Logarithmic Tuning: A Novel Approach to Integer Factorization

An interactive research document exploring a classical algorithm with conjectured \(O(\log \log n)\) complexity.

Abstract

The factorization of large semiprimes \(n = p_1 p_2\) is the cornerstone of modern public-key cryptography and remains a computationally intractable problem for classical computers. Current state-of-the-art algorithms, like the General Number Field Sieve (GNFS), have sub-exponential complexity. This document presents a novel classical algorithm, Logarithmic Tuning (LT), which reframes factorization as a signal processing problem. By modeling factors as "spectral nulls" of the probe function \(T(k) = n \pmod k\), LT employs a hyper-efficient sparse search to achieve a conjectured complexity of \(O(\log \log n)\). We present the theoretical framework, an interactive implementation, and outline the critical mathematical conjectures that, if proven, would represent a revolutionary leap in computational number theory.

1. The Core Idea: Factorization as Spectral Null Detection

The fundamental insight is to move away from traditional algebraic approaches and treat the integer \(n\) as a parameter defining a discrete signal domain.

We can think of \(T(k)\) as a "spectral" measurement at probe point \(k\). A factor \(p\) of \(n\) is uniquely identified by the condition \(T(p) = 0\). This is a perfect "spectral null." The challenge is not in detecting the null, but in finding its location \(k=p\) efficiently without testing all possibilities up to \(\sqrt{n}\).

Analogy: This is akin to finding the resonant frequencies of an object. Instead of exciting it at every possible frequency, we can use a sparse set of probes. If a probe causes even a slight resonance, we can quickly fine-tune our frequency in that region to find the exact peak. Here, we search for nulls instead of peaks.

2. The Logarithmic Tuning (LT) Algorithm: A Hybrid Approach

Based on empirical testing, the LT algorithm has evolved into a robust, multi-stage hybrid method.

Stage 1: Heuristic Checks

For efficiency, the algorithm first employs methods that are highly effective for specific classes of numbers.

• Dense Sweep (for small \(n\)): For smaller numbers (e.g., \(n < 10000\)), a simple trial division with odd numbers is often faster than the full LT sweep. This addresses cases where the sparse sweep might "jump over" factors.
• Fermat's Factorization: A limited run of Fermat's method is performed. This is extremely effective if the factors \(p_1, p_2\) are close to each other (i.e., close to \(\sqrt{n}\)).

Stage 2: Sparse Logarithmic Sweep

If heuristic checks fail, the core LT sweep begins. It uses a sparse, exponentially growing sequence of probes to minimize the number of computationally expensive modulo operations:

Stage 3: Adaptive Refinement ("Q-Notch Filter")

If the sweep finds a \(k_i\) where \(T(k_i)\) is below a certain threshold, it triggers a local, adaptive search. This refinement process iteratively adjusts its search position and step size to rapidly converge on the exact null where \(T(k)=0\). It now includes logic to prevent getting stuck in local minima.

Enter a semiprime and click 'Factorize!' to see the algorithm in action.
Test cases to try:
- 143 (11 * 13) - Now handled by the dense sweep for small n.
- 1003 (17 * 59) - A case where the sparse sweep gets a direct hit.
- 10001 (73 * 137) - A case where Fermat's method succeeds quickly.
- 1000000016000000063 (1000000007 * 1000000009) - A larger number to test BigInt and Fermat's.
        

4. The Mathematical Challenge: The Path to Proof

The extraordinary claim of \(O(\log \log n)\) complexity is currently a conjecture. Proving it requires solving two fundamental mathematical problems:

1. Conjecture 1: Sweep Effectiveness. Prove that for any semiprime \(n\), the sparse sweep sequence \(k_i = 2^i + 1\) will reliably identify a region near a factor \(p_j\) within \(O(\log \log n)\) probes. This requires deep results from analytic number theory on the distribution of \(n \pmod k\) for this specific sequence.
2. Conjecture 2: Refinement Convergence. Prove that the adaptive refinement algorithm, once triggered, converges to a factor in a number of steps that does not dominate the overall complexity. This involves analyzing the discrete dynamical system defined by the refinement process.

The Goal: Finding \(f^{-1}(n)\). Successfully proving these conjectures would be equivalent to constructing an efficient numerical algorithm for \(f^{-1}(n)\), where \(f(p_1+p_2) = p_1 p_2\). This would provide a classical, polynomial-time solution to integer factorization.

5. Conclusion and Vision

Logarithmic Tuning presents a paradigm shift in approaching the integer factorization problem. By leveraging analogies from signal processing, it offers a concrete, testable, and potentially revolutionary classical algorithm. While the path to rigorous proof is challenging, the framework itself provides a powerful new lens through which to view the deep structures of number theory. The ultimate goal is to either prove the conjectures and reshape cryptography, or to understand precisely why they fail, which in itself would be a profound insight into the nature of computational complexity.

This document and its interactive tool serve as a call to action for mathematicians, computer scientists, and cryptographers to explore, test, and challenge this framework. The "best shot" is not just in the algorithm itself, but in the collaborative scientific process required to validate it.