** Elementary Number Theory in Nine Chapters** (Second Ed.) by James J. Tattersall

Rating: 10/10

This is an excellent (and readable) intro text to number theory with proofs to many important number-theoretic formulas and theorems. Aside from theory, there is also an emphasis on applications such as calendrics, representations, and cryptography. From front to back, there are many interesting historical notes that connect the subject to several cultures (including Chinese, Muslim and Indian). I highly recommend this text to anyone who enjoys challenging and enlightening problems.

** Explorations in Geometry** by Bruce Shawyer

Rating: 8/10

This book draws many problems from the IMO and other math contests, with supplementary information and commentary. Certainly, this book is great for expanding one's horizons to solving challenging geometry problems in the spirit of competitions.

*The Works of Archimedes*

Rating 10/10

Forget Euclid's unnecessarily long treatments on trifles, Archimedes is the greater teacher. This book is very readable to the modern mathematician (unlike Newton's *Principia*) despite being over 2000 years old. The math is rigorous and by no means elementary, though its archaic qualities will make one gasp at how brilliant this man was.

** Ancient Puzzles: Classic Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries** by Dominic Olivastro

Rating 10/10

This book is great for three reasons:

A) You learn a lot about problem solving in discrete math

B) It is a fun read because the problems are presented in recreational and novel ways

C) The book introduces tons of ancient and medieval math outside the Greco-Roman tradition (which is a breath of fresh air)

**Three Must Read Books about e and i !!!**

** e: the Story of a Number** by Eli Maor

Rating 10/10

** An Imaginary Tale: The Story of \(\sqrt -1\)** by Paul J. Nahin

Rating 10/10

** Dr. Euler's Fabulous Formula** by Paul J. Nahin

Rating 10/10

These classic books on the number e, i and their pivotal importance to modern math will both captivate and inspire future generations. Any reader will learn a lot about math and the elegance of this long forsaken language.

** How to Solve It** by G.Polya

Rating 10/10

Anybody interested in the teaching of mathematics, or teaching in general should read this book! This classic book teaches people that anyone can do mathematics and problem solve, all you need is clear thinking, patience and the right type of guidance (nature and/or nuture).

** A History of Chinese Mathematics** by Jean-Claude Martzloff

Rating 10/10

For those interested in mathematics in other cultures, this book is a perfect window into Ancient Chinese Mathematics. Concise but very informative, this book will open new eyes to mathematics from the layman to the expert.

** Janos Bolyai: Non-Euclidean Geometry and the Nature of Space** by Jeremy J. Gray

Rating 10/10

This book goes into the details of how Janos Bolyai obsession with Euclid's Fifth Postulate ushered a new age of geometry. The book also comes with Bolyai's groundbreaking dissertation on the Fifth Postulate.

## Comments

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TopNewestIf I have one hour per day to study Combinatorics till up to level 3, how many days will it take and what books would you suggest? Level 4? (I'm a fast learner). For curiosity, how long till level 5 (not restricted to the same book as to lvl 3)

If I have half an hour per day to study Number Theory, which level would you suggest studying up to (or topic) so that it remains within boundaries of application? Which book? Should I study it at all? And for curiosity, how long till level 5 (no book restrictions)?

Replies appreciated (but don't struggle - if you don't know it, don't research it).

Cheers – John Muradeli · 2 years, 1 month ago

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– Steven Zheng · 2 years, 1 month ago

It depends what you are good at. I always choose problems with high points that I am most familiar with first.Log in to reply

– John Muradeli · 2 years, 1 month ago

By level I meant an approximate strength of Brilliant problems, and the books I could read that'd help me solve those problems.Log in to reply

Is hsm coxeter or jeremy gray better for noneuclidean geomety? – Subrata Saha · 2 years, 1 month ago

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Is it me or is Number Theory only a pure-mathematical subject with no real-world applications? – John Muradeli · 2 years, 1 month ago

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– Milind Prabhu · 2 years, 1 month ago

What about cryptography?Log in to reply

– Steven Zheng · 2 years, 1 month ago

There is a chapter on cryptography in the Number Theory Book. Sorry, I should have replied John Muradeli.Log in to reply

Thanks – John Muradeli · 2 years, 1 month ago

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– Krishna Ar · 2 years, 1 month ago

What would you say about the wonderful properties and day-to-day applications of eccentric numbers like \(\pi, \phi\) etc?Log in to reply

– Steven Zheng · 2 years, 1 month ago

Honestly I think (e) is the most useful number. Most people are familiar with (\pi), so that number gets overrated.Log in to reply

– John Muradeli · 2 years, 1 month ago

Can you briefly state why \(e\) is more useful than \(\pi\)? ThanksLog in to reply

– Steven Zheng · 2 years, 1 month ago

Differential equations. A lot of simple physical systems have solutions with exponential functions in base \(e\).Log in to reply

\(\int {e^xdx}=e^x=\frac{d}{dx}e^x\)

(and don't be a party-pooper with your "oh what about \(+C\)?") – John Muradeli · 2 years, 1 month ago

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– John Muradeli · 2 years, 1 month ago

I don't need to know number theory for thatLog in to reply