AI Tutor
If a mathematical pattern holds true for the first million numbers, does that guarantee it will work forever?
Surprisingly, no matter how many positive examples you collect, a single failure completely destroys a universal statement.
A universal claim states that a property P(x)P(x) is true for every single element in a domain SS, written formally as ∀x∈S,P(x)\forall x \in S, P(x).
To disprove this claim, we only need to show its logical negation ∃x∈S,¬P(x)\exists x \in S, \neg P(x) by finding a single counterexample—a specific element where the premise holds but the conclusion fails.
How do mathematicians uncover failures in formulas that seem unbreakable at first glance?

The Power of a Single Counterexample

In 1640, Pierre de Fermat noticed that Fn=22n+1F_n = 2^{2^n} + 1 yielded prime numbers for n=0,1,2,3,n = 0, 1, 2, 3, and 44, leading him to conjecture that all Fermat numbers are prime.
In 1732, Leonhard Euler proved the entire conjecture false with a single counterexample at n=5n = 5: F5=232+1=4,294,967,297F_5 = 2^{32} + 1 = 4{,}294{,}967{,}297, which factors into 641×6,700,417641 \times 6{,}700{,}417.
You do not need numbers in the billions to disprove most claims; instead, you need a systematic strategy for testing.
Where should your search begin when you are given an unfamiliar algebraic statement?

The Boundary Point Strategy