Fermat's theorem on sums of two squares
E146190
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Fermat's sum of two squares theorem | 1 |
| Fermat's theorem on sums of two squares canonical | 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]
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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.
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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.
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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.
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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.
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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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Fermat's theorem on sums of two squares Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Fermat's sum of two squares theorem