number-theory

The ancient Egyptians had a terrible notation for fractions. They had notations for for each , for , but everything else was written as a sum of these, with repeats forbidden, so that for example had to be written as . ( Wikipedia ) In an older article about Egyptian fractions and the Rhind Mathematical Papyrus , I said: Getting the table of good-quality representations of is not trivial, and …

Given a $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$-conjugacy class of $q$-Weil numbers, we obtain the minimal polynomial $m_\pi(x)$ of Frobenius. However, the Weil polynomial is not ...

This manuscript proves the Birch and Swinnerton-Dyer endpoint by fixed-carrier exclusion of analytic-arithmetic mismatch. The earlier structural BSD role-compression paper established the role architecture: elliptic-curve carrier, analytic L-function standing, Mordell-Weil rank readout, and the analytic-arithmetic bridge. The present paper moves from role architecture to endpoint closure. It prov…

We give a single consolidated account of what the analytic structure of the Riemann zeta function does and does not settle about the location of its non-trivial zeros, and we settle the relation between the two exactly. The work has four parts and we mark the grade of each. First, a geometric reformulation, stated at the strength it earns. The Berry-Keating operator H = −i(x d/dx + 1/2) is unitar…

We introduce WayFinder, a framework for generalizing the Delfs-Galbraith and SuperSolver algorithms for the supersingular isogeny problem. Our framework extends the search for elliptic curves with an orientation by an order containing $\mathbb{Z}[\ell \sqrt{-p}]$ to more general orders, and we derive a cost model for such generalisations. Our cost model not only works in a more general context, b…

In the comment section of a YouTube video yesterday, someone opined that the fundamental theorem of arithmetic was needed if you wanted to show that a rational number could always be expressed in ...

According to the tag wiki: A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". I have always wondered why we don't call ...

In this video: https://youtu.be/JUrACHCmJXs?si=7z7Dw80M4jS4Jg8S at 2:15 he shows the graph of the difference between $\pi(x)$ and $\mathrm{li}(x)$. When I tried to graph it its nothing alike. How did ...

This problem is due to my own curiosity. It deals with sequences. Suppose, $a_1, a_2, \dots, a_k$ is a sequence of positive integers, where $k$ itself is a positive integer. We arrange them in a ...

I ran across a cranky formula for π based on physical constants here and decided to play around with it. The source describes λ as “wavelength (chosen in the microwave region)” and I thought perhaps you could chose a value of λ to make the equation work. But as a comment pointed out, the bracketed […] The post A crank formula for π first appeared on John D. Cook .

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 + ...