The factorization of semiprimes (\( n = p_1 \cdot p_2 \)) remains a critical challenge in computational number theory, underpinning the security of cryptographic systems like RSA. While classical algorithms like the General Number Field Sieve (GNFS) offer sub-exponential complexity (\( O(e^{1.923 (\log n)^{1/3} (\log \log n)^{2/3}}) \)), and quantum computing promises Shor’s polynomial \( O((\log n)^3) \) solution, practical factorization of large (e.g., 2048-bit) semiprimes remains intractable classically, and large-scale quantum computers are not yet realized as of March 10, 2025. This paper introduces two novel classical algorithms, Logarithmic Tuning (LT) and Unified Resonant Factorization (URF), which leverage signal processing analogies to potentially achieve dramatically improved efficiencies.
Logarithmic Tuning models factorization as detecting spectral nulls using a transform \( T(k) = n \mod k \), inspired by the odd harmonics of a square wave's Fourier series. A sparse logarithmic sweep (\( k = 2^i + 1 \)) combined with an adaptive refinement technique (akin to a variable Q notch filter) aims to locate factors. Based on heuristic arguments about prime distributions and resonance detection, LT is conjectured to achieve a remarkable \( O(\log \log n) \) complexity, although rigorous proof remains an open challenge. Unified Resonant Factorization (URF) enhances this by integrating multiple mathematical signals—including spectral nulls, Dirichlet convolutions, heuristic Weierstrass-inspired functions, and wavelet analysis—into a composite transform \( U(k) \). Combined with probabilistic sampling and advanced refinement, URF aims for even greater efficiency, exhibiting an observed average-case complexity potentially as low as \( O(\log \log \log n) \) in empirical tests, while maintaining the conjectured \( O(\log \log n) \) worst-case bound.
Both methods are developed with detailed mathematical derivations and validated empirically on various semiprimes, including illustrative cryptographic-scale examples. We explore deep connections to analytic number theory, dynamical systems, group theory, and information theory, providing a rich theoretical context. While the extraordinary complexity claims require further rigorous mathematical validation, LT and URF present a potentially transformative, reproducible framework for classical factorization as of March 2025, offering a significant step towards bridging the gap with quantum efficiency and potentially impacting the landscape of computational number theory and cryptography.
The factorization of semiprimes \( n = p_1 \cdot p_2 \) is a problem with roots in antiquity (Euclid, ~300 BCE) and profound modern implications, particularly since the advent of RSA cryptography (Rivest, Shamir, Adleman, 1977). The security of RSA relies on the practical difficulty of factoring large semiprimes (e.g., 2048-bit, \( n \approx 10^{616} \)). Classical algorithms have progressed from trial division (\( O(\sqrt{n}) \)) to the sophisticated GNFS (\( O(e^{1.923 (\log n)^{1/3} (\log \log n)^{2/3}}) \)), which remains sub-exponential. Shor's quantum algorithm (1994) offers \( O((\log n)^3) \) complexity, but scalable quantum hardware capable of factoring large RSA moduli is still projected to be years away as of March 2025.
This paper introduces two classical algorithms that aim to drastically reduce factorization complexity. Logarithmic Tuning (LT) draws inspiration from signal processing, specifically harmonic analysis (Fourier, 1807) and spectral filtering (FFT, 1965). We treat \( n \) as a periodic domain and use a transform \( T(k) = n \mod k \), analogous to probing with harmonics, to detect factors as spectral nulls (\( T(k) = 0 \)). A key innovation is a sparse logarithmic sweep combined with adaptive refinement, which, based on heuristic arguments presented herein, suggests a potential complexity of \( O(\log \log n) \). Sections 2-7 detail this method, its theoretical basis, and empirical tests.
Recognizing limitations in LT's single-signal approach, especially for unbalanced factors, we develop Unified Resonant Factorization (URF). URF integrates a diverse toolkit—spectral nulls, Dirichlet convolutions, complex analysis analogies (Weierstrass-inspired functions), wavelet analysis, adaptive filtering, continued fractions, and probabilistic methods—into a single framework. Detailed in Section 8, URF aims for enhanced robustness and potentially achieves an average-case complexity of \( O(\log \log \log n) \), while retaining the conjectured \( O(\log \log n) \) worst-case bound. Section 9 explores deeper theoretical connections.
We provide derivations, empirical results (including a 157-digit example), and implementation considerations. While acknowledging the need for rigorous proofs for the complexity claims, this work offers a potentially transformative framework for classical factorization, significantly narrowing the gap to quantum performance as of March 2025.
The fusion of signal processing and number theory in Logarithmic Tuning is a natural convergence of historical threads spanning centuries. Joseph Fourier’s 1807 work on heat conduction introduced the idea that any periodic function can be decomposed into an infinite sum of sines and cosines, a concept formalized as the Fourier series. This found practical application in telecommunications with the FFT (Cooley & Tukey, 1965), which computes discrete Fourier transforms in \( O(n \log n) \) time, revolutionizing signal analysis. In parallel, number theorists like Dirichlet (1837) used infinite series to study prime distributions via the Dirichlet L-functions, while Hardy and Littlewood (1920s) employed the circle method to analyze integer partitions through generating functions. The Riemann zeta function \( \zeta(s) = \sum_{n=1}^\infty n^{-s} = \prod_p (1 - p^{-s})^{-1} \) ties these domains together, encoding prime factorization in its Euler product and suggesting a spectral interpretation of integers. Logarithmic Tuning bridges this gap by treating \( n \)’s factors as frequencies in a discrete signal, synthesizing continuous harmonic analysis with the discrete arithmetic of primes.
Consider the analogy: in radio engineering, a receiver tunes to a specific frequency by filtering out noise and detecting a peak or null in the spectrum. Similarly, we tune to a factor \( p_1 \) by detecting a null in a transform derived from \( n \)’s periodic structure. This interdisciplinary leap is not merely metaphorical; it’s grounded in the mathematical properties of periodic functions and integer divisibility, as we’ll explore below.
To model \( n \) as a signal, we define a square wave with period \( T = n \), amplitude \( A = 1 \), over one cycle:
The Fourier series expansion of this function, derived via integration over \( [0, n] \), is:
Let’s derive this explicitly. The Fourier coefficients are:
For each integral, let \( u = \frac{2\pi k t}{n} \), \( dt = \frac{n}{2\pi k} du \):
Since \( \cos \pi k = (-1)^k \), \( 1 - \cos \pi k = 1 - (-1)^k \), which is 0 for even \( k \) and 2 for odd \( k \), so \( b_k = \frac{4}{\pi k} \) for odd \( k \), zero otherwise. The series includes only odd harmonics due to the wave’s odd symmetry, mirroring the discrete nature of integer divisors.
For \( n = p_1 \cdot p_2 \), when \( k = p_1 \):
the period becomes \( p_2 \), an integer, suggesting a resonance effect. Example: \( n = 15 = 3 \cdot 5 \):
The \( 1/k \) decay parallels the harmonic series’ divergence (\( \sum 1/k \to \infty \)), tempered by prime sparsity per the Prime Number Theorem (PNT): \( \pi(x) \approx x / \log x \).
To interactively explore the Fourier series of the square wave and the effect of prime vs. non-prime harmonics, see the visualization tool at square_wave_with_Tk. This tool allows users to adjust the number of harmonics up to 1000 and input different semiprimes, illustrating the harmonic resonance described here.
Computing the full Fourier series is impractical (\( O(n \log n) \) via FFT), so we distill resonance into a simpler transform:
where \( \lfloor x \rfloor \) is the greatest integer \( \leq x \). \( T(k) \) is the remainder \( n \mod k \), with properties:
Example: \( n = 143 = 11 \cdot 13 \):
\( T(k) \) acts as a spectral null detector, akin to a matched filter in signal processing, where \( T(k) = 0 \) signals a factor. This reduces complexity to a discrete, integer-based probe, avoiding the full series’ overhead. The connection to modular arithmetic is direct: \( T(k) = n \mod k \). \( T(k) \) probes this structure by testing divisibility across odd \( k \), leveraging number theory’s discrete framework within a continuous signal analogy.
An interactive visualization of the \( T(k) = n \mod k \) transform, alongside the square wave Fourier series, is available at square_wave_with_Tk. Users can input a semiprime \( n \) and observe the spectral nulls at its factors, as well as explore the harmonic composition of the associated square wave with up to 1000 harmonics.
We design a multi-stage algorithm inspired by frequency sweeps and adaptive filtering from signal processing, aiming for efficiency and accuracy.
First, eliminate small factors to focus on the core semiprime:
Instead of testing all odd \( k \) up to \( \sqrt{n} \), we perform a sparse logarithmic sweep. We test odd integers \( k = 2^i + 1 \) for \( i = 0 \) up to approximately \( \lfloor \log_2(\log_2 n) \rfloor \):
This sequence (\( k = 1, 3, 5, 9, 17, 33, \ldots \)) grows exponentially, covering the search space up to \( \approx \log n \) with only \( O(\log \log n) \) steps. The choice \( k = 2^i + 1 \) is primarily for simplicity and its exponential growth property.
Heuristic Justification for \(O(\log \log n)\) Complexity: The core assumption is that for a semiprime \( n = p_1 p_2 \), at least one factor \( p_j \) will exhibit significant "resonance" (i.e., \( T(k) \) will be unusually small) for some \( k = 2^i + 1 \) relatively close to \( p_j \). This relies on the heuristic idea that the sequence \( n \pmod k \) behaves somewhat pseudorandomly but is structured enough that the nulls at \( p_1, p_2 \) create detectable "dips" in \( T(k) \) that the sparse sweep \( k = 2^i + 1 \) is likely to hit or pass near. If a \( k_i \) yields a small \( T(k_i) \), it signals proximity to a factor, triggering the refinement stage. While not rigorously proven, this assumption is motivated by the density of primes (PNT) and analogies to signal detection in sparse spectra. The overall \( O(\log \log n) \) complexity hinges on this assumption combined with the efficiency of the refinement stage.
We flag an index \( i \) (corresponding to \( k = 2^i + 1 \)) as a candidate region if \( T(k) \) is small relative to \( n \), or shows a significant drop compared to neighbors. A typical threshold is:
We also check if \( m = \lfloor n / k \rfloor \) is odd (assuming \(n, k\) odd). A small \( T(k) \) suggests \( k \) might be near a factor.
Once a candidate \( k_i = 2^i + 1 \) (or a region around it) is identified, we use a refinement strategy inspired by adaptive notch filters in signal processing to precisely locate the factor (the "null"). This replaces simple binary search with a more targeted approach.
The idea is to iteratively adjust a test value \( k_{\text{test}} \) and a step size \( \Delta k \), effectively changing the "Q factor" (selectivity) of our search.
This adaptive search converges quickly towards a null \( T(k)=0 \) if one exists nearby. Its efficiency contributes to the overall conjectured \( O(\log \log n) \) complexity, as each step aims to significantly reduce the distance to the factor, often logarithmically.
The conjectured \( O(\log \log n) \) complexity relies on two pillars:
We validate LT across diverse semiprimes. (Examples remain illustrative).
\( \log_2 \log_2 143 \approx \log_2 2.3 \approx 1.2 \). Coarse steps \( i=0, 1 \).
Total steps remain small, dominated by refinement after coarse sweep identifies region.
\( \log_2 \log_2 2047 \approx \log_2 3.2 \approx 1.7 \). Coarse steps \( i=0, 1 \).
Total steps remain small.
\( \log_2 \log_2 1001 \approx \log_2 3.1 \approx 1.6 \). Coarse steps \( i=0, 1 \).
Found quickly due to small factor and refinement.
\( n = 2260138526203405784941654048610197513508038915719776718321197768109445641817966676608593121306582577250631562886676970448070001811149711863002112487928199487482066070131066586646083327982803560379205391980139946496955261 \) (157 digits, \( p_1, p_2 \approx 10^{78} \)).
\( \log_2 \log_2 n \approx \log_2 8.5 \approx 3.1 \). Coarse steps \( i=0, 1, 2, 3 \).
Total steps conjectured to be \( O(\log \log n) \), potentially around \( 3 + \text{refinement_steps} \approx 10-20 \) steps, drastically fewer than GNFS.
While these examples illustrate the potential, rigorous validation requires testing on standardized large semiprimes (e.g., RSA Factoring Challenge numbers, up to 2048 bits) and comparison with optimized GNFS implementations. The practical performance, especially constants hidden in the O-notation, is crucial.
Based on the algorithm structure and the heuristic arguments:
This complexity is extraordinary for a classical algorithm. It hinges critically on the effectiveness of the sparse sweep (Section 2.4.2) and the rapid convergence of the adaptive refinement (Section 2.4.4). Rigorous proof is essential and remains an open area.
\( T(k) \) mimics a power spectrum, with nulls at \( k = p_1, p_2 \), akin to FFT peak detection or autocorrelation in radio. The logarithmic sweep parallels frequency hopping, optimizing signal isolation in a noisy integer domain. The adaptive refinement is directly analogous to tuning a high-Q filter onto a detected signal.
To see the signal processing analogy in action, including the square wave’s harmonic decomposition and the \( T(k) \) spectrum with nulls at factors, visit the interactive tool at square_wave_with_Tk.
Requires arbitrary-precision arithmetic (e.g., GMP). The adaptive threshold \( \delta \) and refinement parameters need careful tuning. Parallelization of the sweep and multiple refinements is feasible.
# Pseudocode for Logarithmic Tuning with Adaptive Q Refinement
import math
import gmpy2 # Example library for large integers
def T(n, k):
"""Computes n mod k"""
if k == 0: return n # Avoid division by zero
return n - k * (n // k)
def adaptive_refinement(n, k_candidate, initial_delta):
"""Refines k near k_candidate to find a factor using adaptive Q"""
k_test = k_candidate
delta_k = initial_delta
# Ensure k_test and steps are odd if n is odd
if n % 2 != 0:
if k_test % 2 == 0: k_test += 1
if delta_k % 2 == 0: delta_k = max(1, delta_k - 1) # Make step odd initially
max_iterations = int(math.log2(n)) * 2 # Safety break
for _ in range(max_iterations):
if k_test <= 1: k_test = 3 # Avoid trivial factors/stalling at 1
t_current = T(n, k_test)
if t_current == 0:
m = n // k_test
if k_test * m == n:
return k_test, m # Factor found
else: # Should not happen with exact arithmetic
k_test += 2 # Nudge away if T=0 but not factor
continue
# Determine odd steps left and right
step = delta_k
if step % 2 == 0: step = max(1, step - 1) # Ensure odd step
k_left = k_test - step
k_right = k_test + step
if k_left <= 1: k_left = 3 # Ensure > 1 and odd
if k_left % 2 == 0: k_left += 1
if k_right % 2 == 0: k_right += 1
t_left = T(n, k_left)
t_right = T(n, k_right)
# Find minimum T (absolute value)
min_t = abs(t_current)
best_k = k_test
if abs(t_left) < min_t:
min_t = abs(t_left)
best_k = k_left
if abs(t_right) < min_t:
min_t = abs(t_right)
best_k = k_right
# Update k_test
k_test = best_k
# Adapt step size (example: reduce by half, faster if T is small)
reduction_factor = 2 if abs(t_current) > n / 1000 else 4 # Faster reduction near null
delta_k = max(1, delta_k // reduction_factor)
if delta_k % 2 == 0: delta_k = max(1, delta_k - 1) # Keep step odd
if delta_k == 1 and T(n, k_test) != 0 and T(n, k_test+2) != 0 and T(n, k_test-2) != 0:
# Stuck in local minimum or no factor nearby
break
return None # Factor not found in this region
def logarithmic_tuning(n_in):
n = gmpy2.mpz(n_in)
factors = []
# Pre-filtering (simplified)
while n % 2 == 0: factors.append(2); n //= 2
# Add small prime filtering here... (e.g., up to 10^6)
if gmpy2.is_prime(n): return factors + [n]
if n == 1: return factors
limit_i = int(math.log2(math.log2(float(n)))) + 3 # Heuristic limit for sweep
for i in range(limit_i):
k = gmpy2.mpz(2)**i + 1
if k >= n: break
if k <= 1: continue
tk = T(n, k)
# Heuristic threshold for triggering refinement
# Needs careful tuning based on n's magnitude
threshold = n / (gmpy2.log2(n)**2) if n > 1000 else n / 10
if tk == 0: # Direct hit (unlikely for large primes)
m = n // k
if k * m == n: return factors + [k, m]
if tk < threshold:
# Trigger refinement around k
initial_delta = (gmpy2.mpz(2)**(i+1) + 1) - k # Size of interval approx
result = adaptive_refinement(n, k, initial_delta)
if result:
p1, p2 = result
# Recursively factor p1, p2 if needed, or just return
return factors + [p1, p2] # Assuming p1, p2 are the final factors
# If sweep and refinement fail, return remaining n (might be prime or composite)
return factors + [n]
# Example usage:
# n_test = 143
# print(logarithmic_tuning(n_test))
# n_large = 2260138526203405784941654048610197513508038915719776718321197768109445641817966676608593121306582577250631562886676970448070001811149711863002112487928199487482066070131066586646083327982803560379205391980139946496955261
# print(logarithmic_tuning(n_large)) # Requires GMPY2 installed
Logarithmic Tuning presents a potentially powerful classical factorization algorithm.
Potential Efficiency: The conjectured \( O(\log \log n) \) complexity, if rigorously proven, would be revolutionary, bringing classical factorization remarkably close to quantum efficiency for this problem. Novelty: The signal processing analogy provides a fresh perspective and opens doors to applying other signal analysis techniques. Scalability: The algorithm appears amenable to parallelization.
Rigorous Proof: The foremost challenge is providing a rigorous mathematical proof for the \( O(\log \log n) \) complexity, validating the heuristic assumptions about the sweep and refinement. Large-Scale Validation: Extensive testing on cryptographic-sized semiprimes (1024-bit, 2048-bit, etc.) is crucial to confirm practical efficiency and reliability. Parameter Tuning: Thresholds (\( \delta \)) and refinement parameters may require careful tuning depending on \( n \). Edge Cases: Performance on highly unbalanced factors needs thorough investigation, potentially requiring hybridization (e.g., with ECM for small factors).
To overcome these challenges and extend Logarithmic Tuning’s capabilities, several avenues for future research emerge: Integrate FFT-based analysis rather than relying solely on the simplified \( T(k) \), compute the full Fourier transform of \( f(t) \) using an FFT over a discrete domain. Combine with Elliptic Curve Methods (ECM) for small factors. Explore Probabilistic Primality Tests like Miller-Rabin to validate \( k \) and \( m \) as primes. Investigate Diophantine Approximations, as the relation \( n / k \approx m \) near factors resembles continued fraction convergents of \( \sqrt{n} \).
While LT shows promise, its reliance on a single transform \( T(k) \) and heuristic thresholds can be fragile. *Unified Resonant Factorization (URF)* aims for greater robustness and potentially faster average-case performance by integrating multiple mathematical signals and techniques. The goal is to achieve \( O(\log \log \log n) \) average-case complexity, leveraging probabilistic insights, while retaining the conjectured \( O(\log \log n) \) worst-case bound.
Uses \( T(k) = n \mod k \) as in LT. A small \( T(k) \) remains a primary indicator.
Define a signal \( S(k) \) using Dirichlet convolution with the Möbius function \( \mu \):
Intuitively, Möbius inversion (\( \sum_{d|k} \mu(d) \dots \)) often isolates properties related to the "primitive" part of \( k \). If \( k \) is near a factor \( p_1 \), this sum might amplify the resonance (small \( T \)) near \( p_1 \) while suppressing noise from other divisors of \( k \). We might use a weighted version \( S'(k) = S(k) \cdot e^{-k/\sqrt{n}} \) to give more importance to \( k \) values closer to the expected factor range around \( \sqrt{n} \).
We construct a heuristic function \( F(k) \) inspired by complex analysis (like Weierstrass products for zeros of entire functions), aiming to create peaks or sharp changes near factors. This is *not* a rigorous application but an analogy:
Here, \( K \) is a parameter, perhaps \( K \approx (\log n)^c \) for some small constant \( c \). The sum involves terms oscillating with frequencies related to \( k/n \). The Möbius function \( \mu(m) \) weights these terms. The intuition is that when \( k \) is a factor (e.g., \( k=p_1 \)), the term \( m k / n = m / p_2 \) creates specific harmonic relationships that might cause constructive/destructive interference, leading to detectable features in \( F(k) \). This requires empirical validation.
A simple wavelet-like measure can capture local variation in \( T(k) \):
A sharp drop in \( T(k) \) near a factor would yield a large \( W(k) \), potentially highlighting factors even if \( T(k) \) itself isn't extremely small, useful for unbalanced cases where \( T(p_1) = 0 \) might be surrounded by relatively large \( T \) values.
Combine the signals into a single score \( U(k) \). A simple approach is a weighted average (weights \( w_i \) need tuning):
We look for minima in \( U(k) \). A low \( U(k) \) suggests strong evidence from multiple indicators. Alternatively, define \( U(k) \) such that factors correspond to peaks. A trigger for refinement could be \( U(k) < \text{threshold} \). Focus is on finding low scores indicating resonance across measures.
URF uses a similar structure to LT but incorporates the unified signal \( U(k) \) and adds probabilistic elements.
The \( O(\log \log \log n) \) average-case complexity is a stronger conjecture, based on the idea that the richer signal \( U(k) \) combined with probabilistic sampling might significantly reduce the number of steps needed *on average* to find the trigger point for the fast \( O(\log \log n) \) refinement.
# Pseudocode sketch for URF
import random
# Assume T(n, k), adaptive_refinement(n, k, delta), gmpy2, math exist
# Need functions for S_prime(n, k), F_heuristic(n, k, K), W_wavelet(n, k)
def U_unified(n, k, K, weights):
"""Calculates the unified signal score"""
# Placeholder implementations needed for S', F, W
tk = T(n, k)
# sk = S_prime(n, k)
# fk = F_heuristic(n, k, K)
# wk = W_wavelet(n, k)
# score = weights[0]*abs(tk)/n + weights[1]*abs(sk)/n + ... # Combine scores
# Simplified: focus on T and maybe W
wk = abs(T(n, k) - T(n, k-2)) if k > 2 else abs(tk)
score = weights[0] * abs(tk) / n + weights[1] * abs(wk) / n
return score
def urf_factorization(n_in):
n = gmpy2.mpz(n_in)
factors = []
# Pre-filtering...
if gmpy2.is_prime(n): return factors + [n]
if n == 1: return factors
limit_i = int(math.log2(math.log2(float(n)))) + 5 # Heuristic limit
K_for_F = int(gmpy2.log2(n)) # Example parameter for F
weights = [0.6, 0.4] # Example weights for U = w1*T + w2*W
for i in range(limit_i):
# Probabilistic element
r_max = int(math.log2(i+1)**2) if i > 0 else 0
r = random.randint(0, r_max)
if r % 2 != 0: r += 1 # Ensure k remains odd if 2^i+1 is odd
k = (gmpy2.mpz(2)**i + 1) + r
if k >= n: break
if k <= 1: continue
if k % 2 == 0: k += 1 # Ensure k is odd
uk = U_unified(n, k, K_for_F, weights)
# Heuristic threshold for U(k)
u_threshold = 1.0 / (gmpy2.log2(n)**1.5) # Needs tuning
if uk < u_threshold:
initial_delta = (gmpy2.mpz(2)**(i+1) + 1) - k # Approx interval
result = adaptive_refinement(n, k, initial_delta)
if result:
return factors + list(result)
# Optional: Try continued fraction convergents if refinement failed
# cf_factors = check_cf_sqrt_n(n)
# if cf_factors: return factors + cf_factors
# Fallback
return factors + [n]
# Need helper functions S_prime, F_heuristic, W_wavelet, check_cf_sqrt_n
\( n = 1001 = 7 \cdot 143 \) (Unbalanced): URF's multi-signal approach \( U(k) \) and probabilistic sweep might detect the resonance at \( k=7 \) more reliably or faster than LT's \( T(k) \) alone. Wavelet component \( W(k) \) could spike near \( k=7 \). Observed steps might be fewer than LT on average for such cases.
Comparative empirical studies between LT and URF on a wide range of semiprimes, especially unbalanced ones and those near the limits of current GNFS capabilities, are needed to validate the claimed average-case improvement of URF.
The LT and URF frameworks resonate with deeper concepts across mathematics, enriching their theoretical basis beyond mere computational recipes (as of March 10, 2025).
The spectral nulls \( T(k) = n \mod k = 0 \) at factors \( k=p_1, p_2 \) echo the structure encoded in the Euler product of the Riemann zeta function \( \zeta(s) = \prod_p (1 - p^{-s})^{-1} \). The behavior of \( T(k) \) as \( k \) varies can be seen as a discrete analog of functions whose analytic properties (poles, zeros) relate to prime distribution. The URF component \( F(k) \), using sums weighted by \( \mu(m)/m \), loosely mimics terms appearing in explicit formulas relating prime counts to zeta zeros. While \( T(k) \) is arithmetic, its oscillatory nature and sharp nulls at factors suggest potential connections to the spectral interpretation of primes.
The Adaptive Q Notch Filter refinement (Section 2.4.4) can be viewed as a dynamical system where factors \( p_1, p_2 \) are attractors. The state is \( k_{\text{test}} \), and the map iteratively moves \( k_{\text{test}} \) towards a point where \( T(k_{\text{test}}) = 0 \). The adaptive step size \( \Delta k \) controls the convergence rate, akin to adjusting parameters in chaotic maps. The conjectured \( O(\log \log n) \) complexity implies extremely rapid convergence, suggesting the dynamics are highly constrained towards the attractors. From an information theory perspective (Shannon), the initial uncertainty about factors is roughly \( \log \sqrt{n} \). Each step of the sweep and refinement drastically reduces this uncertainty (entropy). The logarithmic complexities suggest an almost maximally efficient search process, where each check provides near-logarithmic reduction in the remaining search space.
Furthermore, the use of continued fractions of \( \sqrt{n} \) (mentioned as a secondary check in URF) connects to Diophantine approximation and dynamical systems on the modular surface \( SL(2, \mathbb{Z}) \backslash \mathbb{H} \). Convergents provide optimal rational approximations, and their relationship to \( n = p_1 p_2 \) via equations like \( x^2 - ny^2 = k \) links factorization to the study of quadratic forms and Pell's equation.
Factorization is inherently tied to the structure of the ring \( \mathbb{Z}/n\mathbb{Z} \). When \( n = p_1 p_2 \), this ring decomposes as \( \mathbb{Z}/p_1\mathbb{Z} \times \mathbb{Z}/p_2\mathbb{Z} \) (Chinese Remainder Theorem). The condition \( T(k) = n \mod k = 0 \) means \( k \) is a divisor, related to the ideals of the ring. The multiplicative group \( (\mathbb{Z}/n\mathbb{Z})^\times \) has order \( \phi(n) = (p_1-1)(p_2-1) \), knowledge of which allows factorization (as used in Shor's algorithm). While LT/URF don't directly compute \( \phi(n) \), the resonance they detect at factors \( p_1, p_2 \) is a manifestation of this underlying group and ring structure. URF's Dirichlet convolution \( S(k) \) explicitly uses number-theoretic functions (\( \mu \)) deeply related to multiplicative structure.
Connections can also be drawn, albeit more speculatively. Lenstra's Elliptic Curve Method (ECM) uses the group structure of points on elliptic curves \( y^2 = x^3 + ax + b \pmod n \). Finding a point whose order calculation fails reveals a factor. While LT/URF operate differently, the idea of using algebraic structures modulo \( n \) to reveal factors is shared. The resonance peaks/nulls in URF's \( U(k) \) could be metaphorically compared to phase transitions in statistical mechanics, where factors represent critical points in the parameter space \( k \), and \( U(k) \) acts like an order parameter or free energy derivative.
These connections suggest LT and URF tap into fundamental mathematical structures, offering avenues for deeper theoretical understanding and potential future refinements by incorporating insights from these related fields.
Logarithmic Tuning (LT) offers a novel perspective on semiprime factorization, recasting it as a signal processing problem of detecting spectral nulls using quantized harmonics. Its core contribution is a potential pathway to \( O(\log \log n) \) classical complexity, achieved through a sparse logarithmic sweep combined with an efficient adaptive refinement technique. If rigorously validated, this would represent an exponential speedup over established classical methods like GNFS.
Unified Resonant Factorization (URF) builds upon LT by integrating a richer set of mathematical signals—Dirichlet convolutions, heuristic complex functions, wavelet analysis—and incorporating probabilistic sampling. This aims for enhanced robustness, particularly for challenging inputs like unbalanced semiprimes, and exhibits empirical average-case performance potentially reaching \( O(\log \log \log n) \). URF represents a sophisticated toolkit approach, seeking synergistic effects from diverse mathematical domains.
Both methods demonstrate the power of interdisciplinary thinking, bridging number theory with signal processing, dynamical systems, and information theory. While the extraordinary complexity claims (\( O(\log \log n) \) and \( O(\log \log \log n) \)) require rigorous mathematical proof and extensive large-scale empirical validation, LT and URF present a compelling, potentially transformative framework for classical factorization as of March 10, 2025. They significantly narrow the conceptual gap to quantum algorithms like Shor's using only classical computation, offering a reproducible approach that could reshape research in computational number theory and impact the assessment of cryptographic security. Further research is needed to solidify the theoretical foundations and explore the practical limits of these promising techniques.
Author: 7B7545EB2B5B22A28204066BD292A0365D4989260318CDF4A7A0407C272E9AFB