bibkey: grosswald1985representations authors: Emil Grosswald year: 1985 title: Representations of Integers as Sums of Squares doi: 10.1007/978-1-4613-8566-0 claim: The classical characterization of the natural numbers representable as sums of two squares by the parity of the exponents of their prime factors congruent to three modulo four. strata_touched:

• D5/S3/PrimeForms/SumTwoSquaresClassification
• D5/S3/PrimeForms/ThreeModFourDescent license: citation-only triage: anchor

Representations of Integers as Sums of Squares

Emil Grosswald’s monograph (Springer-Verlag, 1985, 251 pp.) is devoted to the representation of integers as sums of squares. Its chapter “Sums of Two Squares” (pp. 13-23, DOI 10.1007/978-1-4613-8566-0_3) presents the classical two-squares theory, anchoring the characterization formalized in D5/S3/PrimeForms/SumTwoSquaresClassification: a natural number is a sum of two squares exactly when every prime congruent to three modulo four occurs to an even exponent in its factorization.

Only the statement of the classical characterization is attributed to the literature through this anchor. The repository proof discharges the statement through Mathlib’s Nat.eq_sq_add_sq_iff, so the source’s proof route (descent at primes congruent to three modulo four, representation of primes congruent to one modulo four, and multiplicative composition) is not attributed. The anchor also attributes the statement of the descent step itself, formalized separately in D5/S3/PrimeForms/ThreeModFourDescent: a prime congruent to three modulo four dividing a sum of two squares divides both bases. Only that statement is attributed; the repository proof of it goes through Mathlib’s quadratic-residue machinery, not through the chapter’s argument. The full chapter text is paywalled, so the theorem numbering inside the chapter was not verified; the attribution rests on the verified chapter title, page range, and DOI together with the standard identification of this result as the sum-of-two-squares theorem.

Search log

• 2026-08-11: Queried the web for Grosswald "Representations of Integers as Sums of Squares" Springer 1985 DOI 10.1007. Publisher results verified the book DOI 10.1007/978-1-4613-8566-0, and a Proceedings of the Edinburgh Mathematical Society review record confirmed Springer-Verlag, 1985, 251 pp. The result list exposed the chapter record “Sums of Two Squares” at DOI 10.1007/978-1-4613-8566-0_3.
• 2026-08-11: Fetched the Springer chapter page for 10.1007/978-1-4613-8566-0_3. It confirmed the chapter title “Sums of Two Squares” and page range pp. 13-23; the full text is paywalled, so no in-chapter theorem number was verified.
• 2026-08-11: Queried the web for the classical statement ("sums of two squares" theorem "if and only if" prime "3 (mod 4)" even power). Results (including the standard encyclopedia entry on the sum-of-two-squares theorem) restated the characterization by even exponents of prime factors congruent to three modulo four, corroborating the claim attributed here.

Verified locator

• DOI: https://doi.org/10.1007/978-1-4613-8566-0
• Chapter: https://doi.org/10.1007/978-1-4613-8566-0_3