All Primes and All Their Squares: The Progression 7 + 6k Without Exceptions up to 10^7
Authors/Creators
Description
This note presents a systematic empirical verification of a dual completeness property for squares of primes in the arithmetic progression 7 + 6k.
Known: for every prime p ≥ 5, p² ≡ 1 (mod 6), so every square of such a prime falls into 7 + 6k with index k = (p² − 7)/6 (OEIS A164097, A110586).
New here: a systematic verification of dual completeness on a large range, with a full table and reproducible codes. By "completeness in both directions" we mean: (i) every square of a prime p ≥ 5 with p² ≤ N is present in the progression (no gaps), and (ii) every square of a prime found in the progression is indeed a square of a prime p ≥ 5 (no "extra" squares of primes). The object of verification is the set of squares of primes and its image in the progression.
Results up to 10⁷: 444 squares of primes, all present and all genuine; the count matches π(√N) − 2. We also study k/p² → 1/6, the distribution of prime k (only 2% of prime k up to 10⁴ yield a square, only for k ≡ 2, 4 (mod 5), plus k = 3), and provide 5 reproducible Python codes and the full 444-row table.
Changes in v2: Added a remark on the equivalence of progressions s + 6k with s ≡ 1 (mod 6), with an empirical check, codes, and a table of k-values. No changes to the main results for 7 + 6k.