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Sum of two squares theorem

The sum of two squares theorem states that a positive integer n can be expressed as the sum of two squares of integers—that is, n = a^2 + b^2 for some integers a and b—if and only if in the prime factorization of n, every prime congruent to 3 modulo 4 has an even exponent.^[1] This general result builds on Fermat's theorem on sums of two squares, which states that an odd prime p can be written as p = a^2 + b^2 if and only if p \equiv 1 \pmod{4}, while the prime 2 is itself a sum of two squares as $2 = 1^2 + 1^2.^[2]^[3] The property extends multiplicatively because the product of two sums of squares is again a sum of two squares, via the Brahmagupta–Fibonacci identity: (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2.^[1] Fermat claimed the prime case in the 17th century without proof. The general theorem was first rigorously proved by Leonhard Euler in 1749, using infinite descent to show that primes congruent to 3 modulo 4 must have even exponents in such representations.^[4] Modern proofs often rely on the ring of Gaussian integers \mathbb{Z}, which is a unique factorization domain where primes congruent to 3 modulo 4 remain prime, while those congruent to 1 modulo 4 factor as sums of two squares, and 2 factors as (1 + i)(1 - i) up to units.^[2] This algebraic perspective highlights the theorem's deep connections to algebraic number theory and quadratic forms. The theorem has significant applications in analytic number theory, including the evaluation of the number of representations r_2(n), which counts the ways n can be written as a sum of two squares (considering order and signs), given by r_2(n) = 4(d_1(n) - d_3(n)), where d_i(n) is the number of divisors of n congruent to i modulo 4, and equals zero precisely when the theorem's condition fails.^[2] It also informs the distribution of Gaussian primes and has implications in cryptography and computational number theory for factoring and primality testing.^[2]

Statement and Properties

Theorem Statement

The sum of two squares theorem provides a complete characterization of the positive integers that admit representation as the sum of two integer squares. A positive integer n is expressible as a sum of two squares if there exist integers a and b (possibly zero or negative, though squares make signs irrelevant) such that n = a^2 + b^2.^[5] The precise statement of the theorem is as follows: A positive integer n can be written as a sum of two squares if and only if, in its prime factorization n = 2^e \prod p_i^{e_i} \prod q_j^{f_j}, where the p_i are distinct primes congruent to 1 modulo 4 and the q_j are distinct primes congruent to 3 modulo 4, every exponent f_j is even.^[5]^[6] This "if and only if" condition is biconditional, meaning the prime factorization criterion is both necessary (no such representation exists if any f_j is odd) and sufficient (a representation is guaranteed otherwise).^[5] The theorem connects the prime factorization of n to the solvability of the quadratic Diophantine equation x^2 + y^2 = n over the integers, forming a cornerstone of the study of binary quadratic forms in number theory.^[6]

Prime Factorization Condition

The prime factorization condition specifies that a positive integer n > 0 can be expressed as a sum of two squares if and only if, in its prime factorization, every prime congruent to 3 modulo 4 has an even exponent.^[7] This condition arises from the behavior of primes under representation as sums of two squares and the closure of such representations under multiplication. Primes are classified based on their residue modulo 4. The prime 2 is representable as $1^2 + 1^2.^[8] An odd prime p is representable as a sum of two squares if and only if p \equiv 1 \pmod{4}; for example, $5 = 1^2 + 2^2.^[9] In contrast, no odd prime p \equiv 3 \pmod{4} can be written as a sum of two squares; for instance, 3 cannot be expressed in this form.^[9] The exponents in the prime factorization determine representability as follows. For the prime 2 or any prime p \equiv 1 \pmod{4}, any non-negative integer exponent is allowed, as powers can be constructed via multiplication of representations. For a prime q \equiv 3 \pmod{4}, the exponent must be even: q^{2m} = (q^m)^2 + 0^2 for integer m \geq 0, but q^{2m+1} introduces an odd power that prevents representation. For example, $9 = 3^2 = 0^2 + 3^2, whereas 3 cannot.^[7] This condition extends to composite numbers through the multiplicative property: the product of two sums of two squares is itself a sum of two squares. This follows from the Brahmagupta–Fibonacci identity,

(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2,

which holds for all integers a, b, c, d.^[10] Consequently, if the prime factors of n satisfy the condition individually, so does n. Any perfect square is also a sum of two squares, via k^2 = k^2 + 0^2 for integer k.^[7]

Historical Context

Early Discoveries

In ancient Greek mathematics, the Pythagorean theorem provided an early geometric foundation for understanding sums of two squares. Euclid's Elements, composed around 300 BCE, includes Proposition 47 in Book I, which states that in a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides, or geometrically, a^2 + b^2 = c^2. This result not only interprets the sum of two squares as another square but also inspired the study of integer solutions known as Pythagorean triples, such as (3, 4, 5), where primitive triples generate further examples through scaling. Euclid's work laid the groundwork for Diophantine problems involving squares, though it focused more on geometric constructions than algebraic representations of arbitrary numbers.^[11] Centuries later, Diophantus of Alexandria (c. 250 CE) advanced these ideas algebraically in his Arithmetica. He posed problems requiring the representation of given numbers as sums of two squares with additional constraints, such as expressing a number already known as a sum of two squares in another pair of squares (possibly rational). For instance, Diophantus sought to divide 13 (noted as $2^2 + 3^2 = 13) into two other squares, yielding rational solutions like \left(\frac{18}{5}\right)^2 + \left(\frac{1}{5}\right)^2 = 13, and explored rational points on circles, implicitly linking to sums of squares via the equation x^2 + y^2 = r^2. These exercises revealed patterns in which numbers could be expressed as such sums, fostering early conjectures about representability without a general theorem.^[12] In ancient Indian mathematics, significant progress occurred with Brahmagupta's Brahma-sphuta-siddhanta (628 CE), where he formulated an identity showing that the product of two numbers, each a sum of two squares, is itself a sum of two squares in two distinct ways: (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 = (ac + bd)^2 + (ad - bc)^2. This composition law allowed generation of larger representable numbers from smaller ones, such as deriving 65 = 1² + 8² = 4² + 7² from factors 5 = 1² + 2² and 13 = 2² + 3². Brahmagupta applied this in solving Pell-like equations, noting patterns in quadratic forms but not a complete classification.^[13] Bhāskara II, in the 12th century, built upon Brahmagupta's work in texts like Lilavati and Bijaganita, using the identity to explore Diophantine equations and infinite series involving squares. He demonstrated applications in astronomical computations and quadratic solutions, observing that certain composite numbers consistently appeared as sums of two squares through iterative multiplication, though he stopped short of a prime-based criterion. These insights highlighted multiplicative closure, influencing later medieval developments.^[13] Medieval European mathematics saw renewed interest through Leonardo of Pisa (Fibonacci), who in Liber Quadratorum (1225) independently presented the same product identity for sums of two squares, likely via Arabic translations of Indian sources. Fibonacci applied it to solve problems like finding congruent numbers (areas of right triangles with rational sides) as sums of two squares, such as showing 5 as both a leg and a sum of squares. His work included conjectural observations on which integers, particularly those related to consecutive numbers or squares, could be decomposed this way, bridging ancient patterns to emerging algebraic traditions without a full proof.^[14]

Contributions by Fermat and Euler

Pierre de Fermat first articulated the core claim of the sum of two squares theorem in a letter to Marin Mersenne dated December 25, 1640, asserting that every prime number of the form $4k + 1 can be expressed as the sum of two integer squares, p = x^2 + y^2, and that he possessed a proof via infinite descent but lacked space to include it. This correspondence with Mersenne, a key figure in disseminating mathematical ideas, highlighted Fermat's assertion without providing details, sparking interest among contemporaries in verifying the claim.^[15] Fermat's statement built on earlier observations but marked the first precise formulation linking the theorem to primes congruent to 1 modulo 4. Leonhard Euler took up Fermat's challenge in the 1740s, successfully proving the claim for primes using a method of infinite descent, first communicated in a letter to Christian Goldbach dated April 12, 1749, demonstrating that assuming a prime p \equiv 1 \pmod{4} cannot be written as a sum of two squares leads to an infinite sequence of decreasing positive integers, yielding a contradiction. Euler's approach relied on algebraic identities and descent arguments. This proof confirmed Fermat's assertion rigorously for the first time. Beyond primes, Euler extended the results to composite numbers in related work, establishing the multiplicative property: if two numbers are each sums of two squares, their product is also a sum of two squares, via the identity (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2. He further showed that a composite number can be expressed as a sum of two squares if and only if all its prime factors of the form $4k + 3 appear to even powers in its factorization. These partial results laid foundational insights into the theorem's broader applicability, influencing later complete proofs.

Proofs

Lagrange's Elementary Proof

Joseph-Louis Lagrange provided an elementary proof of the sum of two squares theorem in 1770, relying solely on congruences, Wilson's theorem, and a descent argument, without algebraic number theory. The approach first establishes the theorem for prime numbers and then extends it to all positive integers via multiplicativity. Specifically, it shows that every prime p \equiv 1 \pmod{4} can be expressed as p = a^2 + b^2 for integers a and b, that $2 = 1^2 + 1^2, and that no prime p \equiv 3 \pmod{4} can be written as a sum of two squares (though even powers p^{2k} = (p^k)^2 + 0^2 can). The product of sums of two squares is again a sum of two squares, via the identity

(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2,

which is multiplicative and allows representation of composite numbers whose prime factors satisfy the condition (no prime \equiv 3 \pmod{4} to an odd power).^[16] To handle primes p \equiv 3 \pmod{4}, suppose p = a^2 + b^2. Squares modulo 4 are 0 or 1, so sums of two squares are 0, 1, or 2 modulo 4, but p \equiv 3 \pmod{4}, a contradiction. Thus, such primes cannot be sums of two squares, and any odd power p^{2k+1} inherits this incongruence modulo 4. Even powers are trivial sums as noted above. For the prime 2, the representation $2 = 1^2 + 1^2 holds directly.^[16] The core of the proof concerns odd primes p = 4k + 1. By Wilson's theorem, (p-1)! \equiv -1 \pmod{p}.^[17] Since p-1 = 4k, (4k)! \equiv -1 \pmod{p}. The factorial decomposes as (4k)! = (2k)! \cdot \prod_{j=1}^{2k} (2k + j). Modulo p, $2k + j \equiv -(2k + 1 - j) \pmod{p} for j = 1 to $2k, so the product \prod_{j=1}^{2k} (2k + j) \equiv (-1)^{2k} \prod_{i=1}^{2k} i = (2k)! \pmod{p}. Thus, (4k)! \equiv [(2k)!]^2 \pmod{p}, implying [(2k)!]^2 \equiv -1 \pmod{p}, or p divides [(2k)!]^2 + 1. Setting u = (2k)! and v = 1, we have integers u, v such that u^2 + v^2 = n p for some positive integer n.^[16] To show n = 1, employ infinite descent on the minimal positive n such that n p = a^2 + b^2 for some integers a, b. If n = 1, done. Otherwise, assume n > 1 is minimal. Since p \equiv 1 \pmod{4}, there exists an integer r modulo p with r^2 \equiv -1 \pmod{p} (from the above congruence, taking r \equiv (2k)! \pmod{p}). Consider integers c \equiv a + b r \pmod{p} and d \equiv b - a r \pmod{p} with |c|, |d| < p/2. Then c^2 + d^2 \equiv (a + b r)^2 + (b - a r)^2 = a^2 + 2abr + b^2 r^2 + b^2 - 2abr + a^2 r^2 = (a^2 + b^2)(1 + r^2) \equiv 0 \pmod{p}, since r^2 + 1 \equiv 0 \pmod{p} and p divides a^2 + b^2. More precisely, the choice ensures c^2 + d^2 = m p for some m < n, contradicting minimality unless m = 0, but adjustments via the identity yield a smaller positive multiple. Alternatively, a direct descent uses cases on parity of n: if n even and a, b same parity, set a_1 = (a - b)/2, b_1 = (a + b)/2, then a_1^2 + b_1^2 = (n/2) p, reducing n. If n odd, find \alpha \equiv a \pmod{n}, \beta \equiv b \pmod{n} with |\alpha|, |\beta| < n/2; then n divides \alpha^2 + \beta^2 (since \equiv a^2 + b^2 \equiv 0 \pmod{n}), and set a_1 = (\alpha a + \beta b)/n, b_1 = (\alpha b - \beta a)/n, yielding a_1^2 + b_1^2 = [(\alpha^2 + \beta^2)/n] p with the new multiplier < n/2. Repeating reduces to n = 1, so p = x^2 + y^2.^[16] This completes the proof for primes, and multiplicativity handles the general case, confirming the theorem's prime factorization condition.^[16]

Proof Using Gaussian Integers

The proof of the sum of two squares theorem using Gaussian integers relies on the algebraic structure of the ring \mathbb{Z} = \{ a + bi \mid a, b \in \mathbb{Z} \}, a subring of the complex numbers where i^2 = -1. This ring is equipped with the norm function N(\alpha) = a^2 + b^2 for \alpha = a + bi, which maps to non-negative integers and satisfies multiplicativity: N(\alpha \beta) = N(\alpha) N(\beta) for all \alpha, \beta \in \mathbb{Z}. The norm plays a crucial role in factorization, as N(\alpha) = 1 if and only if \alpha is a unit (specifically, the units are $1, -1, i, -i), and for non-units, the norm measures "size" in a way that enables the Euclidean algorithm, proving \mathbb{Z} is a Euclidean domain and hence a unique factorization domain (UFD). In a UFD, every non-zero, non-unit element factors uniquely into irreducible elements (Gaussian primes) up to units and order.^[18] The factorization behavior of rational primes in \mathbb{Z} is determined by Dedekind's theorem on the ramification of primes in quadratic number fields. For the field \mathbb{Q}(i) with ring of integers \mathbb{Z} and discriminant -4, the theorem classifies the splitting of odd rational primes p as follows: p ramifies if p divides the discriminant (only p = 2, which factors as $2 = -i (1 + i)^2 up to units, where $1 + i is a Gaussian prime of norm 2); p splits completely if the Legendre symbol (-4/p) = 1, equivalent to p \equiv 1 \pmod{4}, so p = \pi \overline{\pi} for distinct Gaussian primes \pi, \overline{\pi} (conjugates) each of norm p; and p remains inert (prime in \mathbb{Z}) if (-4/p) = -1, equivalent to p \equiv 3 \pmod{4}. This classification ensures that Gaussian primes, up to units, are either associates of $1 + i, inert rational primes p \equiv 3 \pmod{4}, or the factors \pi of split primes p \equiv 1 \pmod{4} with N(\pi) = p.^[19] To connect this to the sum of two squares theorem, consider a positive integer n. Viewing n as an element of \mathbb{Z}, its norm is N(n) = n^2. Since \mathbb{Z} is a UFD, n factors uniquely into Gaussian primes. The norm n^2 is then the product of the norms of these prime factors, raised to even powers due to the square. Thus, n can be written as N(\alpha) for some \alpha \in \mathbb{Z}, say \alpha = a + bi, if and only if the prime factorization of n in \mathbb{Z} involves only the prime 2 and primes p \equiv 1 \pmod{4} (which have even total exponent in the Gaussian factorization via splitting) or primes p \equiv 3 \pmod{4} to even exponents (pairing inert primes). In other words, n = a^2 + b^2 if and only if every prime p \equiv 3 \pmod{4} in the factorization of n has even exponent. This equivalence directly yields the theorem, as the Gaussian factorization guarantees the representation whenever the condition holds.^[15]^[18] This proof, leveraging the unique factorization in \mathbb{Z} and Dedekind's ramification theorem, provides an elegant algebraic perspective on the theorem's characterization, distinct from elementary descent methods.^[19]

Examples

Basic Representations

The sum of two squares theorem provides a criterion for determining which positive integers can be expressed as a^2 + b^2 where a and b are integers, and basic examples illustrate its application to small numbers. For instance, $1 = 1^2 + 0^2 and $2 = 1^2 + 1^2, showing the smallest cases including the role of zero and equal squares. Further small sums include $5 = 2^2 + 1^2, $10 = 3^2 + 1^2, and $13 = 3^2 + 2^2 (or equivalently $2^2 + 3^2).^[20]^[21] Composite numbers that satisfy the theorem's conditions also yield representations, often in multiple ways due to their factorization. Consider $25 = 5^2 + 0^2 = 4^2 + 3^2, where the prime power $5^2 allows distinct decompositions. Similarly, $50 = 5^2 + 5^2 = 7^2 + 1^2, reflecting the influence of the factor $2 \cdot 5^2.^[20]^[16] Primes congruent to 1 modulo 4, as per the theorem's prime factorization condition, are particularly illustrative. Examples include $17 = 4^2 + 1^2 and $29 = 5^2 + 2^2 (or $2^2 + 5^2). These representations highlight how such primes contribute to the theorem's scope.^[21]^[16] A key pattern emerges in products of numbers expressible as sums of two squares, which themselves admit such forms, often multiply. For example, $65 = 5 \cdot 13 = 8^2 + 1^2 = 7^2 + 4^2, demonstrating two distinct ways derived from the individual factorizations $5 = 2^2 + 1^2 and $13 = 3^2 + 2^2.^[16]

Non-Representable Numbers

A fundamental reason certain positive integers cannot be expressed as the sum of two squares stems from quadratic residues modulo 4. The square of any integer is congruent to 0 or 1 modulo 4: even integers square to 0 mod 4, while odd integers square to 1 mod 4. Consequently, the sum of two squares can only be congruent to 0, 1, or 2 modulo 4, never 3 modulo 4. Thus, any integer congruent to 3 modulo 4, such as 3, 7, 11, 15, or 19, cannot be written as x^2 + y^2 for integers x and y.^[1]^[16] This modulo 4 obstruction directly implies that all prime numbers of the form $4k + 3 are non-representable. Examples include the primes 3, 7, 11, and 19, none of which can be decomposed into two squares despite exhaustive checks of small integers. For instance, 3 exceeds $1^2 + 1^2 = 2 but falls short of $1^2 + 2^2 = 5, and similarly for the others up to relevant bounds. These primes serve as the building blocks for broader non-representability.^[1]^[16]^[15] The presence of any prime congruent to 3 modulo 4 raised to an odd power in a number's prime factorization precludes representation as a sum of two squares. For powers of such primes, even exponents allow representation (e.g., $9 = 3^2 = 0^2 + 3^2), but odd exponents do not (e.g., $27 = 3^3 cannot be expressed as two squares, as it would require factoring out the extra factor of 3, which violates the condition). Composites follow suit: 15 = 3 × 5 includes 3 to the first power, and 21 = 3 × 7 includes both 3 and 7 to odd powers, rendering them non-representable. This necessity condition underscores the theorem's precision, as altering exponents can toggle representability.^[1]^[16]

Jacobi's Two-Square Theorem

Jacobi's two-square theorem provides an exact formula for the number of integer solutions to the equation n = x^2 + y^2, where n is a positive integer and the solutions count all ordered pairs (x, y) including different signs and orders. Specifically, let r_2(n) denote this number of representations. Then,

r_2(n) = 4 \left( d_1(n) - d_3(n) \right),

where d_1(n) is the number of positive divisors of n congruent to 1 modulo 4, and d_3(n) is the number congruent to 3 modulo 4.^[5] This formula was established by Carl Gustav Jacob Jacobi in 1829 using properties of elliptic functions.^[5] For prime numbers, the theorem yields simple cases that align with Fermat's theorem on sums of two squares. The prime p = 2 has r_2(2) = 4, corresponding to the representations (\pm 1, \pm 1) and permutations. An odd prime p \equiv 1 \pmod{4} has r_2(p) = 8, reflecting two distinct positive pairs up to order, each extended by signs and swaps. In contrast, a prime p \equiv 3 \pmod{4} has r_2(p) = 0, as it cannot be expressed as a sum of two squares.^[5]^[22] A representative example is n = 65 = 5 \times 13, where the positive divisors are 1, 5, 13, and 65, all congruent to 1 modulo 4, so d_1(65) = 4 and d_3(65) = 0. Thus, r_2(65) = 4(4 - 0) = 16, accounting for the pairs $1^2 + 8^2 and $4^2 + 7^2, each with 8 signed and ordered variants.^[5] The theorem connects to algebraic number theory through the quadratic field \mathbb{Q}(\sqrt{-1}), whose ring of integers (the Gaussian integers) has class number 1. This unique factorization property underpins the representation counts, as the prime factorization in Gaussian integers directly determines the ways n factors into norms of elements, yielding the divisor difference in the formula.^[23]

Fermat's Theorem on Sums of Two Squares for Primes

Fermat stated his theorem on sums of two squares in a letter to Marin Mersenne dated December 25, 1640.^[24] The theorem asserts that an odd prime p can be expressed as the sum of two squares of integers, p = a^2 + b^2 with a, b \in \mathbb{Z} and a > 0, b > 0, if and only if p \equiv 1 \pmod{4}.^[8] Additionally, the prime p = 2 can be written as $1^2 + 1^2.^[8] The "only if" direction follows from properties modulo 4: the square of any integer is congruent to 0 or 1 modulo 4, so the sum of two squares is congruent to 0, 1, or 2 modulo 4, but never 3 modulo 4; thus, no odd prime congruent to 3 modulo 4 can be a sum of two squares.^[25] For the "if" direction, the existence proof combines quadratic reciprocity with a descent argument. Quadratic reciprocity implies that -1 is a quadratic residue modulo p if and only if p \equiv [1](/page/1) \pmod{4}, so there exists an integer u such that u^2 \equiv -1 \pmod{p}, or u^2 + [1](/page/1) = k p for some k with [1](/page/1) \leq k < p.^[25] If k > [1](/page/1), the Brahmagupta–Fibonacci identity (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 is applied iteratively to descend to a smaller multiple until a representation p = a^2 + b^2 is obtained.^[25] The representation of such a prime p as a sum of two squares is unique up to the order and signs of a and b.^[9] Examples include $5 = 1^2 + 2^2, $13 = 2^2 + 3^2, $17 = 1^2 + 4^2, $37 = 1^2 + 6^2, and $41 = 4^2 + 5^2, all primes congruent to 1 modulo 4.^[9] In contrast, primes like 3, 7, 11, 19, and 23, which are congruent to 3 modulo 4, cannot be expressed as sums of two squares.^[8]

References

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Aug 22, 2008 · Theorem 3.13. (Two Squares Theorem) A positive integer n is the sum of two squares if and only if each prime factor p of n such that p ≡ 3 (mod ...
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Our goal is to prove the following result formulated by Fermat. Theorem 1. A prime p can be written as the sum of two squares if and only if p = 2 or.
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The product of two integers, each of which is the sum of two squares, is the sum of two squares. Nothing like that could be said about three squares. But in ...Missing: multiplicative | Show results with:multiplicative
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1. ▫ He proved what is now called the Brahmagupta identity: Page 10. The Bhavana and Chakravala. ▫ He realized that this identity can be used to generate all ...
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None
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... sum of two squares is necessarily divisible by 4n + 1, and thus the prime number 4n + 1 is a sum of two squares. 7. Since the difference of order 2n depends ...Missing: 1740s | Show results with:1740s
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It says that if two numbers are sum of two squares, so is their product. For example 493 = 17 ∗ 29 and 17 = 12 + 42, 29 = 22 + 52 give 493 = (1 ∗ 2 ...
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None
### Summary of Examples of Numbers as Sums of Two Squares
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[PDF] Primes as the sum of two squares
Odd primes p can be written as a sum of two squares if and only if p is congruent to 1 (mod 4q).
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A simple proof of Jacobi's two-square theorem. 1. In a recent note, John A. Ewell [1] derives Fermat's two-square theorem: A prime p = 4n + 1 is the sum of ...
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[PDF] From sum of two squares to arithmetic Siegel-Weil formulas
A prime p 6= 2 is the sum of two squares if and only if ... In his book Fundamenta nova theoriae functionum ellipticarum (1829), Jacobi proved the follow-.
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In this section, we prove Fermat's theorem, which says that any prime number con- gruent to 1 (mod 4) can be expressed as the sum of two squares. We will use ...