Congruence Properties of Minimal-Diameter Prime Triplets and Two Prime-Valued Conjectures
Authors/Creators
Description
This note studies congruence properties of minimal-diameter prime triplets of the forms (p, p+2, p+6) and (p, p+4, p+6).
For p > 5, we prove that the sum S = p + q + r is congruent to 1 or 9 modulo 10 and is not divisible by 3. We also prove the main modulo-30 result:
p² + q² + r² is congruent to 9 modulo 30.
Consequently, (p² + q² + r²)/3 is congruent to 3 modulo 10.
The paper includes computational verification for 5 < p <= 1,000,000, together with a table of prime-triplet counts and the observed cases where the sum or normalized sum of squares is prime.
Two prime-value infinitude conjectures are formulated concerning the triplet sum and the normalized sum of squares. These conjectures are clearly identified as conjectures and are not claimed to be proved.
The work provides a concise study of elementary congruence structures associated with minimal-diameter prime triplets.