A Fermat-Quotient Coordinate for Mersenne Primes

This note records a simple structural observation connecting Mersenne primes and the Fermat quotient in base 2. If M_e = 2^e - 1 is prime, then its canonical Mersenne divisor form M_e = 2ek + 1 has coefficient k equal to the Fermat quotient Q_e(2) = (2^{e-1}-1)/e. The note presents the identity, examples, and a personal mathematical reflection.
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