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3 Notation, Terminology, and Some Useful Elementary Number Theory

3 Notation, Terminology, and Some Useful Elementary Number Theory

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1.3 Notation, Terminology, and Some Useful Elementary Number. . .



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then [m, n] is the set of all integers at least m and no more than n, listed in

increasing order, [m, ∞) is the set of all integers exceeding m − 1, also listed

in increasing order, and gcd(m, n) is the greatest common divisor of m and n.

If n ∈ [2, ∞) then U (n) will denote the set {m ∈ [1, n − 1] : gcd(m, n) = 1}.

If z is an integer then π(z ) will denote the set of all prime factors of z . If A is

a set then |A| will denote the cardinality of A, 2A is the set of all subsets of A,

and ∅ denotes the empty set. Finally, we will refer to a quadratic residue or

quadratic non-residue as simply a residue or non-residue; all other residues of

a modulus m ∈ [2, ∞) will always be called ordinary residues. In particular,

the minimal non-negative ordinary residues modulo m the elements of the

set [0, m − 1].

We also recall some facts from elementary number theory that will be

useful in what follows. For more information about them consult any standard

text on elementary number theory, e.g., Ireland and Rosen [30] or Rosen [48].

If m is a positive integer and a ∈ Z, recall that an inverse of a modulo m

is an integer α such that aα ≡ 1 mod m.

Proposition 1.2 If m is a positive integer and a ∈ Z then a has an inverse

modulo m if and only if gcd(a, m) = 1. Moreover, the inverse is relatively

prime to m and is unique modulo m.

Theorem 1.3 (Chinese Remainder Theorem) If m1 , . . . , mr are pairwise relatively prime positive integers and (a1 , . . . , ar ) is an r -tuple of integers

then the system of congruences

x ≡ ai mod mi , i = 1, . . . , r ,

has a simultaneous solution that is unique modulo



r

i=1



mi . Moreover, if



mi ,



Mk =

i=k



and if yk is the inverse of Mk mod mk (which exits because gcd(mk , Mk ) = 1)

then the solution is given by

r



x ≡



r



mi .



ak Mk yk mod

i=1



k =1



Recall that a linear Diophantine equation is an equation of the form

ax + by = c,

where a, b, and c are given integers and x and y are integer-valued unknowns.

Proposition 1.4 Let a, b, and c be integers and let gcd(a, b) = d . The

Diophantine equation ax + by = c has a solution if and only if d divides c. If



8



1



Introduction: Solving the General Quadratic Congruence Modulo a. . .



d divides c then there are infinitely many solutions (x , y), and if (x0 , y0 ) is a

particular solution then all solutions are given by

x = x0 + (b/d )n, y = y0 − (a/d )n, n ∈ Z.

Given the Diophantine equation ax + by = c with c divisible by d =

gcd(a, b), the Euclidean algorithm can be used to easily find a particular

solution (x0 , y0 ). Simply let k = c/d and use the Euclidean algorithm to

find integers m and n such that d = am + bn; then (x0 , y0 ) = (km, kn)

is a particular solution, and all solutions can then be found by using

Proposition 1.4. The simple first-degree congruence ax ≡ b mod m can thus

be easily solved upon the observation that this congruence has a solution x

if and only if the Diophantine equation ax + my = b has the solution (x , y)

for some y ∈ Z.



Chapter 2



Basic Facts



In this chapter, we lay the foundations for all of the work that will be done

in subsequent chapters. Section 2.1 defines the Legendre symbol and verifies

its basic properties, proves Euler’s criterion, and deduces some corollaries

which will be very useful in many situations in which we will find ourselves.

Motivated by the solutions of a quadratic congruence modulo a prime which

we discussed in Chap. 1, we formulate what we will call the Basic Problem

and the Fundamental Problem for Primes in Sect. 2.2. In Sect. 2.3, we state

and prove Gauss’ Lemma for residues and non-residues and use it to solve

the Fundamental Problem for the prime 2.



2.1



The Legendre Symbol, Euler’s Criterion,

and Other Important Things



In this section, we establish some fundamental facts about residues and nonresidues that will be used repeatedly throughout the rest of these notes.

Proposition 2.1 In every complete system of ordinary residues modulo p,

there are exactly (p − 1)/2 quadratic residues.

Proof It suffices to prove that in [1, p −1] there are exactly (p −1)/2 quadratic

2

residues. Note first that 12 , 22 , . . . , ( p−1

2 ) are all incongruent mod p (if 1 ≤

2

2

i, j < p/2 and i ≡ j mod p then i ≡ j hence i = j or i ≡ −j , i.e., i + j ≡ 0.

But 2 ≤ i + j < p, and so i + j ≡ 0 is impossible).

Let S denote the set of minimal non-negative ordinary residues mod p of

2

12 , 22 , . . . , ( p−1

2 ) . The elements of S are quadratic residues of p and |S| =

(p − 1)/2. Suppose that n ∈ [1, p − 1] is a quadratic residue of p. Then

there exists r ∈ [1, p − 1] such that r 2 ≡ n. Then (p − r )2 ≡ r 2 ≡ n and



© Springer International Publishing Switzerland 2016

S. Wright, Quadratic Residues and Non-Residues, Lecture Notes

in Mathematics 2171, DOI 10.1007/978-3-319-45955-4 2



9



10



2



Basic Facts



{r , p − r } ∩ [1, (p − 1)/2] = ∅. Hence n ∈ S, whence S = the set of quadratic

residues of p inside [1, p − 1].

QED

Remark The proof of Proposition 2.1 provides a way to easily find, at least

in principle, the residues of any prime p. Simply calculate the integers

2

12 , 22 , . . . , ( p−1

2 ) and then reduce mod p. The integers that result from this

computation are the residues of p inside [1, p − 1]. This procedure also finds

the modular square roots x of a residue r of p, i.e., the solutions to the

congruence x 2 ≡ r mod p. For example, in just a few minutes on a hand-held

calculator, one finds that the residues of 17 are 1, 2, 4, 8, 9, 13, 15, and 16,

with corresponding modular square roots ±1, ±6, ±2, ±5, ±3, ±8, ±7,

and ±4, and the residues of 37 are 1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25,

26, 27, 28, 30, 33, 34, and 36, with corresponding modular square roots

±1, ±15, ±2, ±9, ±3, ±11, ±14, ±7, ±4, ±13, ±5, ±10, ±8, ±18, ±17,

±12, ±16, and ±6. Of course, for large p, this method quickly becomes

impractical for the calculation of residues and modular square roots, but

see Sect. 4.9 of Chap. 4 for a practical and efficient way to perform these

calculations for large values of p.

N.B. In the next proposition, all residues and non-residues are taken with

respect to a fixed prime p.

Proposition 2.2

(i) The product of two residues is a residue.

(ii) The product of a residue and a non-residue is a non-residue.

(iii) The product of two non-residues is a residue.

Proof

(i) If α, α are residues then x 2 ≡ α, y 2 ≡ α imply that (xy)2 ≡ αα mod p.

(ii) Let α be a fixed residue. The integers 0, α, . . . , (p − 1)α are incongruent

mod p, hence are a complete system of ordinary residues mod p.

If R denotes the set of all residues in [1, p − 1] then by Proposition 2.2(i), {αr : r ∈ R} is a set of residues of cardinality (p − 1)/2,

hence Proposition 2.1 implies that there are no other residues among

α, 2α, . . . , (p − 1)α, i.e., if β ∈ [1, p − 1] \ R then αβ is a non-residue.

Statement (ii) is an immediate consequence of this.

(iii) Suppose that β is a non-residue. Then 0, β, 2β, . . . , (p −1)β is a complete

system of ordinary residues mod p, and by Proposition 2.2(ii) and

Proposition 2.1, {βr : r ∈ R} is a set of non-residues and there are

no other non-residues among β, 2β, . . . , (p − 1)β. Hence β ∈ [1, p −

1] \ R implies that ββ is a residue. Statement (iii) is an immediate

consequence of this.

QED

The following definition introduces the most important piece of mathematical technology that we will use to study residues and non-residues.



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