Fermat’s Last Theorem
No integer solutions of xⁿ+yⁿ=zⁿ for n>2.
Key idea
No integer solutions of xⁿ+yⁿ=zⁿ for n>2.
In depth
There are no positive integers with aⁿ + bⁿ = cⁿ for n > 2. Fermat claimed a proof in 1637; Andrew Wiles proved it in 1994–95 using elliptic curves and modular forms.
Worked example
n = 2 has many solutions (3, 4, 5), but n = 3 has none.
Formula
Real-world uses
- A famous result explaining why no whole-number solutions exist for xⁿ + yⁿ = zⁿ when n is above 2
Common mistake
Believing that Fermat’s own marginal proof was found.
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 Fermat’s Last Theorem?
No integer solutions of xⁿ+yⁿ=zⁿ for n>2. It is a topic at the expert level in Discrete Math & Number Theory.
What is the formula for Fermat’s Last Theorem?
The key formula is xⁿ + yⁿ = zⁿ.
Where is Fermat’s Last Theorem used in real life?
Real-world examples: A famous result explaining why no whole-number solutions exist for xⁿ + yⁿ = zⁿ when n is above 2.
What should I learn before Fermat’s Last Theorem?
Make sure you are comfortable with Pell’s Equation, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Fermat’s Last Theorem?
Believing that Fermat’s own marginal proof was found.
How can I practise Fermat’s Last Theorem?
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 Fermat’s Last Theorem?
Move on to Goldbach and Twin Prime Conjectures next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.