Euclid’s Infinite Primes Proof and How Stanford’s Mathematical Thinking Course Can Transform Your Mindset

Euclid’s Proof That There Are Infinite Prime Numbers

The idea that prime numbers never run out was first proven by the ancient Greek mathematician Euclid, who lived around 300 BCE. His proof appears in Elements, one of the most influential mathematical works of all time.

The Proof (By Contradiction)

Euclid’s argument is both elegant and simple, using a proof by contradiction:

1. Assume there are only a finite number of prime numbers.

   • Let’s say there are only ( n ) primes: (p1, p2, p3, …, pn).

2. Construct a new number that is not in the list.

   • Multiply all known primes together and add 1:

     

3. Check whether ( N ) is prime or composite.

   • If ( N ) is a prime, then it is a new prime not in our original list—contradicting our assumption.
   • If ( N ) is not a prime, then it must be divisible by some prime number. However, when we divide ( N ) by any of the original primes ( p1, p2, …, pn ), we always get a remainder of 1. This means ( N ) is not divisible by any known prime.

4. Conclusion: Our assumption was wrong.

   • Since every attempt to list all prime numbers leads to the discovery of a new prime, the number of primes must be infinite.

This proof remains one of the most famous arguments in mathematics and continues to inspire number theorists today.

Course Review: Stanford University’s “Introduction to Mathematical Thinking” on Coursera

Instructor: Dr. Keith Devlin
Platform: Coursera
Duration: Approx. 7 weeks

What Is the Course About?

“Introduction to Mathematical Thinking” is a course designed by Stanford mathematician Dr. Keith Devlin. Unlike traditional math courses that focus on calculations, this course emphasizes how to think like a mathematician—developing logical reasoning, problem-solving skills, and abstract thinking.

Who Is It For?

This course is ideal for:

• Students who struggle with the transition from high school math to university-level mathematical thinking.
• Professionals who want to develop critical thinking skills for data analysis, programming, or logic-based fields.
• Anyone curious about how mathematicians approach complex problems.

Key Topics Covered

1. The Language of Mathematics

   • Understanding symbols, logic, and formal notation.

2. Logical Reasoning & Proofs

   • How to construct and evaluate mathematical arguments.
   • Types of proofs: direct proof, proof by contradiction, and induction.

3. Number Theory & Real-world Applications

   • Concepts related to primes, divisibility, and rationality.

4. The Power of Abstract Thinking

   • How to generalize patterns and ideas beyond numbers.

How This Course Can Help Students and Professionals

• Bridges the Gap to Advanced Mathematics: Many students struggle when transitioning to formal mathematics. This course prepares learners for proof-based courses like real analysis and abstract algebra.
• Improves Problem-Solving Skills: Learning to think logically helps in areas beyond math, including coding, finance, and artificial intelligence.
• Great for Competitive Exams: Logical reasoning is crucial in standardized tests like the GRE, GMAT, and actuarial exams.

Pros & Cons

✅ Pros:

• Clear explanations from an experienced professor.
• No advanced math background needed.
• Encourages deep thinking rather than memorization.
• Free to audit, with a certificate option available.

❌ Cons:

• Requires patience and effort—some learners may find it challenging.
• Not focused on calculations, which may disappoint those looking for traditional math problems.

Final Verdict: Highly Recommended!

“Introduction to Mathematical Thinking” is more than just a math course—it’s an eye-opening journey into logical reasoning and problem-solving. Whether you’re a student or a working professional, this course will challenge you to think differently and become a better problem solver.

Would you like to develop a mathematician’s mindset? Enroll in the course today and experience the power of mathematical thinking!