Number Theory Handbook - archive.org

pe Or en Mathematics and Computing: A Third Level Course M381NTHB eee M381 Number Theory and Mathematical Logic

M381

Number Theory Handbook

NOTE: This Handbook may be taken into the M381 examination. However, no annotation is permitted other than the correction of errors and the highlighting or underlining of passages.

CONTENTS

Notation 3 Glossary 4 Results 9

Unit 1 Foundations 9 Unit 2 Prime Numbers 10 Unit 3 Congruence 12 Unit 4 Fermat's and Wilson's Theorems 14 Unit 5 Multiplicative Functions 15 Unit 6 Quadratic Reciprocity 17 Unit 7 Continued Fractions 19 Unit 8 Diophantine Equations 21

Appendix: Table of Primes 24 Cumulative Index of Units 26

Copyright © 1996 The Open University SUP 30666 6 1.4

NOTATION

Some of the notation used in the course is listed below. A reference in the right-hand column refers to the unit (bold) and page number where the term first appears. This referencing system is followed throughout the Handbook.

Fy the nth Fibonacci number 2 28 Fa the nth Fermat number 5 16 Mp the Mersenne number 2? — 1 5 12 Q the set of rational numbers 1 13 R the set of real numbers 1 13 Rn the repunit consisting of n 1’s 5 22 T the nth triangular number 1 7 2 the set of integers 1 12 z^ the set of positive integers 1 12 E the set of equivalence classes modulo n, i.e. the 3 10

set (0,1,2,...,n — 1) with addition and multiplication modulo n

a|b a divides b 1 22 alb a does not divide b 4 22 a =b (mod n) a is congruent modulo n to b 3 6 à £ b (mod n) à is incongruent modulo n to b 3 6 (a/p) the Legendre symbol; see Glossary 6 12 In] the absolute value of n 1 13 frac(z) the fractional part of x T 9f gcd(a, b) the greatest common divisor of a and b 1 23 int(z) the integer part of x 6 18 Icm(a, b) the least common multiple of a and b 1 28 (2) the logarithmic integral 2 25

min{a, b} = { b s the minimum of a and b 2 12

max{a, b} = fo Scone the maximum of a and b 2 13 т(х) the number of primes less than or equal to x 2 25 [20] Euler’s ¢-function; see end of Glossary 5 18 a(n) the o function; see end of Glossary 5 6 T(n) the r function; see end of Glossary 2 13 [a1 a2... an] the finite continued fraction with partial T g

quotients a1,à2,...,Q lai, a2,a3,...] the infinite continued fraction with partial 7 20

quotients a1, a2, a3, ... [a1,@2,...,@n, (b, 05, . .—., bm) a periodic continued fraction with a cycle of 7 24

length m

GLOSSARY

abundant numbers An integer n is abundant if o(n) > 2n.

amicable pair Integers m and n form an amicable pair if o(m) = o(n) =m+n.

arithmetic progression/arithmetic series An arithmetic series is a series of

the form

a+ (a+d) + (a+ 2d) + (a+ 3d)+---

with a common difference d from one term to the next. The terms are said to be in arithmetic progression.

complete set of residues A complete set of residues modulo n consists of one

element from each of the n residue classes.

composite An integer n > 1 which is not prime is said to be composite.

congruence modulon An integer a is said to be congruent modulo n to integer

b if the difference a — b is divisible by n. We write a = b (mod n). If the difference a — b is not divisible by n then a is incongruent modulo n to b.

convergent The convergents of the FCF [a1, a2, 4@3,...,@n] are the values of the

finite continued fractions

[01], [01,9], [а1,а2,аз], ..., [a1,@2,43,.--, Gn].

These are given by [а1,а2, аз,..., ак] = Рв. where the numerators pi, po, . .., Dn

and denominators q1, q2, ..., q, are given by

рі = а, р2=аа2 +1, рь = аһрь-1 + рь-2, for3<k<n;

Ф =1, Фф= а, к = акд-1 + 9-2, {013 <Е < п.

The first n convergents of the ICF [а1,а2,аз,...,аһ,...] аге the convergents of [a1, a2, a3, . .., a4].

cycle (of a decimal fraction) In a decimal fraction, a block of digits which repeats indefinitely is called a cycle of the fraction. The number of digits in the cycle is the cycle length.

cycle (of a periodic continued fraction) The block of partial quotients which repeats indefinitely in a periodic ICF is called the cycle of the ICF. The number of partial quotients in the cycle is the cycle length.

deficient number An integer n is deficient if c(n) « 2n.

degree of a polynomial congruence If

Р(х) = сих" + спа” + +++ +14 + C0

is an integral polynomial then P(x) has degree k modulo n if, after removing all terms, where c; = 0 (mod n), the leading term remaining is c,z^.

If every coefficient c; = 0 (mod n), then the polynomial is not assigned a degree modulo n.

The degree of the polynomial congruence P(x) = 0 (mod n) is the degree of P(x) modulo n.

divisor The integer a is divisible by the positive integer b if there exists an integer q such that a = bq. In this case b is said to be a divisor of a, and that a is a multiple of b.

FCF FCF is an abbreviation for simple finite continued fraction.

Fermat numbers The Fermat numbers are those integers of the form

2?" +1, forany integer n > 0.

5 34

5 34

1 7

5 34

4 22

7 8

5 16

Fibonacci numbers The sequence of Fibonacci numbers

1,1,2,3,5, 8, 13, 21,34... =

is defined by

A=1, Б=1, Е, = Е,_1+Е,2, огъур 2.

finite continued fraction A finite continued fraction is a finite expression (which can be evaluated)

1 а + ї ,

а +

1 аһ-1 + —

an

where a; is any real number and the others, a2, a3,-..,@n, are positive real numbers. The finite continued fraction is written symbolically as [a1, 2, a3, . .., a4].

fractional part The fractional part of a real number z, denoted by frac(z), is defined by

x = int(«) + frac(zx)

where int(z) is the integer part of x.

fundamental solution The fundamental solution of Pell’s equation,

22 — пу? =1,

is the unique solution in which z and y take their smallest. positive values.

geometric series A geometric series

à 4 ar t ar? 4 ar? 4...

is one in which the ratio of successive terms is constant.

greatest common divisor (gcd) The greatest common divisor of two integers a and b, not both of which are zero, is the positive integer n satisfying

(a) n divides a, and n divides b;

(b) if d divides both a and b then d € n.

hexagonal numbers The hexagonal numbers form the sequence

1l, 1+5, 1+5+9, 1+5+9+13,

arising from arrangements of dots forming hexagons.

ICF ICF is an abbreviation for simple infinite continued fraction.

infinite continued fraction Extending the notation for finite continued fractions, an infinite continued fraction is an unending expression [a1,a2, a3, . ..] where a; is any real number and the others, a5,a3,..., are positive real numbers. The value of the infinite continued fraction is the limit of the sequence of values

[ai], [0,95], [@1, 42,03], .:., [a1,@2,a3,..., an],

integer combination If a and b are integers, then any integer of the form

та +пь, т,пє 2.

is an integer combination of a and b.

integer part The integer part of a real number z, denoted int(z), is the largest integer which does not exceed z. That is,

int(x) < x < int(x) +1.

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integral polynomial An integral polynomial is a polynomial

Р(х) = сый" es amu porco,

in which the coefficients co, c1, c2, ..., c4, and the variable z are integers.

leading term The leading term of a polynomial is the non-zero term involving . the highest power of z.

least absolute residue The least absolute residue in a residue class modulo n

is the unique integer z in the class which lies in the range -— < = < B The n

such numbers x form the set of least absolute residues modulo n.

least common multiple (Icm) The least common multiple of the non-zero integers a and b is the positive integer n satisfying

(a) a divides n, and b divides n;

(b) if a divides m and b divides m then n < |m].

least positive residue The least positive residue in a residue class modulo n is the smallest non-negative integer belonging to the class. The set of least positive residues modulo n is the set (0,1,2,...,n — 1).

Legendre symbol If gcd(a, p) = 1, where p is an odd prime, then the Legendre symbol (a/p) is defined by

tate) = 1, if a is a quadratic residue of p, ALD —1, ifa is a quadratic non-residue of p.

linear congruence A linear congruence is a congruence of the form

ax = b (mod n),

where a and b are integers and n is a positive integer.

linear Diophantine equation The linear Diophantine equation is the equation

ал + бу = с, where a,b, and c €Z,

which is to be solved in integers.

Mersenne prime The numbers of the form M, = 2? — 1, where p is prime, are called Mersenne numbers. If M; happens to be prime, it is a Mersenne prime.

method of infinite descent The method of infinite descent is a method of proof which uses the fact that a strictly decreasing sequence of positive integers must terminate. Typically it is used to prove that a Diophantine equation can have no solutions by showing that the existence of any positive solution leads to a second, strictly smaller, positive solution; an impossible situation.

multiple The integer a is said to be a multiple of integer b if there exists an integer q such that a — bq; that is, if b is a divisor of a.

multiplicative function A number-theoretic function f is said to be multiplicative if, for any number with prime decomposition pP pk ...pkr we have

S(t"? PE) = Кр) (р)... non-negative A real number g is non-negative if it satisfies x > 0.

number-theoretic function A function f :Z* —+ Z, with the set of positive integers as domain and which takes integer values, is said to be a number-theoretic function.

order of an integer modulo n If gcd(a,n) — 1, then the order of a modulo n is defined to be the least positive integer c such that a^ z 1 (mod n).

parity The parity of an integer depends on whether it is divisible by 2. It is either an odd integer (odd parity) or an even integer (even parity).

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partial quotient Ina continued fraction [a;, a2, a3, .. .], whether finite or infinite, the numbers a), a@2,@3,..., are called the partial quotients.

pentagonal numbers The pentagonal numbers form the sequence

1, 1+4, 1+4+7, 1+4+7+10, asit

which arise from arrangements of dots in pentagons.

perfect number A perfect number is one which is equal to the sum of its divisors, excluding itself. To be precise, n is perfect if, and only if, a(n) = 2n.

periodic continued fraction An infinite continued fraction in which, from some point onwards, a block of partial quotients repeats indefinitely, is said to be periodic. It is denoted by

[@1,а2,...,@т, (61,62,...,0т)],

where the repeating block, (b1,b2,...,m), is shown in angled brackets. The block is called the cycle of the continued fraction.

polynomial congruence A polynomial congruence is a congruence of the form P(x) € 0 (mod n), where P(x) is an integral polynomial.

prime number An integer n > 2 is said to be prime if it has no positive divisor other than itself and 1.

prime decomposition The prime decomposition of an integer n > 2 is its unique representation as

kı ko kr n = pi Po -Prs

where, for 1 < i € r, the p; are distinct primes and k; is a positive integer. The primes are normally listed in ascending order.

prime triplet If, of four consecutive odd integers, three are prime then they are said to form a prime triplet.

prime quartet If, of five consecutive odd integers, four are prime then they are said to form a prime quartet.

primitive Pythagorean triple A Pythagorean triple is a triple (2, y, z) of positive integers which give a solution of the Diophantine equation 2? + y? = 22. If, in addition, ged(z, y) — 1 then the triple is said to be primitive. In this case x is conventionally chosen to be the unique even member of the triple.

primitive root The integer a is a primitive root of prime p if the successive powers of a run through all the p — 1 non-zero residues modulo p.

The integer a is a primitive root of n if the successive powers of a run through a reduced set of residues for n. In this case there will be ¢(n) distinct powers of a modulo n, where ¢ is Euler’s ¢-function.

pseudoprime A pseudoprime is a composite integer n with the property that n divides 2" — 2.

purely periodic continued fraction A periodic continued fraction,

[(21,a5,... ,)],

in which the cycle begins at the first partial quotient, is said to be purely periodic.

Pythagorean equation The Pythagorean equation is the Diophantine equation а? фу? = 22, requiring positive integer solutions.

quadratic character If pis an odd prime and gcd(a, p) = 1, the quadratic character of a modulo p refers to whether a is a quadratic residue or a quadratic non- residue of p.

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quadratic congruence A quadratic congruence is a polynomial congruence of degree 2; that is, it is a congruence of the form az? + br +c =0 (mod n) where a, b and c are integers and a # 0 (mod n).

quadratic residue (non-residue) If pis an odd prime and gcd(a, p) = 1, then , @is either a quadratic residue or a quadratic non-residue of p. If the congruence

z? = a (mod p)

has a solution, then a is a quadratic residue of p; otherwise, a is a quadratic non- residue.

rational number A rational number, expressed in its lowest terms, is a number of the form ; where a and b are integers, b > 1 and gcd(a,b) = 1.

reduced set of residues — A reduced set of residues modulo n is a set of integers (21,2, . .., ax) each of which is relatively prime to n, no two of which are congruent modulo n, and such that every integer relatively prime to n is congruent modulo n to one of them. The number, k, of integers in a reduced set of residues is ф(п), where ¢ is Euler’s ¢-function.

The reduced set of residues in which each integer is positive and less than or equal to n is the reduced set of least positive residues.

relatively prime Two integers a and b, not both 0, are said to be relatively prime whenever gcd(a, b) — 1.

repunit A repunit is a positive integer each of whose (decimal) digits is 1.

residue class The residue class modulo n of an integer a consists of all integers which are congruent modulo n to a.

simple continued fraction A continued fraction [a1 a2, a5, . . .], finite or infinite, is said to be simple if each partial quotient is an integer.

solutions of a polynomial congruence Let S$ = {b1,b2,...,bn} be a complete set of residues modulo n. The number of solutions of the polynomial congruence P(x) & 0 (mod n) is the number of integers b € S for which P(b) 2 0 (mod n). Any solution of the congruence is congruent modulo n to one of these.

square-free An integer n > 1 is said to be square-free if it is not divisible by the square of any prime. The largest exponent in the prime decomposition of a square-free integer n will be 1.

triangular numbers The triangular number T,, for n > 1, is the sum of the first п, natural numbers:

_ n(n +1) Tp =1t+24+3+4---40 5

twin primes If two consecutive odd positive integers are both prime they are said to be twin primes.

$-function For any integer n > 1, ¢(n) is defined to be the number of positive integers not exceeding n which are relatively prime to n. (4(1) = 1.)

c function For any integer n > 1, o(n) is defined to be the sum of the distinct divisors of n, including 1 and n itself. (o(1) = 1.)

T function For any integer > 1, 7(n) is defined to be the number of distinct divisors of n, including 1 and n itself. (r(1) — 1.)

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RESULTS

Unit 1 Foundations

The Well-Ordering Principle for Z*

Every non-empty subset of Z* has a least member. That is, if S is a non-empty subset of Z* then there exists b € S such that b € n for all n € S.

Theorem 3.1. Principle of Finite Induction

If S is a set of positive integers with the following two properties:

(a) 1 is a member of S;

(b) if k € S, then the next integer k -- 1€ S;

then S — Z*.

Principle of Mathematical Induction

Let P(n) be a proposition depending on a positive integer n. If:

(a) P(1) is true;

(b) for any integer k > 1, if P(k) is true then P(k +1) is true;

then P(n) is true for all n € Z*.

Principle of Mathematical Induction (generalized)

Let P(n) be a proposition depending on an integer n. If:

(a) P(ng) is true;

(b) for any integer k > no, if P(k) is true then P(k +1) is true;

then P(n) is true for all integers n > no.

Second Principle of Mathematical Induction

Let P(n) be a proposition depending on an integer n. If:

(а) P(no) is true;

(b^) for any integer k > no, if P(no), P(no + 1), ..., P(k) are all true, then P(k 4- 1) is true;

then P(n) is true for all integers n > no.

Theorem 4.1 The Division Algorithm

For any two integers a and b, where b » 0, there exist unique integers q and r such that

& —bq-r, whereO x r « b.

Theorem 4.2. Properties of division

Let a and b be positive integers and c and d be any integers.

(a) Ifa |c then a | (c + na) for any integer n.

(b) If c #0 and a | c then a < |].

(c) If a| b and b | a then a — b.

(d) If a | b and b | c then a | c.

(e) If a | c and a | d then a | (mc 4- nd) for any integers m and n.

Theorem 4.3

Given any integers a and b, not both zero, there exist integers m and n such that

gcd(a, b) = ma + nb.

In other words, gcd(a, b) is an integer combination of a and b.

Theorem 4.4

Integers a and b are relatively prime if, and only if, there exist integers m and n such that 1 = ma + nb.

Theorem 4.5

b For any integers a and b, not both zero, if gcd(a, b) = d then 5 and — are integers d

a b a b 75,35] 1. Thati i ——— —— such that gcd ( d а) 1 at is, the integers soia and Зе В) аге

relatively prime.

Theorem 4.6

If a | c and b | c, with gcd(a, b) — 1, then ab | c.

Theorem 4.7 Euclid's Lemma

If a | bc, with gcd(a, b) — 1, then a | c.

Theorem 4.8

For any pair of positive integers a and b,

lem(a, b) x gcd(a, b) = ab.

Result

If a=qb+r, then gcd(a, b) = gcd(b,r).

Theorem 5.1. Solution of linear Diophantine equations

The linear Diophantine equation az + by = c has solutions if, and only if, gcd(a, b) divides c.

If this condition holds with gcd(a, b) > 1 then division by gcd(a, b) simplifies the equation to a'z + b'y = c', where gcd(a',b') = 1.

If zo, yo is one solution of this equation then the general solution is

t=a tk, y=y-ak, kez.

Unit 2. Prime Numbers

Theorem 1.1

Each integer n > 2 is divisible by some prime number.

Theorem 1.2

If the integer n > 2 is composite, then it is divisible by some prime р< уп.

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Theorem 1.3 Euclid’s Lemma for prime divisors

If p is prime and p divides the product @a2...@, then p divides a;, for some integer i, where 1 c i € n.

Corollary to Euclid's Lemma

If p, p1, p», ..., pa are primes such that

р | рірә...рп, then p = pj for some i, where 1 « i <n.

Theorem 1.4. The Fundamental Theorem of Arithmetic

Any integer n > 2 can be written uniquely in the form

R= py" py? pr, where pi, i — 1,...,r, are primes with Di X po €: € p, and each kj, i = 1,...,r, is a positive integer.

Theorem 2.1. Divisors of an integer

Let n be an integer greater than 1, with prime decomposition kik kp n= pipi pi.

Then the set of all divisors of n is the set of integers

{pipt ... phe :0 h € ki fori ,2,...,r].

Prime decomposition of gcd(m,m)

gcd(m, n) de prints hi) min{ka,la} Ln prins)

Prime decomposition of lcm(m, n) lem(m,n) — бы рака, Т орик),

Theorem 2.2. The formula for T (n)

For any integer n > 2, with prime decomposition

n=pi pk pi, we have

T(n) 2 (ki 4 1) (ka -- 1) - - (kr + 1).

Theorem 3.1 Euclid's Theorem

There are infinitely many primes.

Bertrand's Conjecture

For each integer n 7 2, there exists a prime p such that n < p « 2n.

Theorem 3.2

Any integer of the form 4k + 3 is divisible by a prime of this same form.

Theorem 3.3 Primes of the form 4k + 3

There are infinitely many primes of the form 4k + 3, where k is an integer.

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Theorem 3.4 Dirichlet’s Theorem

If a and b are positive integers with gcd(a, b) = 1, then there are infinitely many primes of the form ak + b, where k is a non-negative integer.

* Goldbach Conjecture

Every even integer from 6 onwards can be written as a sum of two odd primes.

The Prime Number Theorem

zls) _ E Sr —

Theorem 5.1. The relative primeness of consecutive Fibonacci numbers

For any positive integer n,

gcd(Frii, Fr) — 1.

A useful relationship between Fibonacci numbers

For all integers m 7 2 and n > 1,

Fintn = Fn-1 Fa + Fm Fati-

Theorem 5.2. Divisibility property of the Fibonacci numbers

For all integers n > 2, Fm is divisible by F, if, and only if, m is divisible by n.

Unit 3 Congruence

Theorem 1.1 Properties of congruence

Let n be a fixed positive integer and let a, b, c and d be any integers. Then the following properties hold.

(a) a =a (mod n).

(b) If a = b (mod n) then b = a (mod n).

(c) If a = b (mod n) and b = c (mod n) then a = c (mod n).

(d) If a =b (mod n) and c = d (mod n) then a +c = b+ d (mod n).

(e) If a = b (mod n) and c = d (mod n) then ac = bd (mod n).

(£) If a =b (mod n) then a” =b" (mod n), for any integer r > 1.

Theorem 1.2 Congruent integers have the same remainders

a = b (mod n) if, and only if, the integers a and b have the same remainder when divided by n.

Theorem 1.3

If a = b (mod m) and a = b (mod n), then

а = b (mod lem(m,n)).

Corollary to Theorem 1.3

Ка = (той т) апда = 0 (той п), where gcd(m,n) — 1, then a = b (mod mn).

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Theorem 1.4 Cancellation rule

If ca = cb (mod n) then a=b (moa a) where d — gcd(c, n).

Corollary to the cancellation rule

If ca = cb (mod n), where gcd(c,n) — 1, then a = b (mod n).

Theorem 1.5 Euclid’s Lemma for prime divisors

If ab = 0 (mod p) then a = 0 (mod p) or b = 0 (mod p).

Theorem 2.1. Divisibility by 9 and 11

Let the integer N be written in decimal notation as N — a4 a4. 4 .. .0501a9 SO that the digits satisfy 0 < a; < 9 with am 40. Then:

(а) М = ад +а + a2 +03 +--+ +am (mod 9);

(b) N & ao — a1 + a2 — a3 +++» + (—1)™am (mod 11).

Theorem 3.1

Let P(x) be an integral polynomial. If a = b (mod n) then P(a) = P(b) (mod n). In particular, a is a solution of the polynomial congruence P(«) = 0 (mod n) if, and only if, b is a solution.

Theorem 3.2 Solution of linear congruences

Consider the linear congruence ax = b (mod n).

(a) The congruence has solutions if, and only if, gcd(a, n) divides b.

(b) If ged(a, n) — 1, the congruence has a unique solution.

(c) If gcd(a, n) — d and d divides b, then the congruence has d solutions which are given by the unique solution modulo 3 of the congruence

E (moa 5)

Strategy: to solve the linear congruence ax = b(mod n)

1 Check that gcd(a, m) divides 6. If it does not the congruence has no solutions. If it does:

2 Cancel any common divisors of all three of a, b and n. The resulting congruence has a unique solution modulo the new modulus.

The resulting coefficients (originally a and b) can then be changed by applying the remaining steps in any order, any number of times, with the goal of reaching a congruence in which the coefficient of x is 1.

3 Cancel any common divisor of the coefficients.

4 Replace either coefficient by any congruent number.

5 Multiply through the congruence by any number which is relatively prime to the modulus.

Theorem 4.1 Chinese Remainder Theorem

Let n1,n2,...,n, be positive integers such that gcd(ni,n;) = 1 for i # j. Then the system of linear congruences

x =b, (mod nı); х= 0 (той пз); ...; 2=b, (mod n,)

has a simultaneous solution which is unique modulo n4n . ..n,.

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Theorem 4.2

The system of linear congruences

x = bı (mod nı); x = be (mod ng)

has a simultaneous solution if, and only if, gcd(m1, n2) divides bz — bı. If a solution «exists then it is unique modulo lem(n;, n2).


The system of linear congruences

x=b; (modm); x= b2 (mod nz); ...; æ= b, (mod n,)

has a simultaneous solution if, and only if, gcd(ni,n;) divides b; — b; for each pair i, j of suffixes. If a solution exists it is unique modulo lem(ni,n», . . . n, ).

Unit 4 Fermat's and Wilson's Theorems

Theorem 1.1 Fermat's Little Theorem

If p is a prime and a is any integer with gcd(a, p) = 1, then

a?-! = 1 (mod p).

Fermat's Little Theorem, an alternative formulation

If p is a prime and a is any integer,

а? = a (mod p).

Theorem 2.1 Terminating decimals

1 The decimal of z terminates if, and only if, n = 2”5°, for some integers r > 0 and

8 > 0.

Theorem 2.2 The cycle length of 5

1 The length of the cycle in the decimal of Е is equal to the order of 10 modulo p.

Theorem 2.3

If a has order c modulo p, where p > 3 is prime, and a* = 1 (mod p) then cis a divisor of k.

In particular, c divides p — 1.

Theorem 3.1 Wilson's Theorem

If p is prime then (p — 1)! z —1 (mod p).

Theorem 3.2 Converse of Wilson's Theorem

If n > 1 is an integer and (n — 1)! = —1 (mod n) then n is prime.

Theorem 4.1 Lagrange's Theorem

Let p be a prime. A polynomial congruence P(x) = 0 (mod p) of degree k > 1 can have at most k solutions.

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Theorem 4.2

Let b be a solution of the polynomial congruence P(x) = 0 (mod p) of degree k > 1. Then P(x) = (x — b) Pi (z) (mod p), where P; (z) is a polynomial of degree k — 1 modulo p.

Theorem 4.3. Factorizing a polynomial modulo p

Let p be prime and let bj, b», ..., b, be incongruent solutions of P(x) = 0 (mod p) which has degree k > r. Then

Р(х) = (2 -6)(= 02)... (s — 5; P, (2) (mod p), where the polynomial P,(x) has degree k — r modulo p.


For any prime p

2? —1z (z - 1)(z — 2)(z — 3)... (z — (p — 1) (mod p).

Theorem 4.4

If p is prime and d is a divisor of p — 1 then the congruence

z^ — 1 — 0 (mod p)

has exactly d solutions.

Unit 5 Multiplicative Functions

Divisor sum formulae for T and a

a(n) = 371 and о(п) = doa ап ат

Theorem 1.1 Equivalent definition of multiplicative functions

The function f is a multiplicative function if, and only if, for every pair, m and n, of relatively prime positive integers,

f(mn) — f(m)f(n).

Theorem 1.2. Generating new number-theoretic functions

If f is a multiplicative function, then the number-theoretic function F defined by

F(n) - Y5 а) ат

is also multiplicative.


The functions 7 and o are multiplicative.

Theorem 1.3 The prime decomposition formula for o

c Ся m (оа +++)... (река)

ki+1 р I i-1 1<i<r Pi

И

15

Theorem 2.1. Classification of even perfect numbers

If k is a positive integer such that 2* — 1 is prime, then n — 2*-1 (2* — 1) is perfect. Furthermore, every even perfect number is of this form for some positive integer k.

Theorem 2.2

If p and 2p + 1 are primes, where p = 3 (mod 4), then 2p + 1 divides Mp.

Theorem 2.3

Any prime divisor of M,, where p is an odd prime, is of the form 2kp +1 for some positive integer k.

Theorem 2.4 Euler’s form for an odd perfect number

If n is an odd perfect number then

k 2k „2k: 2k, n-pp p^..p.*,

where the p;s are distinct primes and p = k = 1 (mod 4).

Theorem 3.1. Euler's Theorem—a generalization of FLT

If n is a positive integer and a is any integer with gcd(a, n) — 1, then a?) z 1 (mod n).

Theorem 3.2

If the integer a has order c modulo n then a^ — 1 (mod n) if, and only if, k is a multiple of c. In particular the order c divides (n).

Repunit property 1

19 2

9 Ry =

Repunit property 2

Вт» = Rm xX 10" + Rp

Theorem 4.1 Multiplicativity of ġ

The function ¢ is multiplicative.

Theorem 4.2. Formula for ó(n)

If n has prime decomposition n — pr pe pF, then

"ubi i-e [eee] $(n) 7 p? py ...p; (: x) ( E we (1 Е

о. Alternative formula for (n)

Ifn= pi pk ... pk, then

$(n) — pp?! ... p (mi - 1)(p2 - 1)... (p. — 1). If p is prime and gcd(a, p) = 1 then the order of a modulo p is the least positive integer c such that a° = 1 (mod p).

16

Theorem 4.3

di 9@) =n d|n

Theorem 5.1

If a is a primitive root of n then

IEN ,а%%}

is a reduced set of residues modulo n.

Theorem 5.2

If a has order c modulo n then, for any k 7 1, a^ has order

c ЕЕ modulo n.


If a is a primitive root of n then a* is also a primitive root of n if, and only if, k is relatively prime to $(n).

Furthermore, if n has a primitive root, then it has exactly ¢(4(n)) primitive roots.

Theorem 5.3

Any integer which can be expressed as the product of two divisors which are relatively prime and each exceeding 2 does not have a primitive root.


If an integer n has a primitive root then it must have one of the following forms.

(a) n = p", p prime, k > 1 (b) n = 2p*, p an odd prime, k > 1

Theorem 5.4

For any k > 3, 2* does not have a primitive root.

Theorem 5.5 Integers with primitive roots

The integer n has a primitive root if, and only if, n = 2, 4, p* or 2p*, for p an odd prime and k > 1.

Unit 6 Quadratic Reciprocity

Theorem 1.1 The quadratic residues of p

- -1 For any odd prime p there are P P,

1 z = : quadratic residues and quadratic

non-residues. The quadratic residues are congruent modulo p to the integers 2

12, 92 32 р 1 23545 5

Theorem 1.2 Euler's Criterion

Let p be an odd prime and a # 0 (mod p). Then a is a quadratic residue of p if, and

only if, a(?-9/? z 1 (mod p) and is a quadratic non-residue of p if, and only if, a(?-)/? z —1 (mod p).

17

Theorem 2.1 Properties of the Legendre symbol

Let p be an odd prime and let a Z 0 (mod p) and b z 0 (mod p). Then the following properties hold.

(a) If a =b (mod p) then (a/p) = (b/p).

(b) (a/p) =1 (c) (ab/p) = (a/p)(b/p) (d) (a/p) = a-Y/? (od p)

1l, ifp=1 (mod 4), (0 (ма = { =. if p z 3 (mod 4).

Theorem 3.1 Gauss’ Lemma

Let p be an odd prime and a Z 0 (mod p). Let S be the set

-1 = {a 20,34,..., 25 a}

consisting of the first В positive multiples of a. If n denotes the number of

members of S whose least positive residue modulo p exceeds 5 then (a/p) = (—1)”.

Theorem 3.2. The quadratic character of 2

If p is an odd prime then

(2/p) = { > ifpz 1 (mod 8) orp 7 (mod 8), —Ь if p=3 (mod 8) or p=5 (mod 8).

Theorem 3.3. (Theorem 2.2 of Unit 5)

If p and 2p + 1 are both primes, where p = 3 (mod 4) then 2p + 1 divides My.

Theorem 3.4

There are infinitely many primes of the form 4k + 1.

Theorem 3.5

There are infinitely many primes of the form 8k — 1.

Theorem 4.1 The Law of Quadratic Reciprocity

If p and q are distinct odd primes then

(p/a)(g/p) (-1)*-2&-1/4.

Theorem 4.2. An alternative formulation of LOR

If p and q are distinct odd primes then

ipfos { (a/p), if either p =1 (шо4 4) or q=1 (mod 4), -(4/p), if p =q = 3 (mod 4).

Theorem 4.3

Let p be an odd prime and a an odd integer not divisible by p. Then

(a/p) = (—1)%%), where

(p-1)/2 ka a(a, p) — У m( 5).

k=1

18

Theorem 4.4 The quadratic character of 3

If p > 3 is prime then

et 1, ifp=+1 (mod 12), (3/p) = { ~1, ifp=+5 (mod 12).

Unit 7 Continued Fractions

First continued fraction identity

1 [a1, 42, 03,...,@n] = a) + ———_-

[a2, 3, ..., a4]

Theorem 1.1 Equivalence of FCFs and rational numbers

Every simple finite continued fraction has a rational value and, conversely, every rational number can be expressed as a simple finite continued fraction.

Second continued fraction identity

[a1,02, . .., 04, 14.1] — [01,02; - ak1, ak + ] ак+1

Theorem 1.2 The value of a finite continued fraction

Let [a1, 5, ..., a4] be any finite continued fraction. Define the numerators pi, po, ps, ++) Pn and the denominators qı, q2, 3, ---, Qn Of this continued fraction by:

Р = а, 1[2=4142+1; рь = акрь-1 +рк-, 3<k<n; Ф = 1, q2 — аә; Gk = QkQk-1 t dk-2; << п.

Then the value of [a;, a5, ..., a4] is Eni m


Let [a1,a2,..-,@n] be any finite continued fraction and the numerators pi, po, . . p,

and denominators q1, q2, ...,q» be as defined in the theorem. Then its convergents are

Сь = DS Tor. ki,2, «$0. dk

Theorem 1.3 Properties of the convergents

Let [a1,a2,...,an] be any FCF, and let p1, p2, --., Pn and qi, qo, ..., Q4 be its numerators and denominators as defined in Theorem 1.2. Then we have the following properties.

(a) Pkqk-1 — Px—1dk — (—1)^; that is,

PR Dei _ (—1)* dk Чк-1 9606-1

, for2<k<n.

(Ь) рьфь-2 = рк-24 = (—1)% аһ; ай іѕ,

рь _ рь-2 _ (1) hax

dk — dk—2 qkdk—2

(c) px and qy are relatively prime for k — 1, 2, ..., n.

» for3<k<n.

19

Theorem 14 The relative sizes of convergents

For any FCF the odd convergents form a strictly increasing sequence C1 < C3 < Cs < ---, while the even convergents form a strictly decreasing sequence Cz > C4 > Cg > ---. Every even convergent is greater than every odd convergent.

Theorem 2.1 Sequences of convergents converge

For any ICF [a1,a2,a3,...] its sequence of convergents {Chn} tends to a limit.


If {Cn} is the sequence of convergents of x then, for each n > 1:

(a) z lies between C, and Cn+1;

©) je- |< х- — r dn Qnn4-1

<

Infinite continued fraction identity

1 [а1, а, аз,...] = а + нет

Theorem 3.1. Irrationality of ICFs

The value of any infinite simple continued fraction is an irrational number.

The Continued Fraction Algorithm

Irrational 21

Express this number as

integer part + fractional part

Integer part is the next Calculate the reciprocal

partial quotient of the fractional part

Theorem 3.2 Uniqueness of the ICF

Given any irrational number z, the ICF determined from x by the Continued Fraction Algorithm converges to x, and no other ICF converges to т.

Theorem 4.1 Accuracy of convergents

If Bi, A Рз -.. is the sequence of convergents of x then, for all k 21, q Ф 8

1 1 e T Pk+1 < < < | Pk .

dk+1 dk+14k+2 7 2969641 i

20


For all k > 1 the convergent Pe of x satisfies dk

1 1 < k- Behet, 2dkQk41 9 | 9+1

о 4.2

Let ̂ — be the nth convergent of the irrational number a, and let $ be any rational

"e with 0 < b < qn41. Then, for n » 1,

|dnt — Pn| € |bx — al,

the equality holding only if a = p, and b = qn.

Corollary to Theorem 4.2 Convergents are best approximations

Тев 21 e Р2

a’ g

number i with 1 «b € qs

. be the convergents of the irrational number x. Then, for any rational

spi]. the M, holding only if a — p, and b = qn.

Theorem 4.3 Which rationals are convergents?

If the rational number P satisfies

-5< a.

then » is a convergent of x.

Unit 8 Diophantine Equations

Theorem 1.1 Solutions of Pell’s equation are convergents

$ ae i a. If x = a, y = b is a positive solution of z? — ny? — 1 then z Ba convergent of yn.

Theorem 1.2 Solution of Pell’s equation

If the continued fraction of y/n has cycle length s then

а о P= e and all solutions of

а? -ny = +1

are given in this way.

Theorem 1.3 All solutions from a fundamental solution

Let x = 21, y = yi be the fundamental solution of x? — ny? = 1. Then, for each integer k > 1, x = £k, Y = Yk İs also a solution, where the positive integers z, апа ур are given by

тк укуп = (21 tun)

Conversely, the solutions given in this way are the only positive solutions.

21

Theorem 1.4

Suppose that the ICF of y/n has a cycle of odd length s. Let £1 = ps, Y1 = qs, where the convergent C, — e and let z, and y; be given by

s

ть + уьУп = (mi iin)"

Then, for all integers k > 1, « = 2026-1), У = Y(2k—1)s is a solution of z? — ny? 2 —1 and z — 22,5, y = yaxs is a solution of z? — ny? = 1. Moreover, all solutions of 2? — ny? — 1 are given in this way.

Theorem 2.1 Primitive solutions of the Pythagorean equation

The primitive Pythagorean triples are the triples

(2mn, m? — n?, m? n?),

where m and n are relatively prime positive integers of opposite parity and with m nm.

Corollary to Theorem 2.1 All solutions of the Pythagorean equation

The Pythagorean triples (z, y, z) are given by

z—2kmn, y-k(m!—m?), z=k(m? + n2),

where k > 1 is any integer and m and n are relatively prime positive integers with opposite parity and m > n.

Theorem 3.1

The Diophantine equation z* 4- y* — 2? has no positive solutions. p. y


The Diophantine equation z* 4- y* — z* has no positive solutions.

Theorem 3.2

The Diophantine equation z* -- 4y* — z? has no positive solution.


The Diophantine equation z* — y* — z? has no positive solution.

Important identity for two squares

(a? + 8°)(c? +d?) = (ac + bd)? + (ad — be)?

Theorem 4.1 Primes expressed as a sum of two squares

A prime p can be expressed as a sum of two squares if, and only if, p = 2 or p € 1 (mod 4).

Theorem 4.2 Uniqueness of representation

The expression of a prime of form 4k + 1 as a sum of two squares is unique except for the order of the two summands.

Theorem 4.3 Sums of two squares

A positive integer n can be expressed as a sum of two squares if, and only if, each of its prime divisors of the form 4k + 3, if any, occurs to an even power.

22

Theorem 4.4 Sums of three squares

A positive integer can be expressed as a sum of three squares if, and only if, it is not of the form 4"(8m + 7) for some n » 0, m 7» 0.

Theorem 4.5 Lagrange's Four Square Theorem

Every positive integer can be expressed as a sum of four squares.

Theorem 4.6 Waring's Problem

For each integer k 7 2 there exists a positive integer g(k) such that every positive integer can be expressed as a sum of at most g(k) kth powers.

23

TABLE OF PRIMES APPENDIX

excluding the multiples of 5. The , The following table lists all the odd integers up to 3999

wise the smallest prime in the table indicates that the number is prime. Other entry ‘P’

ged in blocks of 100, with the units ,

divisor of the number is given. The numbers are arran

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25

CUMULATIVE INDEX OF UNITS abundant 5 34

accuracy of convergents 7 33 algebraic numbers 8 19 amicable pair 5 34 arithmetic functions 55 arithmetic progression 1 7 arithmetic series 1 7

basis for the induction 1 15 Bertrand's Conjecture 2 16 best approximations 7 36

cancellation rule 3 13 casting out nines 3 14 Chinese Remainder Theorem 3 23 classification of even perfect numbers 5 11 complete set of residues 39 composite 2 5

congruence class 39

congruent modulon 36 consecutive composite integers 2 23 Continued Fraction Algorithm 7 26 convergent 7 11 converse of Wilson’s Theorem 4 18 coprime 1 25

Corollary to Euclid’s Lemma 2 9 cycle 410

cycle length 4 10

cycle of a periodic ICF 7 24

deficient 5 34 degree of a polynomial congruence 4 22 degree of a polynomial modulo n 4 22 denominators 7 12

density of primes 2 25 descent step 8 20 Diophantine equation 1 34

Dirichlet’s Theorem 2 19 discriminant 65

divides 1 22 divisibility by 11 3 15 divisibility by 9 3 15

Division Algorithm 1 20 divisor 1 22 divisor sum formulas for rand a 56 divisors of an integer 2 11

equivalence of FCFs and rational numbers 7 9 equivalent congruence 3 20 Euclid's Elements 8 13

Euclid’s Lemma 1 26 Euclid’s Lemma for prime divisors 2 8,314 Euclid’s Theorem 2 15 Euclidean Algorithm 1 32 Euler’s ¢-function 5 18 Euler’s Criterion 6 7

Euler’s Theorem 5 19

even convergents 7 17

factor 1 22

26

factorizing a polynomial congruence 4 23 FCF 78 Fermat numbers 5 16 Fermat’s Last Theorem 8 18 Fermat’s Little Theorem 4 6 Fermat’s Little Theorem, an alternative formulation 4 7 Fibonacci numbers 2 28 Fibonacci numbers, divisibility property 2 31 Fibonacci numbers, relative primeness of consecutive

numbers 2 29 Fibonacci primes 2 28 Fibonacci sequence 2 27 finite continued fraction 7 7 formula for ¢(n) 5 26 formula foro 5 10 formula for r(n) 213 fractional part 7 27

function r 213

fundamental solution 8 9 Fundamental Theorem of Arithmetic 2 10

Gauss’ Lemma 6 15

geometric series 1 16 Goldbach’s Conjecture 2 24 golden ratio 75 greatest common divisor 1 23

heptagonal numbers 1 10 heptagonal-based pyramidal numbers 1 12 hexagonal numbers 19 highest common factor 1 23 hypoteneuse 8 12

ICF 7 20 ideal prime divisors 8 19 important identity for four squares 8 33 important identity for two squares 8 25 incongruent modulon 3 6 induction hypothesis 1 15 induction step 1 15 infinite continued fraction 7 20 integer combination 1 24 integral polynomial 3 16 irrationality of ICFs 7 26

Jacobi symbol 6 28

Lagrange’s Four Square Theorem 8 33 Lagrange’s Theorem 4 22 Law of Quadratic Reciprocity 6 21 leading term of a polynomial 4 22 least absolute residues 3 9 least common multiple 1 28 least positive residues 3 9

Legendre symbol, (a/p) 6 12 length of a cycle 4 10 linear congruence 3 18 linear Diophantine equation 1 34 LQR 621

LQR an alternative formulation 6 21

Mathematical Induction 1 15 Mersenne number 5 12

Mersenne prime 5 12

method of infinite descent 8 20 multiple 1 22

multiplicative functions 5 7

non-negative integers 1 33 non-terminating decimals 4 10

number of solutions of a polynomial congruence 3 17 number-theoretic functions 5 5 numerators 7 12

octagonal numbers 1 10 odd convergents 7 17 order of an integer modulo a prime 4 13 order of an integer to a composite modulus 5 20

parity 8 13

partial quotients 7 7 parts of a positive integer Pell’s equation 8 5 pentagonal numbers 1 9 perfect number 5 11

perfect squares 18 periodic 7 24 polygonal numbers 1 10 polynomial congruence 3 16, 4 19 prime 25

prime decomposition 2 10

prime decomposition of gcd(m,n) 212 prime decomposition of lem(m,n) 2 13 prime number theorem 2 26 prime quartets 2 24

prime triplets 2 24

primes generated by polynomials 2 20

primes in arithmetic progression 2 16 primes of the form 4k +3 218 primitive Pythagorean triple 8 12 primitive root 4 13, 5 29

Principle of Finite Induction 1 14

Principle of Mathematical Induction 1 15

Principle of Mathematical Induction (generalized) 1 16 proper fraction 4 16 properties of congruence 3 7

properties of the convergents 7 16 pseudoprimes 48

5 11

purely periodic 7 24

pyramidal numbers 1 11 Pythagorean equation 8 12

Pythagorean triple 8 12

quadratic character 65

quadratic character of 2, (2/p) 618 quadratic character of 3, (3/p) 6 27 quadratic congruence 4 18, 6 4

quadratic irrationals 7 30 quadratic non-residue 65

quadratic residue 6 5

rational approximations 7 32 rational number 7 9

reduced set of least positive residues modulo n 5 18 reduced set of residues modulo n 5 18 regular primes 8 19 relatively prime 1 25 repunit 5 22

residue class 39 ring 819

Second Principle of Mathematical Induction 1 18 sieve of Eratosthenes 2 7 simple continued fraction 7 7 simple infinite continued fraction 7 20 simultaneous linear congruences 3 22 simultaneous solution 3 23 size of the nth prime 215 solution by exhaustion 3 18

solutions of a polynomial congruence 4 19 square number 18 square-based pyramidal numbers 1 11 square-free 5 33

squares 18

sums of squares 8 24

sums of three squares 8 32

terminating decimals 4 10 triangle-based pyramidal numbers 1 11 triangular numbers 1 7 twin primes 2 22

value of gcd(n,p) 28 value of a continued fraction 7 7

value of an infinite continued fraction 7 21

Well-Ordering Principle 1 13 Wilson's Theorem. 4 17

27

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