Infinitude of Primes
Euclid’s proof that there are infinitely many primes.
Key idea
Euclid’s proof that there are infinitely many primes.
In depth
Euclid’s proof: assume finitely many primes p₁, ..., pₙ. Then N = p₁...pₙ + 1 has a prime factor not in the list, a contradiction. Thus there are infinitely many primes.
Worked example
With primes 2, 3, 5: N = 31, a new prime; with 2, 3, 5, 7, 11, 13: N = 30031 = 59 × 509.
Real-world uses
- Explaining why new prime numbers keep appearing, a key idea in number theory
Common mistake
Thinking N itself must be prime. It only has a prime factor outside the list.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Infinitude of Primes?
Euclid’s proof that there are infinitely many primes. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Infinitude of Primes used in real life?
Real-world examples: Explaining why new prime numbers keep appearing, a key idea in number theory.
What should I learn before Infinitude of Primes?
Make sure you are comfortable with Fundamental Theorem of Arithmetic, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Infinitude of Primes?
Thinking N itself must be prime. It only has a prime factor outside the list.
How can I practise Infinitude of Primes?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Infinitude of Primes?
Move on to Sieve of Eratosthenes next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.