Student & engineer view
Learn from the graph itself
Nothing here is a simplified retelling that can drift from the truth. Every page below is generated from a verified node, and every learning path is a real dependency path: the exact order in which the mathematics is built. Start with intuition; descend to rigor whenever you choose.
Learning paths
8 steps · axioms → theorem
The primes never run out
Eight steps from bare axioms to Euclid's 2,300-year-old jewel — every step machine-checked for structure.
1
What 'never runs out' means
Definition · Infinite set
2The rules of integer arithmetic
Axiom · Integers form a commutative ring
3What 'divides evenly' really means
Definition · Divisibility
4The unbreakable numbers
Definition · Prime number
5Common divisors survive mixing
Lemma · Divisibility respects linear combinations
6There is always a smallest one
Axiom · Well-Ordering Principle
7Every number contains an atom
Lemma · Every integer > 1 has a prime divisor
8The primes never run out
Theorem · Infinitude of primes (Euclid)
10 steps · axioms → theorem
Every number has one recipe
Build up Bezout's identity and Euclid's lemma, then assemble the Fundamental Theorem of Arithmetic.
1
The domino principle
Axiom · Axiom of Induction
2The rules of integer arithmetic
Axiom · Integers form a commutative ring
3What 'divides evenly' really means
Definition · Divisibility
4The unbreakable numbers
Definition · Prime number
5There is always a smallest one
Axiom · Well-Ordering Principle
6The biggest shared factor
Definition · Greatest common divisor
7The gcd is reachable
Lemma · Bezout's identity
8Primes can't be fooled by products
Lemma · Euclid's lemma
9Every number contains an atom
Lemma · Every integer > 1 has a prime divisor
10Every number has exactly one recipe
Theorem · Fundamental Theorem of Arithmetic
Everything in the corpus
Axiom
The domino principle
If the first domino falls, and every falling domino knocks over the next one, then all the dominoes fall.
AxiomThe rules of integer arithmetic
Adding and multiplying whole numbers follows dependable rules: order doesn't matter, grouping doesn't matter, and multiplication distributes over addition.
AxiomThere is always a smallest one
Any non-empty collection of counting numbers has a smallest member.
DefinitionWhat 'never runs out' means
A set is infinite when no list, however long, can contain all of it — there is always one more.
PropositionTwo axioms, one idea
The domino principle and 'there is always a smallest one' are secretly the same fact in different clothes.
DefinitionWhat 'divides evenly' really means
a divides b when b is a times some whole number — division with nothing left over.
DefinitionThe biggest shared factor
The gcd of two numbers is the largest number that divides both of them.
DefinitionThe unbreakable numbers
A prime is a number bigger than 1 that only 1 and itself divide evenly.
LemmaCommon divisors survive mixing
If d divides both a and b, it also divides any whole-number mix of them, like 5a - 3b.
CounterexampleWhy 1 is not a prime
1 is deliberately excluded from the primes — otherwise every number would have infinitely many 'factorizations'.
DefinitionNumbers built from smaller pieces
A composite number is one bigger than 1 that isn't prime — it can be split into a product of smaller numbers.
ExampleSeven, checked by hand
7 can only be divided evenly by 1 and 7 — a prime, verified directly.
LemmaThe gcd is reachable
You can always combine a and b with whole-number multipliers to land exactly on their gcd: gcd(a,b) = ax + by.
LemmaEvery number contains an atom
Every whole number bigger than 1 has at least one prime hiding inside it as a factor.
ExampleSix, taken apart
6 = 2 × 3, so it is composite — built from smaller pieces.
LemmaPrimes can't be fooled by products
If a prime divides a product a×b, it must divide a or divide b — it can't 'split itself' across the two.
TheoremThe primes never run out
No matter how many primes you've found, there is always another one. Proven ~2,300 years ago and still airtight.
CorollaryYou can always ask for the next prime
Because primes never run out, 'the 100th prime' or 'the billionth prime' is always a meaningful thing to ask for.
TheoremEvery number has exactly one recipe
Every whole number bigger than 1 breaks into primes in exactly one way. 60 is 2×2×3×5 — always, and nothing else.