The Palindromic Nature of Prime Numbers

We discover a novel characterization of prime numbers through the periodic behavior of thelinear recurrence a(n) = −a(n−1) + a(n−2) + a(n−3) modulo m. We conjecture that the normalized sumof lengths of palindromic cycles equals 1 if and only if m ≥ 2 is prime. Computational verificationup to m = 101 shows perfect agreement with this criterion. For prime p, we derive the weightformula from an observed three-class structure of palindromic cycles. Additionally, we conjectureself-similarity and weight preservation properties analogous to those found in our previous workon Fibonacci recurrences. This result establishes a connection between palindromic symmetryin dynamical systems and prime number structure.