SOLID STATE PRESS
← Back to catalog
Prime Numbers: The Atoms of Arithmetic cover
Coming soon
Coming soon to Amazon
This title is in our publishing queue.
Browse available titles
Mathematics

Prime Numbers: The Atoms of Arithmetic

The Sieve of Eratosthenes, Unique Factorization, and Euclid's Infinity Proof — A TLDR Primer

Your teacher says every number is either prime or built from primes, and you nodded along — but can you actually explain why 1 doesn't count, or prove there's no biggest prime? This TLDR primer clears up the confusion in plain language, built for students who need the concept to click before the test, not a semester later.

You'll learn what makes a number prime versus composite, why mathematicians exclude 1 (it's not just a rule — there's a real reason), and how the Fundamental Theorem of Arithmetic guarantees every integer has one unique prime factorization. From there, the book walks through two hands-on ways to actually find primes — the Sieve of Eratosthenes and trial division — with worked examples and the square-root shortcut that saves you from checking every possible divisor.

The heart of the book is Euclid's proof that primes never run out, presented step by step so you see exactly why the logic works, not just that it does. A final section tours the open questions that still stump professional mathematicians — twin primes, Goldbach's conjecture, the Riemann Hypothesis — and shows how the difficulty of factoring large primes is the literal foundation of RSA encryption protecting your data online.

Written for high schoolers, early college students, and any parent or tutor who wants a number theory study guide for students that's concise, direct, and free of textbook padding. No filler, no unnecessary jargon — just what you need to understand primes and walk into class or an exam ready.

Grab it, work the examples, and stop guessing on number theory.

What you'll learn
  • Define prime and composite numbers and identify them confidently up to 100
  • State and apply the Fundamental Theorem of Arithmetic to factor integers and compute GCDs and LCMs
  • Use the Sieve of Eratosthenes and trial division to find primes efficiently
  • Reproduce Euclid's proof that there are infinitely many primes
  • Explain at a high level how primes underpin RSA encryption and modern number theory
What's inside
  1. 1. What Is a Prime Number?
    Define primes and composites carefully, address the '1 is not prime' question, and build intuition with small examples.
  2. 2. The Fundamental Theorem of Arithmetic
    Every integer greater than 1 factors uniquely into primes; use this to compute GCDs, LCMs, and reason about divisibility.
  3. 3. Finding Primes: The Sieve of Eratosthenes and Trial Division
    Two practical methods for identifying primes, with worked examples and the square-root shortcut.
  4. 4. Euclid's Proof That the Primes Never End
    Walk through the classical proof by contradiction that there are infinitely many primes, one of the most elegant arguments in mathematics.
  5. 5. Patterns, Gaps, and Famous Unsolved Problems
    Survey the distribution of primes: twin primes, the Prime Number Theorem, Goldbach's conjecture, and the Riemann Hypothesis.
  6. 6. Why Primes Matter: From RSA to Modern Number Theory
    Show how the hardness of factoring large primes powers RSA encryption and why primes remain central to research mathematics.
Published by Solid State Press
Prime Numbers: The Atoms of Arithmetic cover
TLDR STUDY GUIDES

Prime Numbers: The Atoms of Arithmetic

The Sieve of Eratosthenes, Unique Factorization, and Euclid's Infinity Proof — A TLDR Primer
Solid State Press

Prime Numbers: The Atoms of Arithmetic

The Sieve of Eratosthenes, Unique Factorization, and Euclid's Infinity Proof — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 What Is a Prime Number?
  2. 2 The Fundamental Theorem of Arithmetic
  3. 3 Finding Primes: The Sieve of Eratosthenes and Trial Division
  4. 4 Euclid's Proof That the Primes Never End
  5. 5 Patterns, Gaps, and Famous Unsolved Problems
  6. 6 Why Primes Matter: From RSA to Modern Number Theory
Chapter 1

What Is a Prime Number?

A prime number is a whole number greater than 1 whose only positive divisors are 1 and itself. A divisor (or factor) of a number is a whole number that divides into it evenly, with nothing left over. So when we say "the divisors of 12 are 1, 2, 3, 4, 6, and 12," we mean each of those numbers divides 12 with no remainder.

Take 7. Try dividing it by every whole number from 2 up to 6: none of them work evenly. The only numbers that divide 7 are 1 and 7 itself. That makes 7 prime.

Now take 12. It's divisible by 1, 2, 3, 4, 6, and 12 — six divisors, not just two. A whole number greater than 1 that has more than two divisors is called a composite number. Every whole number bigger than 1 is either prime or composite; there's no third option and no overlap. Composite numbers are, in a sense, "built" out of smaller pieces multiplied together — 12=2×6=3×4 — while primes resist being broken down any further. This is why the book's subtitle calls primes the "atoms" of arithmetic: like atoms in chemistry, they're the indivisible building blocks that everything else is assembled from. (Section 2 makes this precise.)

Example. Is 15 prime or composite? Is 17? Solution. Check 15 against small divisors: 15=3×5, so 3 and 5 are divisors besides 1 and 15. That's four divisors total, so 15 is composite. Now check 17: it isn't even, isn't divisible by 3 (1+7=8, not a multiple of 3), and 17÷5 and 17÷7 both leave remainders. Since no whole number between 2 and 16 divides it evenly, 17 is prime.

Why isn't 1 prime?

This trips up almost everyone at first, and it's worth understanding why, not just memorizing the rule. Going strictly by the "only two divisors" definition, 1 fails automatically: its only divisor is itself, so it has just one divisor, not two. But that can feel like a technicality, so here's the deeper reason mathematicians deliberately excluded 1.

About This Book

If you're a high school student in Algebra or an intro Number Theory unit, a freshman taking a discrete math course, or a parent trying to help with homework, this book gets you oriented fast. Maybe you're prepping for a test on primes and factoring, or you just want prime numbers explained simply after a confusing class lecture.

This guide walks through what makes a number prime, why 1 is not a prime number (and why that distinction matters), and the fundamental theorem of arithmetic guide every factoring problem depends on. You'll get the Sieve of Eratosthenes explained step by step, Euclid's proof of infinite primes laid out in plain language, and a look at how RSA encryption uses primes to secure the internet. It doubles as a compact number theory study guide for high school and early college — concise, with no filler.

Read it straight through first. Work through the examples with pencil and paper. Then try the problem set at the end to check what actually stuck.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters — readable in one sitting.

Coming soon to Amazon