number-theory

I was visualizing infinite products of complex numbers, specifically looking at numbers of the form $z=1+\frac{i}{k}$ for integer sequences of $k$. Then I used odd-indexed Fibonacci numbers greater ...

A seemingly simple set of rules kicks off a kind of mathematical magic trick, which has kept great minds busy since the 1930s. Columnist Jacob Aron explores the origins of the Collatz conjecture, why it is so addictive to mathematicians and whether AI could help us solve it once and for all

I am trying to wrap my head around Dedekind cuts and the construction of the Real numbers from Rational numbers. From my understanding, let us assume a set $\beta \subseteq \mathbb{Q}$ (following all ...

In this blog post I will give my personal view on the recent counterexamples to the unit distance conjecture and sum-product conjecture over the reals (see [90] and [52] respectively). My goal is to sketch the constructions and try and give some intuition as to where they came from and why they work. My main target audience is the me-of-a-month-ago, who did not know much algebraic number theory, …

Let $F_N(x)$ be the $N$-th order Fibonacci polynomial over $GF(2)$. I conjecture that $F_{6M+3}(x)=(x^2+1)G^2_1(x)...G^2_K(x)$ where $G_k(x)$ are distinct irreducible factors. Example: ...

This paper examines the validity of “1 = 0.999...” and points out the concept of infinity and the unconscious extension of the concept of equal sign. To view “1 = 0.999...” as valid, it is necessary to adopt the concept of infinity. However, infinity must be handled by replacing it with the concept of the finite. The meaning of the equal sign is also extended from "identical," which always held t…

LLMs have solved Erdős problems before, but the one Price chose wasn’t just any Erdős problem. It was one that human mathematicians had worked on for 60 [Ed:80] years without success. The nature of the solution was also unusual. While previous LLM solutions to Erdős problems used standard... Read more

The recent work [Devadas-Hopkins-Kalai-Kothari-Lombardi-Mathialagan, STOC 2026] proposed a low-norm Nullstellensatz hypothesis for the "AND code": every polynomial $f$ vanishing on the "AND-code ideal'' should admit a Nullstellensatz decomposition over the local AND constraints whose total coefficient \(\ell_1\)-norm is only polynomially larger than the \(\ell_1\)-norm of $f$. We give a countere…

_Zenodo_. 2025A finite, exact approach to digit symmetry and self-similarity with implications for the foundations of mathematics and even stochastics. Especially in regard to brownian motion. Every finite natural number other then one digit numbers have a Mirror-Pair. Thus any finite natural two digit number has a Mirror-Pair. If the pair is plugged in into the function M_b(x) it becomes it’s Mi…

Daniel Shanks introduced the "simplest cubic fields" in a 1974 paper [1]. These have cyclic Galois group of order 3, all roots are real, and a straightforward parameterization: $$f = x^3 + ...

We know that the Selberg normality conjecture for the L-function $\sum_{n=1}^{\infty} a_L(n)n^{-s}$ which belongs to Selberg class with polynomial Euler product is $$ \sum_{p\leq x} |a_L(p)|^2=(\kappa ...

We introduce a new geometric framework for arithmetic functions belonging to the class \mathcal{V},where each function f\in\mathcal{V} is associated with the discrete orbit A_n=(n,f(n))\in\mathbb{Z}^2. From this perspective, arithmetic properties of fare interpreted through Euclidean geometric configurations generated by points on the discrete graph of the function, and one studies whether such c…

Why a proved theorem still needs reproducible claim custody On May 20, 2026, OpenAI announced that an internal reasoning model had produced a counterexample to the Erdős planar unit-distance conjecture. The problem is easy to state: given $n$ points in the plane, how many pairs of points can be exactly distance $1$ apart? For nearly eighty years, the prevailing expectation was that square-grid-ty…

Let ξ(s) = ½s(s−1)π^(−s/2)Γ(s/2)ζ(s) be the completed Riemann zeta function and let {γ_ρ} denote the positive imaginary parts of its nontrivial zeros. We first prove a universal product formula: for all s∈ℂ (under the Riemann Hypothesis), ξ(s)/ξ(½) = ∏_{γ_ρ>0} (1 + (s−½)²/γ_ρ²), expressing the ratio ξ(s)/ξ(½) as a product over the Riemann zeros with coupling a(s) = |s−½|. The formula has two regi…

The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum. What does this mean/imply?

In this note, we study decomposition of the Ate pairing on certain elliptic curves defined over finite fields. As an application, we reduce a generalized pairing inversion to root findings of an element of the affine coordinate ring appearing in the decomposition. For a supersingular curve $E / {\bf F}_q$ satisfying $\sharp E( {\bf F}_q ) = q+1$, heuristic observation suggests that a number of ca…

We formalize a structural perspective on the Riemann zeta function, highlighting a natural balance between the "inside" (integer counting) and "outside" (density/reflection) perspectives of the number system. Using divisor pairings, information measures, and the functional equation, we show that s = 1/2 is a unique symmetry axis where information is balanced. While this framework does not prove t…

5 $s = 2 + {1\over 2}\sqrt 2 = \Nn{2.70710678118654}$ Rigid. Proved by Frits Göbel in early 1979. 10 $s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$ Found by Frits Göbel in early 1979. Proved by Walter Stromquist in 2003. Explore group 11 $s = {}^{8}🔒 = \Nn{3.87708359002281}$ $s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s - 6865 = 0$ Rigid. Found by Walter Trump in 1979.…

It seems to me that the formulation of Schinzel's hypothesis H is related to Pólya's results on integer-valued polynomials (1) and my question is if this conjecture can be generalized as below? Given ...

The Collatz process is traditionally approached as an unsolved numerical conjecture concerning iterative parity operations. This paper proposes a different interpretation. Rather than treating Collatz solely as a destination-oriented puzzle asking whether all trajectories converge to 1, this paper examines the process as a minimal parity-gated recursive architecture composed of alternating expans…