Analysis and Status of the Logarithmic Tuning Remainder Conjecture

This document analyzes the conjecture that the Logarithmic Tuning (LT) sweep, using \( k_i = 2^i + 1 \), finds a remainder \( T(k_i) = n \pmod{k_i} < n^{1/4} \) within a small number of steps for a semiprime \( n = pq \).

Conjecture

For \( n = pq \) (\( p \leq q \), odd primes) and \( k_i = 2^i + 1 \), there exists an integer \( i \) in a range such as \( O(\log n) \) or \( O(\log \log n) \) such that \( T(k_i) < n^{1/4} \).

Analysis Components

Lemma 1: Conditional Remainder Bound (Generalized)

Let \( k_i = 2^i + 1 \). If integers \( i \geq 0 \) and \( m \geq 1 \) exist such that the following three conditions hold:

Proximity: \( \quad 0 < mp - k_i < \frac{n^{1/4}}{q} \)
Size: \( \quad mp > n^{1/4} (1 + 1/q) \)
Bound: \( \quad k_i > m n^{1/4} \)

Then the remainder \( T(k_i) = n \pmod{k_i} \) satisfies \( T(k_i) < n^{1/4} \).

Proof:

Let \( r = mp - k_i \). Condition (1) implies \( 0 < r < n^{1/4}/q \).

Establish that \( qr < k_i \): The inequality \( qr < mp - r \) is equivalent to \( r(q+1) < mp \). Since \( r < n^{1/4}/q \), this holds if \( \frac{n^{1/4}}{q} (q+1) < mp \), which is Condition (2). Thus, \( 0 < qr < k_i \).

From \( p = (k_i + r) / m \), we have \( n = pq = q(k_i+r)/m \), leading to \( mn = qk_i + qr \). This implies \( mn \equiv qr \pmod{k_i} \). Since \( T(k_i) \equiv n \pmod{k_i} \), we have \( mT(k_i) \equiv qr \pmod{k_i} \).

Let \( T(k_i) \) be the unique remainder in \([0, k_i-1] \). The congruence means \( mT(k_i) = qr + tk_i \) for some integer \(t\). Since \(m, T(k_i), k_i\) are non-negative and \(qr > 0\), we must have \(qr + tk_i \ge 0\). As \(qr < k_i\), this forces \( t \ge 0 \).

Assume for contradiction that \( t \ge 1 \). Then \( T(k_i) = (qr + tk_i)/m \ge (qr + k_i)/m \). Using Condition (3), \( k_i > m n^{1/4} \), we get:

\( T(k_i) > (qr + m n^{1/4})/m = qr/m + n^{1/4} \).

Since \(qr/m > 0\), this implies \( T(k_i) > n^{1/4} \). This contradicts the goal of finding \(T(k_i) < n^{1/4}\). Therefore, the only possibility consistent with the conditions leading to the desired outcome is \( t = 0 \).

If \( t = 0 \), then \( mT(k_i) = qr \), so \( T(k_i) = qr/m \). Since \( qr < n^{1/4} \) (from Condition 1) and \( m \ge 1 \), we have \( T(k_i) = qr/m \le qr < n^{1/4} \).

Thus, the three conditions together guarantee \( T(k_i) < n^{1/4} \). \(\square\)

Note: Condition (3) ensures the result holds rigorously for all \(m \ge 1\) by forcing \(t=0\).

Analysis of the Existence Hypothesis

The conjecture requires that the three conditions of Lemma 1 hold simultaneously for some integer \( i \) within a target range (\( O(\log n) \) or \( O(\log \log n) \)). This requires demonstrating the existence of \(i \ge 0\) and \(m \ge 1\) such that:

\( \qquad 0 < mp - (2^i + 1) < \frac{n^{1/4}}{q} \quad \) (Proximity)

\( \qquad mp > n^{1/4} (1 + 1/q) \quad \) (Size)

\( \qquad 2^i + 1 > m n^{1/4} \quad \) (Bound)

Status: This combined existence requirement is an unproven Diophantine approximation hypothesis.

Theoretical Obstacles:

Proving existence requires strong control over the distribution of \( k_i = 2^i + 1 \) modulo \( p \), specifically showing it enters the narrow Proximity interval \( (mp - \delta, mp) \), where \( \delta = n^{1/4}/q \), while satisfying the Size and Bound conditions.

Required Precision vs. Known Bounds: The relative width of the target interval compared to the full range \([0, p-1]\) is \( \delta/p = (n^{1/4}/q) / p = n^{1/4} / (pq) = n^{-3/4} \) (since \( n = pq \)). Proving existence requires \( k_i \pmod{p} \) to have discrepancy \( D(I) < n^{-3/4} \) within \( I \approx \log n \) or \( \log \log n \) steps. Standard bounds (e.g., Erdős–Turán, exponential sums) yield \( D(I) \approx 1/\log n \) (e.g., \( 0.036 \) for \( n = 10^{12} \)) or worse, vastly exceeding the required \( 10^{-9} \) (for \( n = 10^{12} \)).
Role of Multiplicative Order: The distribution depends on \( \text{ord}_p(2) \). Small orders (e.g., \( < \log n \)) may cycle early, but don’t ensure the precise hit needed. Large orders (e.g., \( \approx p-1 \)) suggest better equidistribution over the full cycle, but fine-scale control over \( O(\log n) \) steps is unproven and likely intractable.
Interaction of Conditions: Condition (3) (\( k_i > m n^{1/4} \)) implies \( p > n^{1/4} \) when \( m \approx k_i/p \), limiting Lemma 1 to \( p \in (n^{1/4}, n^{1/2}] \). While the absolute width \( \delta = n^{1/4}/q \) is larger for smaller \( p \) (since \( q \approx n/p \)), the relative width \( \delta/p = n^{-3/4} \) remains constant, maintaining a uniform precision challenge across \( p \).

Conclusion on Existence: Simple probabilistic heuristics fail to support the hypothesis reliably, and tools like Diophantine approximation or equidistribution theory lack the precision to guarantee all three conditions within the required steps.

The Existence Hypothesis remains the central barrier, demanding significant theoretical advances.

Numerical Exploration Strategy

Numerical experiments can test the Existence Hypothesis’s plausibility, focusing on \( k_i = 2^i + 1 \) behavior and Lemma 1’s conditions.

Key Questions:

Order Frequency: How often is \( \text{ord}_p(2) < \log n \) or \( < \sqrt{n} \)?
Order Impact: Does small \( \text{ord}_p(2) \) yield earlier success (smaller \( i \)) in satisfying Lemma 1's conditions?
Distribution: Is \( k_i \pmod{p} \) uniform enough over \( i \leq \frac{1}{2} \log n \)? Compare discrepancy to \( n^{-3/4} / p \).
Hit Rate: Probability of \( 0 < mp - k_i < n^{1/4}/q \) for some \( m \)? Scaling with \( n \), \( p/q \)?
Full Success: Frequency of all three conditions met simultaneously? Typical min \( i \)? Dependence on \( p/q \)?

Experiments:

1: Order Stats: Test \( n = 10^6, 10^9, 10^{12} \), \( p/q = 10^{-2}, 10^{-1}, 1 \). Compute \( \text{ord}_p(2) \), histogram ratios.
2: Small Orders: Compare success rates and minimum \( i \) for satisfying Lemma 1's conditions for \( \text{ord}_p(2) < \log n \) vs. \( > \sqrt{n} \).
3: Discrepancy: For \( p = 10^3, 10^5 \), \( n \approx p^2 \), bin \( k_i \pmod{p} \) for \(i \le \frac{1}{2}\log_2 n\), test uniformity, estimate discrepancy \( D(I) \).
4: Success Rate: Sample many \( n \), vary \( p/q \), count simultaneous successes for Lemma 1's conditions, plot min \( i \) vs. \( n \), \( p/q \).

Feasibility: Requires efficient modular arithmetic and factoring (e.g., GMP, ECM). Feasible for \( n \leq 10^{12} \) or higher with optimization and cluster computing.

Results may reveal \( \text{ord}_p(2) \)’s role, distribution limits, and practical success rates, guiding a refined conjecture or theoretical approach.

Conclusion on the Conjecture's Status

This analysis provides a rigorous conditional result:

Proven (Lemma 1): If the sequence \(k_i = 2^i+1\) produces a value satisfying the specified Proximity, Size, and Bound conditions relative to a multiple of \(p\), then the remainder \(n \pmod{k_i}\) is guaranteed to be less than \(n^{1/4}\).

However, the conjecture requires these conditions to be met simultaneously within a bounded number of steps. This relies on:

Unproven (Existence Hypothesis): The necessary Diophantine approximation property – that \(2^i+1\) *must* simultaneously satisfy all three conditions for some \(i, m\) within \(O(\log n)\) or \(O(\log \log n)\) steps – lacks theoretical proof and faces significant hurdles related to the required precision and known distribution bounds.

Overall Status: The Logarithmic Tuning remainder conjecture remains unproven due to the unresolved Existence Hypothesis. Lemma 1 provides a solid foundation by establishing sufficient conditions for the desired outcome, but demonstrating these conditions are met remains an open problem. Numerical exploration offers a path to gain further insight and test plausibility.

Directions for Future Work

Resolving the conjecture requires tackling the Existence Hypothesis, potentially guided by numerical results. Key directions include:

Proving the simultaneous satisfaction of the three conditions in Lemma 1 for some \(i, m\) within the target ranges, perhaps for specific classes of \(p\) identified numerically.
Developing new results on the fine-scale distribution of \(2^i \pmod p\) for small \(i\), potentially focusing on structures observed in simulations.
Analyzing the interplay between the three conditions more deeply – can Condition (3) be relaxed or are there scenarios where it's automatically satisfied when the others are?
Employing more sophisticated probabilistic models or average-case analysis informed by empirical distributions.
Conducting the proposed extensive numerical experiments to test the frequency of simultaneous condition satisfaction and guide theory.
Investigating alternative sequences \(k_i\) that might possess provably better Diophantine properties relevant to Lemma 1's conditions.