v3 (May 2025): Added §7 heuristic analysis of Bessaoudi super-pairs (E < 5×10⁻⁴ under Wagstaff and Bateman-Horn conjectures). AddedCyclotomic Theorem 6. 1. Corrected base-3 table. n=53 identified asterm of A396249 but not a super-pair. Two new references. OEIS sequence A396249 submitted. We introduce the Bessaoudi-Germain conjecture, a geometric reformulation ofMersenne primality through the 6k+/-1 matrix. For any prime n, the indexk= (4ⁿ-1) /3 places 6k+1=2^ (2n+1) -1 in the right column as a Mersenne candidate. Three results are proved: (i) k always exhibits a pure binary wave 10101. . . 01; (ii) the left column 2^ (2n+1) -3 is systematically divisible by 5 for all odd primes n; (iii) Euler's theorem (1750) provides the fifth criterion n=1 (mod 4). NEW: theBinary Lock Corollary shows that a Sophie Germain prime n is Euler-locked if andonly if its binary representation ends in '11', giving an O (log n) eliminationtest. A base-9 family k= (9ⁿ-1) /8 with right column (3^ (2n+1) +1) /4 is proposed asan unstudied quasi-Mersenne family. Computational verification for n<=1000 confirmsonly n=2 and n=3 satisfy all criteria.
BESSAOUDI HAMZA (Fri,) studied this question.