Fermat's theorem on sums of two squares

E146190

result in arithmetic theorem in number theory

Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.

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All labels observed (3)

Label	Occurrences
Fermat's sum of two squares theorem	1
Fermat's theorem on sums of two squares canonical	1
sum of two squares theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T1281483 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Fermat's theorem on sums of two squares
Context triple: [Pierre de Fermat, notableWork, Fermat's theorem on sums of two squares]

A. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
B. Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
C. Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
D. An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
E. Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Über die Anzahl der Primzahlen unter einer gegebenen Grösse is Bernhard Riemann’s seminal 1859 paper that introduced the Riemann zeta function and laid the foundations of analytic number theory, including the famous Riemann Hypothesis.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Fermat's theorem on sums of two squares
Target entity description: Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.

A. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
B. Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
C. Disquisitiones Arithmeticae
Disquisitiones Arithmeticae is a foundational 1801 treatise on number theory that systematically developed the subject and introduced many of its central concepts and methods.
D. An Introduction to the Theory of Numbers
An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
E. Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Über die Anzahl der Primzahlen unter einer gegebenen Grösse is Bernhard Riemann’s seminal 1859 paper that introduced the Riemann zeta function and laid the foundations of analytic number theory, including the famous Riemann Hypothesis.
F. None of above. chosen

Statements (51)

Predicate	Object
instanceOf	result in arithmetic ⓘ theorem in number theory ⓘ
appliesTo	positive integers ⓘ prime numbers ⓘ
characterizes	integers that are sums of two squares ⓘ primes that are sums of two squares ⓘ
concerns	prime numbers expressible as sum of two squares ⓘ representation of integers as sums of two squares ⓘ
excludesPrimes	odd primes congruent to 3 modulo 4 from being sums of two squares ⓘ
field	number theory ⓘ
firstProofBy	Leonhard Euler ⓘ
generalizes	classification of norms from Q(i) ⓘ
hasAlternativeProofBy	Carl Friedrich Gauss ⓘ Joseph-Louis Lagrange ⓘ
hasConsequence	classification of norms of Gaussian integers ⓘ infinitely many primes congruent to 1 modulo 4 ⓘ
hasCounterexample	3 ≡ 3 (mod 4) and 3 is not a sum of two squares ⓘ 7 ≡ 3 (mod 4) and 7 is not a sum of two squares ⓘ
hasExample	13 = 2^2 + 3^2 and 13 ≡ 1 (mod 4) ⓘ 5 = 1^2 + 2^2 and 5 ≡ 1 (mod 4) ⓘ
hasIntegerCaseStatement	A positive integer n is a sum of two squares if and only if every prime factor q ≡ 3 (mod 4) occurs with even exponent in the prime factorization of n ⓘ
hasPrimeCaseStatement	An odd prime p can be written as x^2 + y^2 with integers x,y if and only if p ≡ 1 (mod 4) ⓘ The prime p = 2 can be written as 1^2 + 1^2 ⓘ
historicalAttribution	first stated by Pierre de Fermat in the 17th century ⓘ
holdsIn	ring of Gaussian integers Z[i] ⓘ
implies	if n is a sum of two squares then primes ≡ 3 (mod 4) divide n to even powers ⓘ if p ≡ 1 (mod 4) is prime then there exist integers x,y with p = x^2 + y^2 ⓘ no prime p ≡ 3 (mod 4) is a sum of two nonzero squares ⓘ
isSpecialCaseOf	theorems on representations by quadratic forms ⓘ theory of binary quadratic forms ⓘ
namedAfter	Pierre de Fermat ⓘ
proofTechnique	algebraic number theory ⓘ geometry of numbers ⓘ infinite descent ⓘ
relatedTo	Dirichlet's theorem on arithmetic progressions ⓘ Gaussian integer unique factorization ⓘ Lagrange's four-square theorem ⓘ Pythagorean triples ⓘ sum of two squares function r_2(n) ⓘ
statementAbout	sum of two perfect squares ⓘ
usedIn	algebraic number theory ⓘ analytic number theory ⓘ elementary number theory courses ⓘ
usesConcept	Gaussian integers ⓘ congruence modulo 4 ⓘ norm in quadratic integer rings ⓘ prime factorization ⓘ unique factorization domain ⓘ
usesProperty	norm multiplicativity in Z[i] ⓘ primes p ≡ 1 (mod 4) split in Z[i] ⓘ primes p ≡ 3 (mod 4) remain inert in Z[i] ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Pierre de Fermat → notableWork → Fermat's theorem on sums of two squares ⓘ

Gaussian integers → associatedWith → Fermat's sum of two squares theorem ⓘ

linked to: Fermat's theorem on sums of two squares

Legendre's three-square theorem → relatedTo → sum of two squares theorem ⓘ