Number Theory

2608 Submissions

[3] ai.viXra.org:2608.0092 [pdf] submitted on 2026-08-29 21:01:19

An Elementary Construction for Goldbach's Conjecture: Limit Expressions, Error Analysis, and Additive Sieves

Authors: Miles Huang
Comments: 12 Pages. Author retains copyright; non-exclusive license to distribute.

This paper synthesizes a series of results on the (even) Goldbach conjecture and completes two previously unaddressed components: textbf{error analysis} and textbf{fractional sequence encoding (additive sieve)}. The main results are as follows. (1) We systematically present limit-type elementary constructions of the Goldbach conjecture (using both exponential and cosine functions) and their equivalent logical formulations, and extend the method to the Collatz conjecture, the twin prime conjecture, and the prime counting function. (2) For the truncated construction with finite error factor $k$, we derive a rigorous error bound and prove that the criterion holds provided $kn,e^{-4k/n^{2}}ln n$. This determines the textbf{critical exponent for the sufficient condition to be $2$}, and rigorously verifies that the empirically discovered choice $k=10n^{e}$ ($eapprox 2.718>2$) satisfies this sufficient condition for all even $nge 4$, yielding a textbf{limit-free elementary closed-form equivalent proposition} for Goldbach's conjecture. Numerical experiments show that for $nle 120$, the difference between the truncated construction $gamma_{k}(n)$ and the true prime pair count $G(n)$ is less than $10^{-9}$ (see Remark~ef{rem:numerics}).(3) We prove the closed form $A_{j}=j,(d(j)-2)$ of the "additive sieve" sequence ($d$ being the divisor-counting function), so that $A_{j}=0$ if and only if $j$ is prime; we prove that Goldbach's conjecture is equivalent to the existence of zero positions in the symmetric superposition sequence, and give the zero-count identity $#{j}=2G(n)-[,n/2inPP,]$.(4) We completely resolve the textbf{carry problem} in fractional sequence encoding: with block length $dgelceillog_{10} (4n^{3/2}) ceil$, the encoding is carry-free and fully faithful; meanwhile we prove that for fixed $d$ the infinite sequence inevitably carries ($sup_j A_j=infty$), thereby clarifying the precise scope of previous objections to the method's feasibility. (5) We provide conditional proofs for related propositions ($p_1+p_2+3$ and $p_a+p_b+p_c-1$ forms are consequences of the strong Goldbach conjecture), relate the $2p+q$ representation to Lemoine's conjecture with numerical verification up to $10^6$, and correct a limit-order error in the original "irrationality test formula" (the single-limit version actually diverges as $tfrac{sqrt{pi}}{2}k^{3/2}$). It is emphasized: all constructions in this paper are textbf{logical equivalents} of Goldbach's conjecture, not proofs; their value lies in recasting the conjecture into elementary analytic or combinatorial forms with precisely controllable errors.
Category: Number Theory

[2] ai.viXra.org:2608.0052 [pdf] submitted on 2026-08-13 12:06:32

A τ-Vanishing Identity in the Collatz Problem: A Duality Between Base-6 Geometry and Base-2 Algebra

Authors: XinXin Li
Comments: 5 pages. CC BY 4.0.

We establish an exact algebraic identity connecting the cumulative near-conjugacyerror Eσ of a base-6 circle-rotation model of the Collatz map (HonarvarShakibaei Asli, 2026) with the terminal correction ε in the classical stopping-timeidentity. For a convergent Syracuse orbit, Eσ = 1 − log6(5n0+1)+ (log2 n0 + ε)·log6 2, in which the Syracuse stopping time τcancels exactly via the change-of-base formula log2 6 · log6 2 = 1.Consequently Eσ = O(1) ⇔ ε = O(1) ⇔ Σ 1/ni−1 = O(1), so theboundedness of the reciprocal sum becomes the central open question. The identity isverified to machine precision (≤ 2×10−15) for every tested n0 ≤ 77031.
Category: Number Theory

[1] ai.viXra.org:2608.0050 [pdf] submitted on 2026-08-12 20:10:56

Prime Egyptian Fractions: A Spectral Threshold and the Four-Term Case

Authors: Khazri Bouzidi Fethi
Comments: 9 Pages.

We study the Diophantine equation 1/p_1 = 1/p_2 + 1/p_3 + 1/p_4 for distinct primes p_1, p_2, p_3, p_4, a special case of the Sierpiński—Schinzel unit-fraction equations. We introduce a spectral coherence threshold R, derived from an exact infimum formula for a normalized exponential sum over the primes involved via the Kronecker—Weyl equidistribution theorem, and show that R = 1/2 is its sharp transition value. For three primes, we establish an exact identity linking R to prime gaps, giving a new proof that no three consecutive primes satisfy the analogous three-term equation. For four primes, we combine this classification with an exact divisor-based algorithm to search over 6 times 10^8 configurations, finding no solution for p_1 le 5431 but reporting the closest known near-solution. We prove that no solution can occur with the three larger denominators in arithmetic progression, and record a short, fully general divisibility argument showing that no such equation admits a solution in distinct primes for any number of terms — clarifying explicitly that this fact concerns single-prime denominators only, not the composite-denominator setting central to the Sierpiński—Schinzel conjectures.
Category: Number Theory