., (3, 5, 7) -> (6, 10, 14)
Also, note that the convention for identifying CPAPs does not require completeness at k-1. I.e, CPAP-4 is part of CPAP-5: (5, 11, 17, 23) is part of (5, 11, 17, 23, 29).
TBC (perhaps)
Edited on November 28, 2022, 9:27 pm
other references found online
