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In 1772, Leonhard Euler discovered that the formula n2+n+41n^2 + n + 41 produces a prime number for every integer from n=0n = 0 all the way up to n=39n = 39.
Forty successful tests in a row might feel like certainty, but plugging in n=40n = 40 gives 402+40+41=40(41)+41=41240^2 + 40 + 41 = 40(41) + 41 = 41^2, which is composite. In mathematics, examples never replace a deductive proof.
How do we break down a written proof to verify that every single link in the logical chain is truly unbroken?

Anatomy of a Proof

Every valid mathematical proof consists of three core components: a premise, a sequence of justified deductions, and a conclusion.
A premise is a starting statement or assumption assumed to be true, while a deduction is a logical step where each new line follows necessarily from the previous line using established definitions, axioms, or algebraic rules.
What does this structural breakdown look like when analyzing a real algebraic proof line by line?

Worked Example: Deconstructing a Proof

Consider this claim: If nn is an odd integer, then n2n^2 is an odd integer. Let's read and critique its formal proof.
Before evaluating the steps, we re-establish our formal definition: an odd integer is any integer expressible in the form 2k+12k + 1 for some integer k∈Zk \in \mathbb{Z}.