Existing in Mathlib · Number theory

Fermat's Two-Square Classification

A natural number is a sum of two natural-number squares exactly when every prime factor leaving residue 3 modulo 4 occurs to an even exponent. Under Mathlib's valuation convention, zero is included.

sum of two squares
prime-factor classification
parity of valuations
quadratic residues
multiplicative representations

Exact theorem

Exact Mathlib statement

theorem Nat.eq_sq_add_sq_iff {n : ℕ} : (∃ x y : ℕ, n = x ^ 2 + y ^ 2) ↔ ∀ q ∈ n.primeFactors, q % 4 = 3 → Even (padicValNat q n)

The theorem at a glance

Fermat's Two-Square Classification — the exact prime-factor criterion

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory number-theory poster balances a two-square equation against a prime-factor ledger in which all 3 mod 4 prime factors are paired, with a separate zero-convention note. — Two-square representability is equivalent to even valuation at every prime factor congruent to 3 modulo 4.

Accessible poster text

Read the headline and panel text

NUMBER THEORY · SUMS OF TWO SQUARES

FERMAT'S TWO-SQUARE CLASSIFICATION

n = x² + y² for some x,y ∈ ℕ

⇔

Every prime factor q ≡ 3 (mod 4) has even exponent in n

REPRESENTATION MEETS PRIME FACTORIZATION

HOW THE CHECKED ROUTE MOVES

1 · Two-square representations multiply

2 · Separate a square part from a squarefree remainder

3 · Connect the remainder to −1 being a square modulo it

4 · Read that modular condition at each prime factor

5 · Exactly the 3 mod 4 prime factors require even exponent

ZERO CONVENTION

n = 0 is included: 0 = 0² + 0², and padicValNat q 0 = 0 by definition.

EXACT SCOPE

Existence only. No uniqueness or representation count; this is the all-natural classification, not only the prime case.

The detailed visual description covers the diagram and its local mathematical labels.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A two-axis square lattice representation is joined by a balanced two-way braid to a prime-factor necklace where one distinguished class of faceted tokens appears only in matched pairs. — A natural number has a two-square representation exactly when every 3 mod 4 prime factor occurs with even exponent.

Why it matters

A mathematical landmark

The two-square theorem is a model example of local-to-global arithmetic: a concrete additive representation is governed exactly by parity conditions in the prime factorization.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's classification of all natural numbers representable as a sum of two natural-number squares. It is not only the prime case: representability is equivalent to every prime factor congruent to 3 modulo 4 occurring with even padicValNat exponent. The convention includes n = 0.

The selected endpoint classifies all natural numbers; it is not merely the prime 1 mod 4 theorem.
It asserts existence of a representation, not uniqueness or a count of representations.
Prime factors congruent to 1 or 2 modulo 4 are not subject to the even-exponent condition.
The exact convention includes n = 0, for which both sides hold.
ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics.

Source and local evidence

Where the theorem comes from

Existing declaration: Nat.eq_sq_add_sq_iff in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.fermat-two-square-theorem.v001
Source: Open the pinned upstream reference