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[14] ai.viXra.org:2610.0070 [pdf] submitted on 2026-10-31 21:56:00
Authors: Ibrahim Amin Nicola Shnoudi
Comments: 18 Pages.
A positive integer n is almost perfect if σ(n) = 2n − 1. Every power of 2 is almost perfect, and the conjecture that there are no others is open. This project does not prove it. It proves the conjecture for odd parts with at most two prime factors, with a machine-checked proof, and it reduces each remaining case to a precisely stated problem.
Category: Number Theory
[13] ai.viXra.org:2610.0064 [pdf] submitted on 2026-10-27 02:39:34
Authors: Chaiya Tantisukarom
Comments: 7 Pages.
We study a deterministic procedure, denoted $mathcal{P}$, built from three operations: $3n+1$ on odd values, division by two ($op{d2}$) on even values, and a correction step $-1$ that is applied whenever the current odd value exceeds the previous odd value. The correction keeps the sequence of odd values below its starting point, so the ``lower part'' of every start is maintained by construction. We prove that for each odd start $nge 3$ the procedure reaches an odd value $G(n)<n$ in a number of operations determined by a $2$-adic valuation, that $G$ has a closed form, and that iterating $G$ reaches $1$ in at most $lfloor log_{4/3}(n-1)floor+1$ descents. We then show that $mathcal{P}$ coincides with the Collatz map only until its first correction, and we apply the minimal-counterexample (contradiction) method. The argument closes if and only if a single merging statement holds: for every $yequiv 2 pmod 3$ the Collatz orbits of $y$ and $2y+1$ meet. This statement is equivalent to the Collatz conjecture, so $mathcal{P}$ locates the difficulty precisely but does not remove it. Worked examples are tabulated.
Category: Number Theory
[12] ai.viXra.org:2610.0059 [pdf] submitted on 2026-10-25 10:53:31
Authors: J. W. McGreevy
Comments: 352 Pages. Volume V (Lemmas 86-118) of the programme Thermodynamics of Arithmetic Action. Previous volumes: ai.viXra:2610.0045, 2610.0054, 2610.0052, 2610.0056, 2610.0055.
This volume collects TAA Lemmas 86-118 of the programme Thermodynamics of Arithmetic Action. Volumes I-IV (ai.viXra:2610.0045v2, 2610.0054, 2610.0052, 2610.0056, 2610.0055) recast the Riemann Hypothesis (RH) as the absence of negative-norm ("ghost") directions for the Weil form, and traced the missing positivity to the Hodge index theorem on C x C, which zeta lacks. Over finite fields Frobenius is a rotation in the Rosati metric; a ghost would be a boost. Volume V asks what a surface Spec Z x Spec Z would have to supply. Central-value positivity (Gross-Zagier) turns out to be a parity statement about real zeros. Tested next: the standard conjectures and correspondence rings, the Connes-Consani self-intersection with synthetic ghost zeros, ample classes, two scalings and weight gradings. Certified (Arb) results: positivity of the Weil form of zeta on explicit spline spaces up to support a = 8, and certified detection of hypothetical off-line zeros. A compatible complex structure commuting with Frobenius exists exactly when the Frobenius torus is compact; the volume then compares Krein, Toeplitz and Newton-over-Hodge signatures, the modular cusp, BRST and thermodynamic geometry, and ends with Lax-Phillips passivity: the doubled modular system is passive unconditionally. A statement at the start lists exactly what is claimed, and where. Three proposed errata await the author's decision; they are listed, not applied. Formalization and computations were AI-assisted, and this is disclosed. RH is NOT proved: structures built from the zeros restate Weil positivity, those built without them are blind to off-line zeros, and an RH-independent ample class for Spec Z x Spec Z is still missing.
Category: Number Theory
[11] ai.viXra.org:2610.0057 [pdf] submitted on 2026-10-26 01:09:52
Authors: Ahcene Ait Saadi
Comments: 22 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We study systems of equations Ni(t)D(t) = ki (i = 1, . . . , n) that have a common solution t0, possibly together with equations Fi(x) = ki in a second unknown x with the same right-hand sides ki. Adding or subtracting these equations produces new systems whose solutions other than t0 (the hidden solutions) are computed by the rule a b Θ c d = a+c b+d (the Farey mediant), provided the orientation −A/B ̸= A/(−B) is respected. A second law, denoted ♢, says how to multiply an equation by a coecient: λ♢a b = λa λb , which is not the usual productof a fraction by a number. We explain these rules by a factorisation theorem, determinethe signs to choose in order to obtain a prescribed hidden solution, and give an inverseconstruction, relations between equations, a barycentric formula in degree m, the complexcase in degree 2, covariance under homographies, the correspondence between the unknownsx and t, two-parameter systems (relations between hidden values in t and in α, elliptic curves of rank ≥ 1), and a study of the rationality of all solutions (conics, elliptic curves). A rank criterion proved with Mazur's theorem gives innitely many fully rational combinations for certain pencils of cubics. Every statement is proved and illustrated by explicit systems. The core is elementary (Vieta's relations and linearity); the contribution is the reading of thesesystems and of their signed combinations.
Category: Number Theory
[10] ai.viXra.org:2610.0056 [pdf] submitted on 2026-10-24 04:26:42
Authors: J. W. McGreevy
Comments: 322 Pages. Thermodynamics of Arithmetic Action, Lemmas 48-65. Volume IV, part 1 of 2 (companion: Volume IVb). RH is not proved.
Volume IVa is the first of two parts of Volume IV of the programme Thermodynamics of Arithmetic Action. It collects TAA Lemmas 48-65; the companion Volume IVb contains Lemmas 66-85. The parts form one volume, split only because of the archive's 5 MB file limit. Volumes I-III (ai.viXra:2610.0045v2, 2610.0054, 2610.0052) recast the Riemann Hypothesis (RH) as the absence of negative-norm ("ghost") directions for the Weil form, measured that obstruction, and asked which structure could supply the positivity. Volume IVa tests three families of candidates. Polarizations as vacua and complex structures: Kramers-Kronig causality, the Betti-de Rham comparison, spinor vacua and Krein collisions, the Hardy-space vacuum and its ghost charge, and the complex structure Frobenius chooses for itself. Saddles and positivity ladders: saddles of |zeta| and xi', the Weil positivity ladder, the derivative tower of xi, and ghosts as dents in a flat cone metric. The prime side: class numbers, a visibility criterion for zeros, helicity and the Moebius function, local versus global positivity, and the field with one element. Certified computer-assisted results concern the Davenport-Heilbronn function (a zero-free box, and a lower bound 0.230607 for its de Bruijn-Newman-type constant) and truncated Weil forms (eigenvalue bounds; the Davenport-Heilbronn form is certified indefinite for A >= 3.43). A statement at the start lists exactly what is claimed across both parts, and where. Formalization and computations were AI-assisted, and this is disclosed. RH is NOT proved: no chapter supplies the missing positivity for zeta; most identify why a candidate cannot.
Category: Number Theory
[9] ai.viXra.org:2610.0055 [pdf] submitted on 2026-10-24 04:26:53
Authors: J. W. McGreevy
Comments: 280 Pages. Thermodynamics of Arithmetic Action, Lemmas 66-85. Volume IV, part 2 of 2 (companion: Volume IVa). RH is not proved.
Volume IVb is the second of two parts of Volume IV of the programme Thermodynamics of Arithmetic Action. It collects TAA Lemmas 66-85; the companion Volume IVa contains Lemmas 48-65. The parts form one volume, split only because of the archive's 5 MB file limit. Volumes I-III (ai.viXra:2610.0045v2, 2610.0054, 2610.0052) recast the Riemann Hypothesis (RH) as the absence of negative-norm ("ghost") directions for the Weil form. Here we test modular forms and Eisenstein anomalies, hyperbolic geometry, 3D gravity, Farey fractions and codes, and isolate the obstruction. The ghost count kappa, the negative index of a Weil-type form, recurs throughout; for a Weil polynomial it equals the number of negative directions of the Rosati trace form, a classical fact we restate and check exhaustively for 13135 genus-2 polynomials. We prove a no-go theorem for positivity built from scaling-commuting multipliers. We show numerically that the Davenport-Heilbronn ghost is visible at finite Connes cutoff, with primes n <= 41. Closed geodesics on Bianchi orbifolds see only Euler-product channels, not the off-line zeros of Epstein zeta functions. Over function fields, the BSD count in a Mordell-Weil lattice is one Frobenius eigenvalue, tied to RH through the algebraicity of Frobenius. Certified computations show that extremal weight enumerators of self-dual codes satisfy Duursma's RH in stated ranges. A statement at the start lists exactly what is claimed across both parts, and where. Formalization and computations were AI-assisted, and this is disclosed. RH is NOT proved: no chapter supplies the missing positivity for zeta. What is missing is a global operator whose spectrum is the zeros, with a positive form from a global ample class.
Category: Number Theory
[8] ai.viXra.org:2610.0054 [pdf] submitted on 2026-10-22 04:55:27
Authors: J. W. McGreevy
Comments: 173 Pages. Thermodynamics of Arithmetic Action, Lemmas 14-29 (compiled volume II). Sequel to ai.viXra:2610.0045.
Volume I (ai.viXra:2610.0045v2) showed that the Riemann Hypothesis (RH) is equivalent to unitarity, meaning the absence of negative-norm ("ghost") directions, of the logarithmic scaling flow for the Weil form Q. This volume measures that obstruction directly in one geometric language: the Weil form is the transverse part of a Lorentzian form. For curves over F_q, the Hodge index theorem makes NS(C x C) Lorentzian, the graph of Frobenius is the diagonal boosted by rapidity (1/2) log q, and Weil's bound is Cauchy-Schwarz on the transverse part. For zeta, the pole plane minus Q is a form L computable without zeros, and RH is equivalent to L having Lorentzian index <= 1 on every finite-dimensional subspace (a restatement of Weil's criterion). We certify positivity of Q on every window of length <= 1.6 (computer-assisted, Arb ball arithmetic); give the arithmetic action a Lagrangian, Noether, Ostrogradsky and transmission-line reading; make the damped (sigma-shifted) window index exact; separate trace bounds from Rosati positivity and read T_p as two boosts; prove no-go theorems for zero-side certificates, prolate detectors, tropicalization and hbar-deformation; correct explicit zero-counting constants and certify zero-sampling inequalities on Platt's zeros; read each zero as a telegrapher line in a Krein space, with a passivity ladder from the functional equation to RH; show that ghost birth is a two-stage event (Krein-space degeneration, then a Krein collision); and prove pole dominance for damped forms with an exact closed-form threshold. Every statement carries a claim label (Theorem, Restatement, Analogy, Conjecture, Open, Numerical). RH is not proved. The positivity source that geometry supplies for curves is not found for Spec Z x Spec Z; several natural candidates are proved not to supply it. Computations still running at submission are marked as placeholders.
Category: Number Theory
[7] ai.viXra.org:2610.0052 [pdf] submitted on 2026-10-22 21:12:00
Authors: J. W. McGreevy
Comments: 296 Pages. Volume III of the series. Volume I: ai.viXra:2610.0045. Volume II: submitted October 2026.
Volume I (ai.viXra:2610.0045v2) showed that the Riemann Hypothesis (RH) is equivalent to unitarity (the absence of negative-norm ("ghost") directions) of the logarithmic scaling flow for the Weil form Q; Volume II measured that obstruction on the Weil form itself. This volume asks which structure could supply the positivity. For curves over F_q, Weil's proof needs a commutative algebra containing Frobenius, an involution with F F^dagger = q, a trace for which the involution is positive, and Albert's algebraic step. Zeta has the first, second and fourth canonically; the third is axiom G5: a positivity of the zero-side trace that holds for a geometric reason, built without the zeros. We then test candidate sources of G5: thermal, modular and Kepler structures; Osterwalder-Schrader reflection positivity; causality, Kramers-Kronig and passivity; de Branges/Krein lines from the prime side, with G5 stated as a passive transmission line; Galois symmetry, Iwasawa theory and Bost-Connes; polarization, Krein unitarity and the two stages of ghost birth; half-sided modular inclusions; holomorphic flows, cavities, boson/fermion second quantizations; and the Verschiebung and the reflection N -> Z. One conclusion recurs as a theorem: a reflection gives the functional equation; positivity needs a polarization. Algebra gives Krein unitarity; a positive polarization gives Hilbert unitarity. An editorial section (from Lemma 50) recasts this in spinor language: in the abstract model, RH for curves is equivalent to a Frobenius-invariant positive compatible complex structure (a positive fermionic vacuum), and second quantization never repairs an indefinite signature. Every statement carries a claim label (Theorem, Restatement, Analogy, Conjecture, Open, Numerical). RH is not proved; no candidate tested supplies G5. Computations still running at submission are marked as placeholders.
Category: Number Theory
[6] ai.viXra.org:2610.0046 [pdf] submitted on 2026-10-21 20:24:57
Authors: J. W. McGreevy
Comments: 6 Pages. TAA working notes.
On C_c^infty(R) (functions of the logarithmic scale u = log x), the Weil explicit-formula pairing Q is a Hermitian form, defined arithmetically by primes, the pole and the archimedean place, and spectrally by the zeros. The scaling flow T_t F(u) = F(u - t) is always Q-isometric (pseudo-unitary). We show: (a) the exponential growth rate of the matrix coefficients t -> Q(T_t F, G) is exactly Theta - 1/2, where Theta = sup Re rho; (b) RH <=> Q >= 0 <=> the flow is realized by a strongly continuous unitary group on a Hilbert space <=> all matrix coefficients are bounded; (c) the number of negative squares of Q equals the number of distinct zeros with Re rho > 1/2. Under RH the generator has simple pure point spectrum equal to the set of zero ordinates. Nothing here proves or disproves RH; parts (b) and (c) are in substance due to Weil and Bombieri, and the paper records exactly what is new. Part of the Thermodynamics of Arithmetic Action (TAA) program.
Category: Number Theory
[5] ai.viXra.org:2610.0039 [pdf] submitted on 2026-10-18 21:56:45
Authors: Brian Scannell
Comments: 10 Pages.
The irrational numbers tan 15◦ = 2 −√3, cot 15◦ = 2 + √3 generate the integer sequence2, 4, 14, 52, 194, . . .when corresponding powers are added. Surprisingly, 15◦ is only one of infinitely many angles that generate this same sequence when the exponents are chosen appropriately. We describe a two-parameter family behind this phenomenon and show how two elementary recurrence relations connect symmetric trigonometric powers with Chebyshev polynomials, Lucas sequences, Pell equations, and, unexpectedly, the Lucas—Lehmer test for Mersenne primes.
Category: Number Theory
[4] ai.viXra.org:2610.0029 [pdf] submitted on 2026-10-11 02:21:41
Authors: Xuda Ye
Comments: 49 Pages.
We prove that infinitely many primes $p = m^2 + n^2$ have $n < p^{0.1129}$, and that infinitely many primes satisfy ${sqrt{p}} < p^{-0.2742}$. This sharpens the exponents $0.119$ and $-0.262$ of Harman and Lewis (2001). Following their approach, we derive both statements from a lower bound of the correct order of magnitude for the number of Gaussian primes of norm between $X/2$ and $X$ in every sector of angle $X^{-0.3871}$, and we prove this bound with Harman's sieve and their arithmetic information. The new ingredients are three groupings of terms that their sieve discards: a factorization of a two-dimensional sum for the linear estimate, and two applications of the bilinear estimate to four-dimensional sums, one of them after a further application of Buchstab's identity. These groupings give the grouped terms asymptotic formulas, so that fewer terms are discarded. In addition, we give every step of the sieve argument in full, filling in the details that Harman and Lewis present briefly and making several of their statements and discarded regions precise, and we evaluate the losses of the discarded sums by two numerical methods.
Category: Number Theory
[3] ai.viXra.org:2610.0022 [pdf] submitted on 2026-10-07 17:04:33
Authors: Xuda Ye
Comments: 9 Pages.
We study three quadratic Collatz-type maps on the non-negative integers. Each of them halves an even number, and they send an odd $n$ to $n(n-1)/2$, to $n(n+1)/2$ and to $(n^2 - 1)/4$, respectively. Along an orbit, the odd terms grow until they reach a number of the form $2^m + 1$ or $2^m - 1$, depending on the map, and an orbit with an odd term greater than $1$ is bounded exactly when this happens. The step that reaches such a number gives a solution of an exponential Diophantine equation, which becomes the equation $2^a pm 2^b + 1 = z^2$ after completing the square. With Szalay's classification of the solutions of this equation, we classify all orbits of the maps. Every orbit either is eventually periodic or tends to infinity, and we determine exactly the initial values with bounded orbits. For the map with odd branch $n(n-1)/2$, Sedaghat stated the classification, and we give a new and complete proof. For this map, we also give an elementary criterion for unbounded orbits, based on divisibility alone. To our knowledge, the classifications for the other two maps are new.
Category: Number Theory
[2] ai.viXra.org:2610.0021 [pdf] submitted on 2026-10-08 13:33:07
Authors: Chaiya Tantisukarom
Comments: 10 Pages.
We reformulate Goldbach's Conjecture in terms of the midpoint $C = N/2$ and a symmetric offset $x$, so that $N = (C-x) + (C+x)$. For prime pairs exceeding $3$, the offset $x$ is confined to explicit residue classes modulo $6$ determined by $C bmod 6$; this is an elementary mod-$6$ wheel restriction, and we state exactly how the restricted search relates to the conjecture. We show that shielding a center $C=P_n$ by the first $n$ primes guarantees prime pairs only for $n le 3$, and we explain why this is a limitation of trial division rather than a demonstration of the sieve parity barrier. Under Hardy--Littlewood heuristics we derive the expected number of valid offsets and a Poisson-type model for the smallest one, and compare both with an exact computation over every center $C in [10^5, 2cdot 10^5]$. The counting prediction is accurate to within $1%$. The leading-order prediction for the mean smallest offset is too large by $11$--$34%$; we trace this to the point offset $x=0$ (the case $C$ prime) and to the regularity of the sifted set of candidates, and a model incorporating both agrees with the data to within $2%$. We also confirm by computation that every center $5 le C le 10^6$ admits a valid offset, and we report the smallest offset at the centers $C = 6cdot 10^{j}$ for $0 le j le 23$, which reaches heights far beyond the exhaustive range. The note is expository; no progress toward a proof is claimed.
Category: Number Theory
[1] ai.viXra.org:2610.0015 [pdf] submitted on 2026-10-05 23:58:35
Authors: Bouchaib Bahbouhi
Comments: 15 Pages.
We introduce a computational and mathematical framework for the study of bilateral prime synchronisation in Goldbach representations. For a fixed even integer E, the method generates one member of a candidate Goldbach pair through quadratic polynomial rails of the form f_C(x) = x² + x + C and defines the complementary member exactly by q = E − f_C(x). Particular attention is given to C = 11, 17, and 41, corresponding to the discriminants −43, −67, and −163. Rather than considering the prime-producing behaviour of these polynomials in isolation, we study the simultaneous event in which both f_C(x) and E − f_C(x) are prime, termed P/P synchronisation.A central component of the method is a reproducible random-sequence protocol in which the indices x are selected independently of subsequent primality outcomes. The same realised sequence is applied to all polynomial rails under identical modular filtering and primality-testing conditions. This permits direct measurement of the first P/P synchronisation, total P/P frequency, and the associated P/P, P/C, C/P, and C/C state distributions. The framework is complemented by the boundary statistic Pmin(E), defined as the smallest prime p for which E − p is prime, thereby linking polynomial synchronisation with the first Goldbach representation encountered from the boundary.The local arithmetic of bilateral synchronisation is described through an exact obstruction count ν_r(E,C), measuring the residue classes modulo an odd prime r for which either f_C(x) or E − f_C(x) is divisible by r. For r not dividing E, this quantity satisfiesν_r(E,C) = 2 + (Δ_C/r) + ((Δ_C + 4E)/r),where Δ_C = 1 − 4C and (·/r) denotes the Legendre symbol. These local factors motivate a finite synchronisation score and a probabilistic prediction framework for the occurrence and location of P/P events.The empirical component is designed to be fully reproducible: the complete Pmin and P/P observations, sampled sequences, rail identifiers, primality status, and associated experimental quantities are provided in a separate supplementary data file. Probable-prime results are distinguished strictly from proved or externally certified primes.The experiments indicate that polynomial enrichment and local modular structure can substantially organise the search for Goldbach prime pairs and suggest a measurable synchronisation phenomenon across different numerical scales. The results are computational and heuristic where explicitly stated and do not constitute a proof of the binary Goldbach conjecture. They instead provide a reproducible experimental framework, exact local identities, and a mathematical prediction problem that may serve as a basis for further analytic investigation of Goldbach synchronisation.
Category: Number Theory