The case of three
The prerequisites are included before the words that depend on them. Skip what you already know.
Start with step one → Return to the storyReading a Lean statement
High school
Separate the objects, their conditions, and the conclusion.
Powers and positive integers
Middle school
A power repeats multiplication; positive integers start at one.
Prime numbers
Middle school
A prime is at least two and has no positive divisors except one and itself.
Comparing remainders
Middle school
Two integers can differ while having the same remainder.
Clock arithmetic
Middle school
Computing inside a fixed finite set of residues is a number world of its own.
Cubes on the mod-nine clock
High school
Every cube lands on 0, 1 or 8 of the nine-residue clock: the first case of the case-of-three story.
Common factors and coprimality
Middle school
Coprime numbers share no positive divisor larger than one.
A wider net with a unit
University
To descend, first prove more: a³ + b³ = u·c³ with an arbitrary unit u.
A hypothetical counterexample
High school
Assume a forbidden solution exists, then keep its evidence together.
Addition and multiplication together
Middle school
A shared set of rules for two operations describes many different number worlds.
How big is a signed number?
Middle school
The absolute value |c| keeps only the size: how far the number sits from zero.
Pick the smallest solution
University
Rank the hypothetical solutions by how often λ divides c: one must be smallest.
Three factors in the Eisenstein world
University
A sum of two cubes splits into three factors once cube roots of unity enter.
Contradiction with the same object
High school
Evidence for f ≠ 0 and f = 0 cannot both hold for the same f.
A stairway that must end
University
A gauge that always strictly decreases cannot decrease forever: Fermat’s descent.