| ai.viXra.org > Number Theory > 2607 |
|
 |
| ai.viXra.org > Number Theory > 2607 |
|
|
[3] ai.viXra.org:2607.0063 [pdf] submitted on 2026-07-26 22:54:26
Authors: Yaakov Abdelhak
Comments: 3 Pages.
We investigate the integer solutions of the Diophantine relation = for non- square integers when n> 2. By projecting parameters and as the functional side lengths of a variable geometric boundary, we leverage the Law of Cosines (LOC) to track rational and irrational distributions. Using the integer constraints inherent to Fermat's hypothesis, we demonstrate a structural field mismatch under acute angular transformations, bypassing perfect square integers handled via classical infinite descent.
Category: Number Theory
[2] ai.viXra.org:2607.0012 [pdf] submitted on 2026-07-07 20:45:06
Authors: Chen Ah Yaw
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a deterministic primality sieve derived from the multiplicative structure of integers modulo 30. All primes greater than 5 are confined to the reduced residue classes R = 1, 7, 11, 13, 17, 19, 23, 29. We prove that the set R forms a multiplicative group modulo 30, and consequently, the product of any two numbers from these columns remains within the same columnar structure. This allows us to construct a complete multiplication table that generates every composite number coprime to 30. Our central theorem states that for any fixed column c ∈ R, the numbers in that column which are not generated by the multiplication table are exactly the prime numbers. Critically, we demonstrate that this structural sieve provides a direct mathematical explanation for the irregularity of prime distribution: primes are the set-theoretic complement of dynamically spaced, overlapping arithmetic progressions. Based on this, we derive an optimized trial-division algorithm that tests only divisors of the form r + 30k, reducing the number of trial divisions by 73.3% compared to naive trial division and 46.7% compared to odd-number-only division.
Category: Number Theory
[1] ai.viXra.org:2607.0003 [pdf] submitted on 2026-07-03 21:58:27
Authors: Haizhu Wu
Comments: 55 Pages. Zenodo DOI: 10.5281/zenodo.21165454.
We present a complete, rigorous, and unconditional proof of Polignac’s conjecture, which asserts that for every even integer y ≥ 2, there exist infinitely many pairs of primes (q, q+y). The twin prime conjecture corresponds to the special case y = 2. The proof introduces a fundamentally new framework that combines five key in novations. First, a geometric primality criterion establishes that in the interval (p, p2), an integer is prime if and only if it is coprime to all primes ≤ p. This deterministic equivalence completely circumvents the parity problem of classical sieve theory. Second, a double arithmetic progression structure with CRT pre-sieving processes small primes r ≤ log p via the Chinese Remainder Theorem, constructing an admissible set A0 modulo M0 = Q r≤log p r. This ensures the crucial condition gcd(M0, d) = 1 for all subsequent moduli d. Third, physical-space period cancellation exploits the fact that the arithmetic progression a + tM0 (mod d) traverses a complete residue system, causing error contributions from complete periods to vanish identically. Fourth, resonance breaking decomposes exponential sums over A0 into 3π(log p) = po(1) explicit terms, eliminating the large factor |A0| from estimates. Fifth, a four-zone modulus partition applies distinct estimation techniques to different ranges of moduli d in the Möbius expansion. All error terms are rigorously shown to be of strictly lower order than the main term C0p2/(log p)2, where C0 > 0 equals the Hardy-Littlewood constant for the pair (0, y). The proof uses only classical Fourier analysis on finite abelian groups, elementary number theory, and standard estimates from analytic number theory. No unproven hypotheses are assumed.
Category: Number Theory