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
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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.