Category Archives: Number theory

To cheer you up in difficult times 25: some mathematical news! (Part 2)

Topology Quasi-polynomial algorithms for telling if a knot is trivial Marc Lackenby announced a quasi-polynomial time algorithm to decide whether a given knot is the unknot! This is a big breakthrough. This question is known to be both in NP … Continue reading

Posted in Algebra, Combinatorics, Geometry, Number theory | 2 Comments

To cheer you up in difficult times 20: Ben Green presents super-polynomial lower bounds for off-diagonal van der Waerden numbers W(3,k)

What will be the next polymath project? click here for our post about it.  New lower bounds for van der Waerden numbers by Ben Green Abstract: We show that there is a red-blue colouring of [N] with no blue 3-term … Continue reading

Posted in Combinatorics, Number theory | Tagged , | 2 Comments

Péter Pál Pach and Richárd Palincza: a Glimpse Beyond the Horizon

  Prologue Consider the following problems: P3: What is the maximum density of a set A in without a 3-term AP? (AP=arithmetic progression.) This is the celebrated Cap Set problem and we reported here in 2016 the breakthrough results by … Continue reading

Posted in Combinatorics, Geometry, Number theory, Open problems | Tagged , | 9 Comments

To cheer you up in difficult times 10: Noam Elkies’ Piano Improvisations and more

For the previous post “quantum matters” click here. Noam D. Elkies piano improvisations Every day since March 27, 2020 Noam Elkies (Noam’s home page) uploaded a new piece of piano improvisation. The Hebrew title of his page is “Music will … Continue reading

Posted in Games, Music, Number theory | Tagged | Leave a comment

To cheer you up in difficult times 5: A New Elementary Proof of the Prime Number Theorem by Florian K. Richter

Here is a piece of news that will certainly cheer you up: Florian Richter found A new elementary proof of the prime number theorem. (I thank Tami Ziegler for telling me about the new result.) From left to right: Atle Selberg, … Continue reading

Posted in Number theory, Updates | Tagged , , | 6 Comments

Hoi Nguyen and Melanie Wood: Remarkable Formulas for the Probability that Projections of Lattices are Surjective

Following a lecture by Hoi Nguyen at Oberwolfach, I would like to tell you a little about the paper: Random integral matrices: universality of surjectivity and the cokernel by Hoi Nguyen and Melanie Wood. Two background questions: Hoi started with … Continue reading

Posted in Algebra, Combinatorics, Number theory, Probability | Tagged , | 6 Comments

The largest clique in the Paley Graph: unexpected significant progress and surprising connections.

The result on Paley Graphs by Hanson and Petridis On May 2019, Brandon Hanson and Giorgis Petridis posed a paper on the arXive: Refined Estimates Concerning Sumsets Contained in the Roots of Unity. The abstract was almost as short as … Continue reading

Posted in Combinatorics, Number theory | Tagged , , , , , | 3 Comments

Dan Romik on the Riemann zeta function

This post, about the Riemann zeta function, which is among the most important and mysterious mathematical objects was kindly written by Dan Romik. It is related to his paper Orthogonal polynomial expansions for the Riemann xi function,  that we mentioned … Continue reading

Posted in Combinatorics, Guest blogger, Number theory | Tagged , , , , | 2 Comments

8866128975287528³+(-8778405442862239)³+(-2736111468807040)³

Update: The result was achieved by Andrew Booker from Bristol. Here is the preprint Cracking the problem with 33. It is a notoriously difficult open problem which integers can be written as the sum of three integer cubes.  Such integers … Continue reading

Posted in Number theory | Tagged | 17 Comments

Test Your Intuition (or knowledge, or programming skills) 36

How much is   The product ranges over all primes. In other words, Just heard it from Avinoam Mann.  

Posted in Number theory, Test your intuition | Tagged | 12 Comments