- Quadratic residues table. Recall that the Legendre symbol . x x2 (mod 15) 4 6 10 1 9 4 1 The table shows that 2 is not a square mod 15. A special case is the Legendre symbol of a square: a quadratic residue. This leads to the quadratic residue graph, which we shall denote by G2(p), coinciding with a Paley graph when p 1 mod 4 and a complete graph when p 3 mod 4 (which also happens to be the underlying graph of a Paley digraph). More general variants of this problem underlie other cryptosystems such as Paillier encryption. Let p be an odd prime, and let . e Dec 6, 2022 · Classically, for each prime number p we can construct the corre-sponding Paley graph using quadratic and non-quadratic residues modulo p. , 1926), pp. Such a b is said to be a square root of a modulo n. From this definition onward throughout the entire handout, let p be an odd prime number. 2) are clearly quadratic residues modulo p. 67) so that the number of quadratic residues (mod n) is taken to be one less than the number of squares (mod n). Quadratic residues are highlighted in yellow — note that no entry with a Jacobi symbol of −1 is a quadratic residue, and if k is a quadratic residue modulo a coprime n, then (k n) = 1, but not all entries with a Jacobi symbol of 1 (see the n = 9 and n = 15 rows) are quadratic residues. We say that a is a quadratic residue mod p if a is a square mod p (it is a quadratic non-residue otherwise). The Jacobi symbol is a generalization of the Legendre symbol, which can be used to simplify computations involving quadratic residues. Graham and Spencer brought forth the idea of using quadratic residues to construct a tournament with p vertices where p 3 (mod 4) is a prime. A second property that might take a little longer to spot is the multiplicativity of quadratic residues: for example 2 and 4 are quadratic residues modulo 7, as is 2 · 4 ≡ 1. Also, the product of two nonresidues may be either a residue, a nonresidue, or zero. In 1744, Euler proved the following criterion for quadratic residues [Lem13]. Lemma. We have already proved that −1 is a quadratic residue modulo p for primes p ≡ 1 mod 4, and a quadratic nonresidue for primes p ≡ 3 mod 4. The choice of q ensures that −1 ∈ S, so S = −S and P (q) is an undirected graph; the fact that S generates the additive group ensures that P (q) is connected. But since , is a quadratic residue, as must be . Plus the values to be distributed consist of a composite number (p 1 p 2) multiplied by a prime (p 3) as a sum of numbers (see column 1, Table I Algorithm and Sequences The x using known quadratic residues in Part IV were generated using the Chinese Remainder Theory on the web page of B. Among the numbers in the set , the numbers are quadratic residues and the numbers are quadratic nonresidues. Table of quadratic residue Quadratic residues are all squares: 1, 2, 3, 4, 5, 8, 10, 14 Quadratic residues are all prime powers: 1, 2, 3, 4, 5, 6, 7, 8, 9, E, 10, 12, 14, 18, 28 Quadratic residues are all non-primes: 1, 2, 3, 4, 5, 8, 10, 13, 14, 20, 24, 34, 40, 48, 50, 60, 74, 94, X0, 120, 174, 180, 1E4, 220, 2X0, 374, 534, 5X0, 920, 10X0 Modular Arithmetic Quadratic Residue Table FIXME Cubic Residue Table FIXME Cyclotomic Fields Gauss and Jacobi Sums in Complex Plane Exhaustive Jacobi Plotter Elliptic Curves Adding points on an elliptic curve Plotting an elliptic curve over a finite field Cryptography The Diffie-Hellman Key Exchange Protocol Other Continued Fraction Plotter Quadratic Residues Definition. We let Definition 16. Quadratic residues are an important part of elementary number theory. Dec 10, 2013 · The quadratic non-residues of $m$ are the numbers $a$ relatively prime to $m$ which are not congruent to a square. If a is in row p, then m is a residue (mod p); if a is not in row p of the table, then m is a nonresidue (mod p). Explore math with our beautiful, free online graphing calculator. . Dirichlet's class number formula yields a number of results about the distribution of quadratic residues, for instance, the well-known fact that the interval $ [0,p/2]$ contains more quadratic residues than nonresidues. See the following two tables. 1 Squares and square roots An integer a is called a quadratic residue (or perfect square) modulo n if a ≡ b2 (mod n) for some integer b. 1: Quadratic Residues and Nonresidues 1. Given that p and q are odd primes. of quadratic residues (non-zero squares) in F. , where the generating set is the set of all quadratic From King's Landing, we give the definition of a quadratic residue modulo n as well as a few examples. But since is a quadratic residue, so is , and we see that are all quadratic residues of . http://www. Thus x - 0 if n is a quadratic residue with respect to p, and x - 1, if n is a quadratic non-residue with respect to p. Mar 28, 2017 · In this paper, we investigate the notion of quadratic exponential invertible graphs whose vertex set is reduced residue system mod n, and there will be an edge between x and y such that x 2 α ≡ The new class of graph called Arithmetic graph was introduced on the basis of Number theory, particularly the Theory of Congruence. When looking for quadratic residues you usually exclude $p=2$ (because everything here is a quadratic residue). Then S* is a symmetric subset of an additive abelian group of (ZP,⊕), of integers 0,1,2,, P-1. ) For example, this means that from Number of Quadratic Residues of Prime the number of quadratic residues of p p is always equal to p − 1 2 p − 1 2, which is the same as the number of quadratic non-residues. In this article, we introduce the generalized Paley graphs. The neighbours of a vertex v are the elements of S + v, so its valency is |S| = (q − 1)/2 Quadratic Residues: Definition, Example and Calculating how many can be found Iqbal Shahid 2. Quadratic residues modulo 11 are 1, 3, 4, 5, and 9. Complete documentation and usage examples. The Legendre symbol is a function that encodes the information about whether a number is a quadratic residue modulo an odd prime. May 21, 2021 · Note that a 2 Z is a quadratic residue mod p if and only if [a]p 6= 0p and [a]p is the square of some element in Fp, and b 2 Z is a non-residue mod p if and only if [b]p is not the square of any element in Fp. Quadratic Residues GraphsRezaei [7], who introduced the first simple graph G, defined it as a quadratic residue graph modulo n if its vertex set is reduced, a residue system modulo n such that two different vertices a and b are nearby, and a 2 ≡b 2 (mod n). The table below shows the status of residue/nonresidue among the integers in 1. Efficiently distinguishing a quadratic residue from a nonresidue modulo N = p q for primes p, q is an open problem. Download an example notebook or open in the cloud. Otherwise, q is called a quadratic nonresidue modulo n. ), 2008). 2 By the way, the terminology is explained by the fact (recall Section 4. Hence, the quadratic residues are 4; 3; 12; 9; 10; 1 (and there are 6 = 13 1 of them). The quadratic residues form a subgroup, but the non-quadratic resid es only for e group Q11 = (F11)2. Quadratic non-residues modulo 11 are 2, 6, 7, 8, and 10. randolphcollege. How do you prove a quadratic residue? A Comprehensive Course in Number Theory - August 2012A summary is not available for this content so a preview has been provided. It shows that there can only be . Given an odd prime p and an integer a with p ∤ a, we say a is a quadratic residue of p iff the congruence x2 ≡ a (mod p) has a solution. Thus the quadratic residues of p = 17 are 1; 2; 4; 8; 9; 13; 15 and 16. When looking for cubic residues you will have to make a similar conditionsince for some primes EVERYTHING is a cubic residue (e. Because the Legendre symbol is so compact and has such useful properties, it is an invaluable tool for doing computations and answering questions related to quadratic residues. Seldom is much material presented on residues of higher order. , if there exists an integer x such that: x 2 ≡ q (mod n) . 4) that the equivalence classes [a] are called residues, so one which is a perfect square is justly called quadratic b otherwise. g. For this reason, we also use the terms \quadratic (non-)residue" in relation to Fp : we say that an element a 2 Fp is a quadratic residue if This graph has vertex set {0, 1, 2, . It is used in the law of quadratic reciprocity to simplify notation. Chapter 11 is much more speci c and essentially attempts to address the question: Sep 14, 2014 · and is a quadratic non-residue modulo $p$ if and only if Introduction to Quadratic Residues Quadratic residues are a fundamental concept in number theory, with far-reaching implications in various fields, including cryptography, coding theory, and more. Then the congruence uble i Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. We shall illustrate the above theorem by taking p = 11 and p = 13 and verify the case of expression as sum of two quadratic residues for p = 11 and expression as sum of two quadratic non-residues for p = 13. Assume that a ≢ 0 (mod p), for p a prime. (i) When (a; m) = 1 and xn a (mod m) has a solution, then we say that a is an nth power residue modulo m. , p-1} with two vertices i and j joined by an edge if and only if i - j is a quadratic residue modulo p. Jesus said don't image worship. Lecture 7: Quadratic Reciprocity Instructor: Chao Qin Notes written by: Wenhao Tong and Yingshu Wang Definition (Quadratic Residue). By doing so, it helps to create the impression of a larger space and improve acoustics within small rooms. Hint: it 4 Use table 3. Unless otherwise speci ed, p is an odd prime. Definition. -1 0 1 81 Reveal the Dec 25, 2023 · Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that, if the context makes it clear, the adjective "quadratic" may be dropped. Bennett, Table of Quadratic Residues, Annals of Mathematics, Vol. Thus the set Up = (Fp )2 of quadratic residues is closed under multiplication. The quadratic residue of a modulo a prime p is 1 if a is This table is complete for odd primes less than 50. Q is the set of quadratic residues mod 23 and N is the set of non-residues. not divisible ; otherwise, For example, the squares modulo 5 are 12 = 1, Sep 25, 2025 · The focus lies on quadratic residues and sixth power residues. Jan 1, 2015 · The Cayley graph was first considered for finite groups by Arthur Cayley in 1878. The first three chapters present some basic facts and the history of quadratic residues and non-residues and discuss 1. Quadratic Residues s a solution modulo m. 27, No. 13. We want to know when there are 0 or 2 solutions. To do so, note that the (p 1)/2 numbers in (4. If the latter congruence is insoluble, then we say that a is a quadr at p is a prime number and (a; p) = 1. The Legendre symbol is defined by Note that is disallowed (since ) even though has a solution. Thus these are the quadratic residues. To check whether a number m is a quadratic residue mod one of these primes p, find a ≡ m (mod p) and 0 ≤ a < p. Wolfram Language function: Compute the quadratic residues of an integer. Considering the p 1 congruences and because each congruence has either no solution or two incongruent solutions, there must be exactly (p 1) / 2 quadratic residues of p among 1, 2,, p 1. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. The following table gives the ways in which each of the elements in the reduced By the way, the terminology is explained by the fact (recall Section 4. Residues Modulo A Prime. In particular, the product of two numbers that are both quadratic residues or quadratic non-residues modulo p is a residue, whereas the product of a residue with a non-residue is a non-residue. In this comprehensive guide, we will explore the definition, properties, and applications of quadratic residues, providing a thorough understanding of this fascinating topic. Nov 1, 2019 · Example of Quadratic Residues The set of quadratic residues modulo $11$ is: $\set {1, 3, 4, 5, 9}$ This sequence is A010375 in the On-Line Encyclopedia of Integer Sequences (N. Given a number a, s. For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1, …, n − 1. 37K subscribers Subscribe Jul 1, 2016 · The study of digraphs provides a proving ground where mathematicians’ ability to bind together multiple disciplines of mathematics becomes evident. This initiates to study the present article, here we introduce a cartesian product of quadratic residue graphs F = G m •H n , where m May 27, 2025 · Dive into the world of quadratic residues, examining the theoretical underpinnings and exploring real-world examples and applications 1. The quadratic reciprocity law is the statement that certain patterns found in the table are true in general. It remains to show that there are at least (p 1)/2 quadratic residues modulo p. 3. 3Quadratic residues Sage can calculate these for us, of course. In this section, we define Legendre symbol which is a notation associated to quadratic residues and prove related theorems. 4) that the equivalence classes [a] are called residues, so one which is a perfect square is justly called quadratic. Sloane (Ed. In this article we discuss basic and advanced properties of these symbols and show how the theory of quadratic residues is applied in Diophantine equations and other types of problems that can hardly be solved otherwise. The entries in the table are the values of the exponent x in the relation, ( ). For which of these moduli do the quadratic residues form a subgroup of Mn ? For which of these values of n is the mapping of M n given by x → x 2, two to one? 5 State the number of quadratic residues modulo 3, 5, 7, 11, 13 and 17 respectively. Although some authors also define this notion for composite moduli (as does Sage, see Sage note 16. In number theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that Otherwise, q is a quadratic nonresidue modulo n. Jun 14, 2025 · Delve into the world of quadratic residues and uncover their hidden patterns and significance in modern computing, from coding theory to cryptography. Ikenaga on Quadratic Residues. (In fact, it s a subgrou a quadrati a quad a group hom f quadratic residues. 3), we will go with the majority of them Lecture notes on quadratic residues, quadratic congruence, the Legendre symbol, Gauss's lemma, and the quadratic reciprocity law. A. In this paper, we will focus on quadratic residue Cayley graphs, i. 1. Jul 7, 2021 · In this section, we define Legendre symbol which is a notation associated to quadratic residues and prove related theorems. 99, and 100. Our result answers the question posed by Ken Ono. Jesus Christ CANNOT be white, it is a matter of biblical evidence. Note that the trivial case q=0 is generally excluded from lists of quadratic residues (e. (Definition) Quadratic Residue: Let p be an odd prime, a 6 0 mod p. , rp –1, each reduced modulo p. The definition is sometimes generalized to have value 0 if p|a, (1) If p is an odd prime, then the Jacobi symbol reduces to the Legendre symbol. The above table shows the least residues of for . Please use the Get access link above for information on how to access this content. $p=3,5,11,$). Since x2 = (−x)2, the second row of the table is symmetric left-to-right. Let p be an odd prime, and consider the congruence x2 = a (mod p) . 85K subscribers Subscribed 3. The question that this section will answer is whether p will be a quadratic residue of q or not. Then a is said to be a kth order residue of p (or modulo p) if the congruence xk ≡ a (mod p) Quadratic Residue And Quadratic Non Residue | Modulo Prime | Cryptography Quick Trixx 5. Since then, a lot of research has been done on various families of Cayley graphs, e. J. Simply put, if squaring some integer \ ( x \) gives a result congruent to \ ( a \) modulo \ ( p \), then \ ( a \) is a quadratic residue under this modulus The product of the residue 3 and the nonresidue 5 is the residue 3, whereas the product of the residue 4 and the nonresidue 2 is the nonresidue 2. The new class of graph called Arithmetic graph was introduced on the basis of Number theory, particularly the This paper discusses advanced topics in physics and mathematics, contributing to the scientific community through theoretical and experimental research. Sequences from Quadratic ResiduesThus, setting up Table I according to the instructions layed out in Part IVa with slight differences, the distribution (Dstr) columns have been eliminated with the ordered pair acting as a "distributed" ordered pair. His rst theorem according to the law of quadratic reciprocity was known as Euler's criterion. They are closely related to the Paley construction for constructing Hadamard matrices from quadratic residues. Quadratic Residues Tristan Shin 29 Sep 2018 In this handout, we investigate quadratic residues and their properties and applications. For those of you who have taken abstract algebra, observe that Property 4 tells Legendre symbol will help you to find the quadratic residue, non-quadratic residue values. Fact: If p p is an odd prime, the residue classes of 0 2, 1 2, 2 2,, (p 1 2) 2 02,12,22,…,(2p−1)2 are distinct and give a complete list of the quadratic residues modulo p p. Jan 4, 2024 · The Quadratic Residue Calculator is a valuable tool used to determine quadratic residues, a significant concept in number theory. The cayley graph of the group (ZP,⊕) is associated with the This book offers an account of the classical theory of quadratic residues and non-residues with the goal of using that theory as a lens through which to view the development of some of the fundamental methods employed in modern elementary, algebraic, and analytic number theory. By applying a specific formula, this calculator computes the quadratic residue of a given number 'a' under a prime modulus 'p'. nethttp://www. , Hardy and Wright 1979, p. Suppose we know whether q is a quadratic residue of p or not. By the way, the terminology is explained by the fact (recall Section 4. • Quadratic residues: 1,3,4, 5, 9, Quadratic non-residues: 2, 6, 7,8,10. If there is a solution of x 2 ≡ a (mod p) we say that a is a quadratic residue of p (or a QR). (In fact, it is a subgro a quadrati residue. 349-356 May 25, 1999 · Quadratic ResidueTherefore, is a quadratic residue of . If there is not a solution of x 2 ≡ a (mod p) we say that a is a quadratic nonresidue of p . It follows that ab must be a quadratic non-residue if b is a quadratic non-residue. t. Discover the intricacies of quadratic residues, from basic definitions to advanced applications in number theory and cryptography. Proof To list the quadratic residues of $11$ it is enough to work out the squares $1^2, 2^2, \ldots, 10^2$ modulo $11$. Jesus Christ is NOT white. Thus the set (Up)2 of quadratic residues is closed under multiplication. Beyond this, images of white 1=a is a quadratic residue if and only if a is a quadratic residue. Let p be an odd prime & let S be the set of quadratic residue modulo p. Mar 30, 2023 · This initiates to study the present article, here we introduce a cartesian product of quadratic residue graphs F = Gm Hn, where m and n are either prime or composite, and Gm and Hn are quadratic residue graphs, respectively. (For the quadratic residues coprime to n, see OEIS: A096103, and for nonzero quadratic residues, see OEIS: A046071. Sage note16. Conclusion: Given any fixed positive integer m, it is possible to determine the quadratic residue by simply listing the positive integers less than and prime to m, squaring them and reducing modm. Suppose a is a quadratic residue. The number of Squares in is related to the 13. 2 to find the quadratic residues modulo 5, 11, 12 and 13. Proof. The general quadratic equation 0 mod m: Assuming that m is odd or that b is even we can always complete the square (the usual way) and so we are reduced to solving an equation of the form x2 a mod m: In fact, we are usually only interested in solving the equation modulo y missing the p Dec 9, 2015 · For example, consider the modulus . michael-penn. If p=1 mod 4 then -1 is a quadratic residue modulo p so this is a bona fide undirected graph. However, other Mar 6, 2023 · In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications Thus, the number of different squares is --- these squares are the quadratic residues, and the other numbers in are quadratic nonresidues. According to the formula when p = 11 the number of residues is 5 and we depict the corresponding x and r values from the residues in Table I: a quadratic residue of p. Basic properties of quadratic residues special prope De nition 1. Using + to denote a quadratic residue and to denote a quadratic nonresidue, the following table says the frequency of the quadratic residue patterns among the triples (a; a + 1; a + 2) in F 101 is ne The quadratic residues are the squares: that is, 1, 7, and 13. A newer algorithm has been found which basically does the same using an ordered pair to generate the x values as ± pairs. Paley graphs are named after Raymond Paley. he congruence x2 a (mod m) is soluble. Jul 7, 2021 · To find all the quadratic residues of p among all the integers 1, 2,, p 1, we determine the least positive residue modulo p of 1 2, 2 2,, (p 1) 2. Fix a prime . If both of a; b, or neither, are quadratic residues, then ab is a quadratic residue; If one of a; b is a quadratic residue and the other is a quadratic non-residue then ab is a quadratic non-residue. , unitary Cayley graphs, Paley graphs, Dihedral Cayley graphs, quadratic residue Cayley graphs, etc. ) 112 2 (mod 17), 72 102 15 (mod 17) and 82 92 13 (mod 17). Exercise 5. As b runs over the non-zero residues mod p, so does ab. As an easy example, , since . It is thus the Cayley graph [15] for the additive group of F, with S as the connection set. Standard theorems on quadratic residues form an integral part of any introductory course on the theory of numbers. For example, from the table for modulus 15 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (residues in bold). Use Property 4 to show that if a, b ∈ Up are units that are both non-quadratic residues, then ab i a quad Remark. Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p. Consider the set S* ={ s, p-s/ ∈ }. We know that ab is a quadratic residue if b is a quadratic residue, and we know that just half the non-zero residues are quadratic residues. The general quadratic equation 0 mod m: Assuming that m is odd or that b is even we can always complete the square (the usual way) and so we are reduced to solving an equation of the form x2 a mod m: In fact, we are usually only interested in solving the equation modulo y missing the p we can replace congruence with equality and odulo p sin a quadratic residue. The investigation extends to the degree of G, Γu, and Γq, illuminating the properties of these subgraphs in the context of quadratic congruences. Nov 10, 2022 · How do you determine if a number is a quadratic residue? We only need to solve, when a number (b) has a square root modulo p, to solve quadratic equations modulo p. We also Math 3527 (Number Theory 1) Lecture #30 Quadratic Residues and Legendre Symbols: Quadratic Congruences Quadratic Residues and Nonresidues Legendre Symbols This material represents x5. 4) that the equivalence classes [a] are called residues, so one which is a perfect square is justly called quadratic Quadratic Residue An integer \ ( a \) is a quadratic residue modulo \ ( p \) if there exists an integer \ ( x \) such that: \ [x^2 \equiv a \pmod {p}\]If no such \ ( x \) exists, then \ ( a \) is not a quadratic residue modulo \ ( p \). The following table (sequence A096008 in the OEIS) lists the quadratic residues mod 1 to 75 (a red number means it is not coprime to n). Quadratic residues are used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers. Otherwise, a is called a quadratic non-residue modulo m. The quadratic residues modulo 7 are 1, 2 ≡ 32 and 4, whereas 2, 5 and 6 are quadratic nonresidues modulo 7. The number a is called a quadratic residue modulo m if there exists x ∈ Z such that a ≡ x2 mod m. It shares many of the properties of the Legendre symbol, and can be used to state and prove an extended version of the law of quadratic reciprocity. This is exploited by several cryptosystems, such as Goldwassser-Micali encryption, or Cocks identity-based encryption. Let p be a prime and let the integer a satisfy 1 ≤ a < p. Up to congruence modulo $m$, we can confine attention to the numbers in the interval $ [0,m-1]$. Apr 8, 2023 · QRD® Diffuser FAQ’s What is a QRD® diffuser? A QRD® diffuser, or Quadratic Residue Diffuser, is an acoustical device designed to break down a strong reflection from a room boundary surface into a series of smaller reflections. Since 2 is a primitive root, the quadratic residues are those 2n with n even. ABSTRACT In this paper the concept of perfect codes to Quadratic Residue Cayley graph modulo an odd prime p, is introduced. , the congruence (1) has a solution, then q is said to be a quadratic residue (mod p). This notation has the advantage of being capable Sep 29, 2025 · If there is an integer 0<x<p such that x^2=q (mod p), (1) i. [1] 1=a is a quadratic residue if and only if a is a quadratic residue. e. Introduction: There's a reasonable reason to jump from Chapter 9 to Chaper 11 which is that both sections concern themselves with solutions to equations. An integer by is a quadratic residue modulo if is a square modulo is a quadratic nonresidue. Here is another way to construct these using the kronecker command (which is also called the “Legendre symbol”): 3 days ago · The Legendre symbol is a number theoretic function (a/p) which is defined to be equal to +/-1 depending on whether a is a quadratic residue modulo p. The quadratic residues mod 15 are 1 and 4, as those are the squares that are relatively prime to 15. Here we explain the definition of a quadratic residue mod p, go through an example of finding quadratic residues, and note one Math 406 Section 11. The third applet takes a prime p as input, and provides the following output: a list of the quadratic residues modulo p; the smallest primitive root r modulo p; and, a table of the values of r 1, r 2, r 3, . The aforementioned work suggests and evaluates the regular graphs that are produced from graph F and its adjacency matrix. 2 from the course notes. = (_-)1, where the left hand member is the Legendre symbol. Feb 10, 2024 · In this work, we investigate essential definitions, defining G as a simple graph with vertices in Zn and subgraphs Γu and Γq as unit residue and quadratic residue graphs modulo n, respectively. We start by setting p = 91 into the product of two odd primes (13×7) and generate Table I where: Aug 10, 2023 · We compute Hirano's formula for the mod 2 arithmetic Dijkgraaf-Witten invariant Zk for the ring of integers of the quadratic field k = Q(p1 ⋯pr− −−−−−√), where pi 's are distinct prime numbers with pi ≡ 1 (mod 4), and give a simple formula for Zk in terms of the graph obtained from quadratic residues among p1, ⋯,pr. Understanding its formula and functionality can aid in various mathematical calculations and cryptographic applications. The Legendre symbol is the expression denoted by a , which is equal to 1 if Remark16. This method is not guaranteed to produce all quadratic residues, but can often produce several small ones in the case of large , enabling to be factored. , gcd (a, p) = 1; a is called a quadratic residue if x2 = a mod p has a solution otherwise it is called a quadratic non-residue. 4 (Jun. The Legendre Symbol Recall. v8ox qoa eiid l5rsmo fgfz xpp n3 d2x r7r 3qp
