number-theory

We develop moment and prime-trace formulations of the Riemann hypothesis and prove two classes of spectral exclusions. The analytic results use explicitly stated entire-function, moment, and operator hypotheses; a separate TCGS--SEQUENTION application formulates the source-to-readout realization problem. Writing $F(w)=\xi(\tfrac12+i\sqrt w)/\xi(\tfrac12)$ by its entire power series, we establish …

It is possible to use various methods to assign finite values to divergent series (e.g., regularization, analytic continuation). See, for instance, this or this Q&As here on MSE. The limits of ...

Ramanujan's 1913 letter looked implausible because it compressed deep mathematics into unsupported claims. Hardy's response mattered because suspicion led to examination rather than dismissal.

The formalization of Fermat’s Last Theorem (FLT) in the Lean proof assistant remains an ongoing community-led effort. It should not be attributed to Claude as a completed, first formalized proof. The distinction matters because formal verification is a demanding process: converting a mathematical argument into machine-checkable code can expose missing assumptions, unclear steps, and dependencies …

Eric Lu announced on X today that he has factored RSA-260, a number N with 260 digits (862 bits) that is the product of two large primes [1]. RSA numbers are challenge problems posed to gauge the security of RSA encryption, which rests on the difficulty of factoring large numbers [2]. The naming scheme is […] The post New RSA number factored first appeared on John D. Cook .

denotes ℤ the spectrum of the commutative ring of integers. Its underlying topological space (also known as the “prime spectrum” or “Zariski spectrum” of ) has the prime ideals of as points and carries the Zariski topology. The closed points are the maximal ideals , for each prime number in ; the non-maximal prime ideal is a generic point as it has as closure the whole of . A subset of is closed …

Alain Connes (born on April 1, 1947) is a French mathematician, Fields medalist (1982), Crafoord prize winner (2001), Professor at IHÉS, Professor at Collège de France and part-time working as a Professor at Vanderbilt University. His interests include geometry, topology, especially K-theory and index theory, operator algebras, the connections between noncommutative geometry and number theory, an…

- Ponder This This puzzle was suggested by Hugo Pfoertner - thanks Hugo! In a now-famous 2004 article, Ben Green and Terence Tao proved that arbitrarily long arithmetic progressions exist in the primes. This holds true analogously for other sets of numbers if their density is sufficiently high, for example, numbers that are the sum of two squares. There are also enough numbers of the form x² + y²…

Starting from the cognitive operation of information collapse, this paper re-anchors the foundational position of mathematics—as the pure grammar of information-collapse operations. Based on a strict correspondence between the four arithmetic operations and the four basic collapse actions, it argues that subtraction is the more primordial first action than addition: extracting the first determina…

This paper extends the information collapse analytical framework established in The Meaning of Mathematics (Part I), clearly distinguishing between two cognitive directions—grasping outward and deconstructing inward—and conducts a complete structural argumentation analysis of Ramanujan's summation of divergent series. Grasping outward extends along the temporal direction, producing determinate ob…

Say you have a dot. That dot explodes into an infinite number of dots. Each of those dots explode into a new infinite number of dots. Each of those dots explode into a new infinite number of dots, ad infinitum. How many dots do you have in total?

I understand that $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}\cdots$$ I understand what an infinite series is. I know n goes on to infinity and that s is the function input. I just don't know how the heck this finds primes? Can someone explain? I have... Read more

Ramanujan’s insights into pi are now guiding scientists toward a deeper understanding of how the universe works.

A paper out today in Nature might interest some folks in this forum: https://www.nature.com/articles/s41586-021-03229-4 Permanent citation: Nature volume 590 , pages67–73(2021) The authors used machine learning to generate a large number of continued fraction expressions... Read more