Mathematicians Uncover a New Way to Count Prime Numbers

Does it? For any number a, a^2 = a (mod 2), and primes greater than 2 are all odd, so if a prime p = a^2 + b^2, doesn't one of a or b have to be odd? Reducing mod 2, p = 1 (mod 2), a^2 + b^2 = 1 (mod 2), a + b = 1 (mod 2), so either a = 0 (mod 2) and b = 1 (mod 2) or vice-versa?

edit:

If Euler proved infinitely many such primes exist then "With this in hand, Green and Sawhney proved Friedlander and Iwaniec’s conjecture: There are infinitely many primes that can be written as p^2 + 4q^2." makes no sense without a further condition of p or q, let (in my notation) a=p be odd and b=2q be even.

Now having finished the article, I think this was just sloppy writing, and the actual accomplishment is related to the post-perhaps clause: one of p or q has to itself be a perfect square? Anyway, I have very little certainty about what was actually accomplished from reading this article.

https://renaissancephilanthropy.org/news-and-insights/renais...

> Proposals should be aligned with one of the following categories:

> Production grade software tools: AI for auto-formalization, proof generation, synthesis of verifiable code, and more

> Datasets: Open-source collections of theorems, proofs, and math problems Field building: Textbooks, courses, and resources to grow the AI-for-math community

> Breakthrough ideas: High-risk, high-reward approaches to AI-driven math research

I wonder how true this statement is but it probably also relies on the understanding of the word 'random'. In the colloquial sense, it is certainly true in the sense that (truly) 'random' means 'occurring without (discernible) rules'.

However, in a stricter definition 'random' means (if I'm not mistaken, hobby-thinker here) just "a given set of numbers S is called random with respect to a set of binary tests T when all individual procedures in T yield a positive outcome". That is, you can only ever a finite set S and test against a finite set T, meaning the algorithm that generated your random-looking set S can, potentially, always be amended to creep closer to overcome the failed tests (and the reverse is also true: one can, potentially, always move the goalposts and add another test to T to make a set fail that used to pass).

Ultimately, then, randomness—where it is not occurring eo ipso as in radioactive decay—is always a relative (S WRT T), social (people must agree), and finite (can't test against unseen members of S and, per precondition, can't give a rule to cover infinity as in 'divisible by eleven') procedure (that may or may not practically terminate). In other words, once we settle on a given set of tests T to determine what is random, one can (potentially) always come up with an algorithm that passes the tests, thus looks random though it is determined.

Concisely:

http://mrob.com/pub/math/numbers.html

The rest of the site it's amazing too.

Green and Sawhney’s work is especially exciting because it shows how tools from one field—Gowers norms—can unlock progress in another. That kind of cross-disciplinary insight is where breakthroughs happen. And yeah, it’s fair to question funding priorities, but basic research has given us antibiotics, GPS, and even computers. Without it, we’d still be in caves.

In this article, neither is informative.

And even after several paragraphs in, you don't know what the general area of the proof is. Just meandering long-winded story-telling.

If anyone of you authors/editors of this magazine are here; please, for the love of all that's holly, put the crux of the matter at the top and then go off to tell your beautiful, humanized, whatever... story.

Nowadays we are wondering if LLM are going to be the next prime numbers. The question is if solving the language problem is going to provide us the key for solving the AGI problem. We still don't know what is the equivalent of a prime for a LLM, that is the smaller independent part that allow it to express some knowledge, the pieces could be embeddings or the topology of the layers or some new insight.

Some more random ideas, just like Norvig see python as a more practical Lisp, the basic ideas from prime numbers impregnate a great part of mathematics. You can be really far from the root but the principles are always with you, primes are a second nature, you have internalized all their properties and dynamically you learn to see primes like features in many fields (prime ideals, spectrum of a ring, points in commutative algebra).

The problem of how many primes (in relation to positive integers) is like wondering whether a given decomposition exists in a general sense that could allow us to solve the general problem. So few primes implies that the theory could solve some kind of problems but not many. What are LLMs number?, that is the question if solving the language problem will allow us to solve some very general problems, like a good approximation to AGI. The problem of whether LLMS will open the way to AGI could be the next Riemann hypothesis once we succeed in defining what is a prime number in relation to a LLM.

We are trying to prove escalating laws for how LLM improve when increasing the number of parameters, this is like trying to guess the convergence of an infinite serie by using a finite sum. The analogous of a Riemann hypothesis could be defining certain kind of LLM and conjecturing if it could obtain AGI pass some threshold for the number of parameters.

Edited: Sorry for being overly verbose this mornig!

Stunning to me that a staff writer for a science magazine would type this sentence without referencing the riemann hypothesis.

Is it that the methods required to do serious research on them ends up helping us discover other things?

Is there some deeper truth about the universe hidden in the prime numbers?