Power Sum Residues Modulo 144: A Four-Value Partition Theorem for Prime Power Sums

This paper proves that the power sum S(p,k) = 1ᵏ + 2ᵏ + ⋯ + (p−1)ᵏ mod 144 is independent of the exponent k for all odd k ≥ 3 if and only if the prime p belongs to one of exactly 32 residue classes mod 144. When this holds, the value is one of only six possibilities: {0, 1, 9, 64, 73, 81}.

The stronger form of the theorem — constant for all odd k ≥ 1 — holds for exactly 8 residue classes, with lock values {0, 1, 64, 81} = {0, 1, 2⁶, 3⁴}. These four values are the pure prime-power skeleton of 144 = 2⁴ × 3²: zero, unity, and the sixth power of 2 and fourth power of 3.

Proof structure: The proof uses three ingredients. First, modular periodicity: S(p,k) mod 144 = S(p mod 144, k) mod 144, reducing the problem to m ∈ {1,...,144}. Second, the Chinese Remainder Theorem decomposition 144 = 16 × 9 (gcd = 1), which allows independent analysis of the mod 9 and mod 16 components. Third, two lemmas:

Lemma 1 (mod 9): S(m,k) mod 9 is constant for all odd k if and only if m ≡ {0,1,2,8} mod 9. The proof uses the complete residue system argument — {1,...,m−1} mod 9 contributes 0 to any power sum when it contains only complete residue systems, since iᵏ + (9−i)ᵏ ≡ 0 mod 9 for odd k.

Lemma 2 (mod 16): S(m,k) mod 16 is constant for all odd k ≥ 3 if and only if m ≡ {0,1,2,15} mod 16. Odd integers mod 16 split into Type A (i² ≡ 1 mod 16: stable for all odd k) and Type B (i² ≡ 9 mod 16: individually unstable). Type B elements come in complementary pairs {a, 16−a} satisfying aᵏ + (16−a)ᵏ ≡ 0 mod 16 for all odd k, since (−a)ᵏ = −aᵏ. The lock condition requires every Type B element in {1,...,m−1} to appear with its pair.

Combining via CRT yields 4 × 4 = 16 universal classes (all odd k ≥ 1) and 32 classes (odd k ≥ 3), with lock values arising as CRT combinations of {0,1} mod 9 and {0,1} mod 16.

Wieferich corollary: The two known Wieferich primes occupy structurally opposite positions. 3511 ≡ 55 mod 144 is a universal class (lock = 9 for odd k ≥ 3). 1093 ≡ 85 mod 144 is a splitting class — S(1093,k) mod 144 takes values 60 and 132 depending on k, with difference 72.

All results are verified computationally for all residue classes mod 144 and all odd k up to 199. Complete Python verification code is included in the appendix.

Discovery note: This theorem was found during investigation of harmonic structure in the CTF frequency identity f₀ = 144.069 Hz. The result is independent of that framework and is stated here purely as a theorem in modular arithmetic.