At its heart, this conjecture rests on a simple, elegant truth from modular arithmetic: to test if a prime \( p \) divides a number \( n \), one need only compute \( n \pmod p \). Viewed as a periodic sequence with period \( p \) and frequency \( 1/p \), this check corresponds to sampling at \( f_s = 1/p \)—exactly once per cycle. For \( n = 35 \), sampling at \( f_s = 1/5 \) and \( f_s = 1/7 \) reveals zeros, identifying \( 5 \) and \( 7 \) as factors. This rate is half the Nyquist rate (\( 2/p \)) for each sequence, a minimalist yet sufficient effort that leverages the sparsity of prime factors. The information recovered—approximately 2.58 bits (\( \log_2(6) \))—representing the choice of \( \{5, 7\} \) from 6 unordered pairs of primes up to 7 (\( \binom{4}{2} \))—underscores how little data is needed when targeting sparse, specific properties. This insight is not just a clever reframing of division; it’s a bridge between number theory and signal processing, suggesting that factorization can be recast as a sampling problem.
The power of this approach lies in sparsity. A number like \( 35 \) has only two prime factors among a vast field of possibilities, making them needles in a haystack. Sampling at \( f_s = 1/p \) is akin to probing with a finely tuned detector, hitting the needles directly rather than scanning the entire stack. This aligns with ideas in compressive sensing, where sparse signals can be recovered from far fewer samples than their full bandwidth demands. For \( 35 \), checking \( p = 5 \) and \( p = 7 \) (2 samples) suffices, a tiny fraction of the effort to reconstruct a full signal representing \( n \). This sparsity-driven efficiency is the conjecture’s most compelling feature, hinting at broader applications if it can scale.
The conjecture’s ambition shines in its extension to complex signals, like the chirp \( s(t) = \cos(2\pi t^2 / 70) \). Here, sampling at \( f_s = 1/5 \) or \( 1/7 \)—far below the signal’s Nyquist rate (\( f_s \ge 0.972 \))—produces aliased data, yet the hypothesis is that factor-related frequencies (\( 1/5, 1/7 \)) might still emerge in an FFT. For \( n = 35 \), \( s(7) = 0 \) and periodic hints at \( t = 5, 10, \ldots \) suggest resonance, but aliasing scrambles the full picture. This leap is tantalizing: if true, it could transform factorization into a signal processing task, potentially faster than trial division for large \( n \). However, it’s a leap of faith—aliasing typically obscures, not reveals, and the claim lacks a proven mechanism or robust simulation. The “rotation independence” idea (stability under sample shifts, e.g., shifting samples from \( t = 0, 7 \) to \( (1, 8) \) preserves periodic hints) adds intrigue but needs rigorous definition and testing.
The parallels drawn to the Riemann Hypothesis (\( \sigma = 1/2 \)) and elliptic curves are striking but speculative. In RH, the critical line balances zeta zeros’ chaotic oscillations with prime distribution, much like \( f_s = 1/p \) might balance aliasing chaos with factor signals. For elliptic curves, \( a_p = 0 \) marking factors in some cases (e.g., for some curve over \( \mathbb{F}_5 \), \( N_5 = 5 \), \( a_5 = 0 \)) mirrors modular zeros, with \( \sigma = 1/2 \) in Weil’s framework suggesting a sparsity threshold. These analogies inspire—both systems involve critical points where minimal information encodes deep structure—but they don’t yet mechanistically tie \( f_s = 1/p \) to factorization beyond modular arithmetic. They’re conceptual sparks, not proofs, awaiting a concrete link.
The conjecture’s scope is its Achilles’ heel. Demonstrated only for \( n = 35 = 5 \cdot 7 \), it faces uncharted territory with larger numbers (e.g., \( n = 1001 = 7 \cdot 11 \cdot 13 \)), repeated factors (\( n = 25 = 5^2 \)), or closely spaced factors (\( n = 77 = 7 \cdot 11 \)). Does \( f_s = 1/5 \) distinguish \( 5^2 \) from \( 5 \)? Can sparse sampling scale logarithmically with \( n \), or does aliasing overwhelm as factors multiply? Practicality is another question—trial division already checks \( n \pmod p \), so what computational edge does a signal-based approach offer? These gaps define the conjecture’s future: without generalization, it’s a curiosity; with it, it could be revolutionary.
If the modular insight scales to complex signals, it might inspire a new factorization paradigm, blending number theory with signal processing tools like FFTs. The RH analogy, while unproven, hints at deeper ties between prime distribution and sampling theory, potentially illuminating zeta zero statistics. Practically, simulating the chirp FFT for \( n = 35, 25, 77 \) could test the aliasing hypothesis, while exploring \( S_N = n e^{i t_n \ln n} \) might quantify how zeta zeros encode factors. The immediate task is clear: validate the speculative extensions with data, then tackle generalization.
This conjecture is a bold fusion of disciplines, grounded in the minimalist beauty of \( n \pmod p = 0 \). Its modular core is unassailable; its signal processing ambition is unproven but electrifying. It’s a “what if” that demands exploration—not yet a theorem, but a seed with the potential to grow into something profound if the gaps are bridged.
Author: 7B7545EB2B5B22A28204066BD292A0365D4989260318CDF4A7A0407C272E9AFB