Next Meetup

An amateur's trip into number theory ... starring Fermat's Two-Squares Theorem
I -- that is, Tom Frenkel, your Organizer for our two recent meetings -- have long been fascinated with "Fermat's theorem on sums of two squares" (to use the name of the Wikipedia article). (Fermat was not the first to come upon this concept, but that's another story.) The theorem states that any odd prime p can be expressed as the sum of two squared integers, if and only if p, when divided by 4, leaves a remainder of 1. Thus, 5 (=4+1) and 13 (=9+4) are examples, while 7 and 11 are not. After trying to come to grips with various methods of proof (including the first that we have, by Euler about 1750), I have finally found one that works for me. It uses "elementary" methods, but makes use of two less commonly used theorems: Wilson's Theorem, and Thue's Lemma. The paper I used to guide me on this path was written by an undergraduate (I believe) at the University of Chicago. Here's the link: . (I only used material through the middle of page 7.) My talk will be an exposition of this proof, preceded by some discussion of number-theory fundamentals (especially modular arithmetic). I don't think any special math background will be required for attendees. A book on number theory that I found excellent (though I am not using its method of proof for the Two Squares Theorem) is The Higher Arithmetic, by H. Davenport. I used the eighth edition.

Hack Manhattan

137 W. 14 suite 201 (second floor, rear of building) · New York, NY

    Past Meetups (66)

    What we're about

    Math is the original liberal art, the first stop on any intellectual journey to understand a little more about the world. It's about trying to answer questions, not because they've been assigned out of some dusty textbook, but because the question itself is interesting and the attempt is more interesting still. It's about learning for the sake of learning, studying for the sake of studying. This group is about math for math's sake.

    The School of Mathematics was founded to cultivate a natural and stress-free environment where anyone can study, discuss, explore, and experience mathematics. No prior knowledge is assumed. Whether you are an avid student of mathematics or have always shied away saying "math is not for me", you are welcome. Our approach allows anyone to naively discover mathematics. All studies at the school are free of charge. Philosophically, we share much in common with the vision of mathematics education Paul Lockhart outlines in his wonderful essay A Mathematician’s Lament. (

    Curious? Take a look at some of our past topics ( or just stop by a meetup.

    Members (1,200)

    Photos (20)