Basic number theory

Abstract

In this chapter we cover basic number theory and set up notations that will be used freely throughout the rest of the book.

Appendices

Exercises

• 2.1 Prove Lemma 2.5.

• 2.2 Show that the alternative definitions in Definition 2.10 are equivalent.

• 2.3 Use the Euclidean Algorithm to give another proof for Theorem 2.12.

• 2.4 Prove Proposition 2.20.

• 2.5 Prove Proposition 2.21.

• 2.6 For the following pairs of integers (a, b), find integers x, y such that $\gcd (a, b) = ax + by$:

  • a. (13, 15);

  • b. (398, 270);

  • c. (162, 65).

• 2.7 ($\maltese$) Find the $\gcd$ of 6437 and 12675. Find integers x, y such that $6437x + 12675y = \gcd (6437, 12675)$.

• 2.8 ($\maltese$) Find the $\gcd$ of 2594876242943772804330 and 11446995929696298.

• 2.9 Write the following number as a fraction $\frac{a}{b}$ with $a, b \in \mathbb N$ and $\gcd (a, b) =1$:

  $$ 10^{59} \left( \frac{1025}{1024}\right) ^5 \left( \frac{1048576}{1048575}\right) ^8 \left( {6560 \over 6561} \right) ^3 \left( {15624 \over 15626}\right) ^8 \left( {9801 \over 9800}\right) ^4. $$

  Determine the prime factorizations of a, b without the use of a computer. Mossaheb [34] attributes this problem to Gauss.

• 2.10 Determine all natural numbers n such that $\prod _{d \mid n} d = n^2$.

• 2.11 Suppose for integers a, m, n, k we have $a^m \equiv 1 \ \mathrm {mod}\ k$ and $a^n \equiv 1 \ \mathrm {mod}\ k$. Show that $a^{\gcd (m, n)} \equiv a^{\mathrm{lcm}(m, n)} \equiv 1 \ \mathrm {mod}\ k$.

• 2.12 Show that if a rational number $\frac{a}{b}$, with $a, b \in \mathbb Z$ and $\gcd (a, b) = 1$, satisfies the equation

  $$ a_n x^n + a_{n-1} x^{n-1} + \dot{+} a_1 x + a_0 =0, $$

  with $a_0, a_1, \dots , a_n \in \mathbb Z$, then $a \mid a_0$ and $b \mid a_n$. Use this result to find the rational roots of the following equations:

  • a. $5x^3 + 8 x^2 + 6x -4 =0$;

  • b. $x^5 - 7x^3 - 12 x^2 + 6x + 36 =$;

  • c. $6x^6 - x^5 - 23 x^4 - x^3 - 2x^2 + 20 x - 8=0$.

• 2.13 Use the previous exercise to show $\sqrt{2} + \root 3 \of {3}$ is irrational.

• 2.14 Show that for all integers a, b, c, d satisfying $ad - bc=1$ we have $\gcd (a+b, c+d) =1$.

• 2.15 Show that for all integers $n > 1$, $1 + 1/2 +1/3 + \dots + 1/n$ is not an integer.

• 2.16 If f is a non-constant polynomial with integer coefficients, f(n) is composite for infinitely many values of n.

• 2.17 Show that if p, q are prime numbers larger than 3, then the remainder of division of $p^2 + q^2$ by each of the numbers 3, 4, 6, 12, and 24 is equal to 2.

• 2.18 ($\maltese$) Show that a number n is prime if and only if it is not divisible by any natural numbers m with $1 \le m \le n^{1/2}$. This result is known as the Sieve of Eratosthenes. Use this idea to list all prime numbers between 1 and 1000.

• 2.19 ($\maltese$) Find five natural numbers k such that $22 + 37 k$ is a prime number.

• 2.20 Show that for all $m, n \in \mathbb N$,

  $$ \frac{\gcd (m, n)}{n}\left( {\begin{array}{c}n\\ m\end{array}}\right) $$

  is an integer.

• 2.21 Suppose $F_n = 2^{2^n}+1$. Show that for all $m >n$, $F_n \mid F_m -2$.

• 2.22 Find necessary and sufficient conditions for the solvability of the system (2.4). Find the general solution of the system.

• 2.23 Solve the system of congruence equations

  $$ {\left\{ \begin{array}{ll} 3x \equiv 1 \mod 4, \\ 3x \equiv 1 \mod 13, \\ 5x \equiv 11 \mod 21. \end{array}\right. } $$

• 2.24 Find the general integral solution of the Diophantine equation

  $$ 239 x - 111y =1. $$

• 2.25 Find all pairs of integers (x, y) satisfying the equation $6x + 9 y = 12$.

• 2.26 ($\maltese$) Find all x such that $85 x \equiv 970 \ \mathrm {mod}\ 6 4322$.

• 2.27 ($\maltese$) Find all solutions of $37 x \equiv 217 \ \mathrm {mod}\ 8 600$.

• 2.28 ($\maltese$) Find all x that satisfy the following system of congruence equations:

  $$ {\left\{ \begin{array}{ll} x \equiv 12 &{} \mod 64; \\ x \equiv 1 &{} \mod 173; \\ x \equiv 5 &{} \mod 715. \end{array}\right. } $$

• 2.29 Show that $5! 25! \equiv 1 \ \mathrm {mod}\ 3 1$.

• 2.30 Show that if $p \equiv 3 \ \mathrm {mod}\ 4$, then

  $$ \left( (\frac{p-1}{2})!\right) ^2 \equiv 1 \mod p. $$

• 2.31 Prove the uniqueness assertion of Lemma 2.37.

• 2.32 Give two different proofs for the statement that for all integers n, $n^5/5 + n^3/3 + 7n/15 \in \mathbb Z$. Generalize.

• 2.33 Find all the solutions to the congruence $x^2 \equiv 1 \ \mathrm {mod}\ 2 64$.

• 2.34 By examining the solutions of the equation $x^2 \equiv 1 \ \mathrm {mod}\ p$, show that for all primes p, $(p-1)!\equiv -1 \ \mathrm {mod}\ p$. Show that if $n >4$ is not prime, $(n-1)!\equiv 0 \mod n$.

• 2.35 Let $n \in \mathbb N$. Compute the product

  $$ \prod _{1 \le d \le n \atop d^2 \equiv 1 \ \mathrm {mod}\ n } d. $$

  Use your formula to determine

  $$ \prod _{1 \le d \le n \atop \gcd (d, n) =1} d. $$

• 2.36 Find the roots of the polynomials $x^2 -x + 1$ and $x^2 -x + 2$ modulo 7.

• 2.37 ($\maltese$) Find an integer x such that $x^2 \equiv 1879121 \ \mathrm {mod}\ 3 698963$.

• 2.38 ($\maltese$) Find the last four digits of $2^{4000}$.

• 2.39 Show that $\phi (n)$ is even if and only if $n>2$.

• 2.40 Determine all n such that $\phi (n)=6$.

• 2.41 Determine all n such that $\phi (n) = 40n/77$.

• 2.42 Show that for every odd integer $n>1$ we have $\phi (n) > \sqrt{n}$.

• 2.43 Determine all n with $\phi (n) \mid n$.

• 2.44 Give a different proof for Theorem 2.32 using the Inclusion–Exclusion Principle.

• 2.45 Prove the following generalization of Theorem 2.32: If $\gcd (m, n) = d$, then

  $$ \phi (mn) = \phi (m)\phi (n) \frac{d}{\phi (d)}. $$

• 2.46 Use Theorem 2.33 to give another proof for Theorem 2.34.

• 2.47 Show that for each complex number $\alpha \ne 1$ and for each natural number n, we have

  $$ \sum _{k=0}^{n-1} \alpha ^k = \frac{1-\alpha ^n}{1-\alpha }; $$

  Show that if $|\alpha |<1$, then

  $$ \sum _{k=0}^\infty \alpha ^k = \frac{1}{1-\alpha }. $$

• 2.48 Show that for each $g>4$, the number $(4.41)_g$ is the square of a rational number. Find its square root. Repeat the same problem for $(148.84)_g$ for $g>8$.

• 2.49 For what values of g, are the numbers $(0.16)_g, (0.20)_g, (0.28)_g$ the consecutive terms of a geometric sequence?

• 2.50 If $25/128=(0.0302)_g$, find g.

• 2.51 Find the base 5 expansion of 2877 / 3125.

• 2.52 Find the base 9 expansion of $(200.211)_3$.

• 2.53 Determine the rational number with base 7 expansion $(0.\overline{130})_7$. Solve the same problem for $(0.1\overline{296})_{12}$.

• 2.54 Find all primitive roots modulo 38.

• 2.55 ($\maltese$) Find all primes $p < 1000$ for which 3 is a primitive root.

• 2.56 ($\maltese$) Find five primitive roots modulo 100003.

• 2.57 ($\maltese$) Find five primitive roots modulo $987654103^2$.

• 2.58 If $p = 4q+1$ and $q = 3r+1$ are prime, show that 3 is a primitive root modulo p.

• 2.59 Let p, q be distinct primes. Find the number of solutions of $x^p \equiv 1 \ \mathrm {mod}\ q$ in terms of p, q.

• 2.60 Without using a computer, prove that $2^{17}-1$ is prime. Hint: Show that if it is not prime, it must be divisible by one of the numbers 103, 137, 239, or 307.

• 2.61 Let p be an odd prime such that $(p-1)/2$ is an odd prime. Prove that if a is a positive integer with $1< a < p-1$, then $p-a^2$ is a primitive root modulo p.

• 2.62 Show that n is a square if and only if d(n) is odd.

• 2.63 Show that for all $n, s, t \in \mathbb N$, with $s \ne t$, $s-t \mid d(n^s) - d(n^t)$.

• 2.64 Show that for all $m, n, s \in \mathbb N$, we have $s \mid d(m^s) - d(n^s)$.

• 2.65 Find necessary and sufficient conditions on integers a, b, c, d so that there are integers x, y, z satisfying the system of Diophantine equations

  $$ {\left\{ \begin{array}{ll} 4x + y + az = b, \\ x + y + cz =d. \end{array}\right. } $$

• 2.66 Let p be an odd prime. Show that if we write $1+1/2+\dots +1/(p-1)$ as a fraction a / b with $a, b \in \mathbb N$, then $p \mid a$. A far more interesting problem is to show that if $p \ge 5$, $p^2 \mid a$.

Notes

1.1 Historical references

The standard reference for the history of classical number theory is Dickson’s History of the theory of numbers in three volumes. Most of the material in this chapter has been reviewed in the first volume [15], especially Ch. III, V, and VIII. A more current reference for the history of mathematics is [9]. As impressive as these books are, like many other books on the history of science, they are unfortunately very Eurocentric. The history of mathematics as told through these and other similar texts runs like this: The Greeks invented mathematics; then as Europe was falling into the Dark Ages, Muslims ran to the rescue; Muslims carefully guarded mathematics for a few centuries; with the arrival of the Renaissance, the Muslims handed mathematics back to the Europeans who gracefully accepted the gift, and who have ever since been championing the progress of mathematics. This Eurocentricity does not stop at the history, and in fact it permeates every aspect of the practice of mathematics. In reality the history of mathematics is far more complicated and far more multicultural than a simple straight line connecting Athens of the antiquity to the North America and Europe of 21st century.

In this book I have made a conscious effort to highlight contributions by non-Europeans to number theory. However—and this is far from an acceptable excuse—because of my lack of expertise as well my own Eurocentric education I am not able to do justice to the subject. Getting the history right is not just a matter of intellectual curiosity. Those of us who work as educators in North America are acutely aware of the fact that a good portion of our students are not of European descent. To many of our students mathematics is a European invention, and will continue to be practiced by Europeans and people of European descent. Nothing could be further from the truth. Mathematics has been practiced on every continent, by all sorts of people, for thousands of years, and there are distinguished mathematicians of every imaginable background today doing fantastic mathematics—and this should be emphasized in our teaching. There is, unfortunately, a shortage of modern, easily accessible texts putting in the correct historical perspective the progress of mathematics through the millennia. Even in cases where a serious mathematician such as van der Waerden [55] has attempted to write a history of mathematics as inspired by the progress made by non-Europeans, the works of these non-Europeans are described in relation to and within the framework of modern European mathematics, or the Greek mathematics of the antiquity, in the sense that, what of the works of non-Europeans that has not been superseded and swallowed by some mathematical work developed by a European mathematician is often not considered worthy of review. The same problem exists in most works written by European or North American historians of mathematics, with a notable exception being Plofker [39]. Writings by historians like Roshdi Rashed, especially the second volume of Encyclopedia of the history of Arabic sciences [40] which covers mathematics, and Joseph [28] are good alternatives to standard Eurocentric narratives that saturate the literature.

On a personal note, growing up in Iran, I never felt that mathematics was a European invention or practice—I knew of Iranian mathematicians like Omar Khayyam, Mohammad Al-Khwarizmi, and Mohammad Karaji, and these were people I identified with. I credit Iranian education pioneers like G. H. Mossahab, M. Hashtroodi, M. Hessabi in the 1940s and 1950s, and more recently S. Shahshahani, P. Shahriari, O. A. Karamzadeh, Y. Tabesh, and others starting in the 1970s, for initiating the effort to instill the notion in the minds of the Iranian youth that mathematics, along with other sciences, was as Iranian as apple pie is American. It is because of their efforts that Iran has enjoyed a revitalization of mathematics in the last 25 years. Culture building takes time, and, as in the case of those Iranian pioneers, one may not live long enough to see the fruits of one’s labor, but with patience and perseverance great things are possible.

1.2 Euclid and his Elements

Euclid (325–265 BCE) was the person who transformed mathematics from a number of uncoordinated and loosely proven theorems into an articulated and surely grounded science. Some of the theorems in Euclid’s Elements were previously known by other mathematicians: Thales (624–546 BCE) who was according to Aristotle the first Greek philosopher, Eudoxus (410–355 BCE), Pythagoras and other Pythagoreans, etc. A predecessor to Euclid was Hippocrates (470-410 BCE) who wrote the first Elements around 430 BCE. Euclid was extremely rigorous in his treatment of mathematics. (Though as noted by David Hilbert [26] , Euclid should have augmented his postulates by adding a few more.) E. T. Bell argues that if the world had followed Archimedes as opposed to Euclid, Calculus would have been discovered before the birth of Christ. This is a harsh criticism of the Euclidean rigor, and of the course of history, but it is nonetheless most likely true that the sort of rigor that Euclid brought into mathematics slowed down progress in some sense. Archimedes was a master problem solver who was interested in the applications of mathematics in the real world. Euclid, on the other hand, was interested in gaining a deep understanding of concepts via systematic study. For what it is worth, almost 2500 years later, we still practice mathematics the way Euclid did mathematics in his magnum opus. An interesting feature of the Elements is that the writing is extremely homogeneous. Euclid makes no distinction between trivial facts and deep theorems, and everything is proved with the same degree of care. Was Euclid really not aware that some of his results are more important than others? We will never know.

The theory of numbers is treated by Euclid in books 7-10 of the Elements. At the beginning of Book 7 Euclid lists definitions: unit, numbers, multiple, even and odd number, prime and composite numbers, square, proportional, perfect number, etc. These are very much in the Pythagorean style, but with some modifications. We refer the reader to the excellent commentary in Sir Thomas L. Heath’s “The Thirteen Books of Euclid’s Elements” [20] published in 1926. In this book Sir Heath compares Euclid’s definitions to those given by his predecessors. In the case of prime numbers, Euclid’s definition varies slightly from the one written by the Pythagorean Philolaus (480–390 BCE) who seems to have been the first person to give a definition of prime numbers.

For all their aura of naturalness, prime numbers almost never appear in nature for reasons of primality. The only example of such a process is the life cycles of a certain genus of cicadas. These insects spend most of their lives underground and emerge to daylight every 13 or 17 years. The fact that 13 and 17 are prime numbers gives these insects a computable but small evolutionary edge over their predators. Over millions of years the evolutionary edge of these insects has helped them not go extinct. Beyond this, we are not aware of any cosmic or earthly processes that produce prime numbers for reasons of primality. Even within mathematics, as practiced by human beings, it appears that prime numbers were an invention of the Greeks, and that no one else in the ancient world had a notion of prime numbers. Mathematicians in Babylon, India, China, and the Americas investigated very sophisticated mathematical theories, including those applicable to astronomy and other sciences, but as far as we can tell none of these mathematicians had a theory of prime numbers.

For more on Euclid’s work on prime numbers, see Notes, Chapter 6.

1.3 Natural Numbers and mathematical induction

In this book we will treat natural numbers in a common sense, intuitive fashion. We assume the set of natural numbers $\mathbb N$ consists of positive integers $1, 2, 3, \dots$, equipped with the standard addition and multiplication operations, enjoying the familiar properties of commutativity and associativity for addition and multiplication, and distribution laws for multiplication over addition. We also know that we can prove statements in the set of natural numbers using mathematical induction, accepted as an axiom. In reality, however, all of these statements are non-trivial and require close examination. The axiomatic study of the set of natural numbers has a long, rich history. We refer the reader to [18, Ch. 1] for an accessible introduction to this beautiful subject.

1.4 Number-theory-based cryptography

Many modern cryptographic methods are based on the material presented in this chapter. Here we will explain two standard techniques. For an elementary treatment of these methods and other number theoretic cryptosystems we refer the reader to [53].

The RSA Cryptosystem, named after Ron Rivest, Adi Shamir, and Leonard Adleman, is based on the notion that while multiplying numbers is easy, finding the prime factors of a large number is difficult. More specifically, if we know the prime factors of a natural number n, then Theorem 2.33 tells us how to compute the value of $\phi (n)$. However, without knowing the prime factors of n, we do not have a fast algorithm to compute $\phi (n)$. Presumably, one can take (2.6) as the definition of $\phi (n)$. This requires going through the list of numbers 1 to n and examining the $\gcd$ of each one with n, which, if the number n is of the order of $10^{500}$, would be impossible.

RSA is an example of a public key cryptosystem. In such a cryptographic scheme an individual A sets up a public key K, which is available to everyone, and keeps a private piece of information S, which is kept secret. The idea is that anyone who wants to communicate with A will encrypt the message using the publicly available key K but decrypting the encrypted message requires the secret information S. In the case of RSA, the public key is a large natural number n which is the product of prime numbers p, q. The prime numbers p and q are kept secret..

This is how RSA works. Suppose Azadeh wants to set up a public key. She picks large prime numbers p, q. She computes $n=pq$, $\phi (n) = (p-1)(q-1)$, and she picks a natural number e such that that $\gcd (e, \phi (n))=1$. She also finds an integer d such that $ed \equiv 1 \ \mathrm {mod}\ \phi (n)$, i.e, $ed = 1 + u \phi (n)$ for some integer u. She will keep p, q, d, and $\phi (n)$ secret, but publishes the pair (n, e). Now suppose Azadeh’s friend, Behnam, wants to communicate with Azadeh. Suppose the message that Behnam wants to send has numerical value m, obtained using ASCII or some other method (technically speaking, Behnam will have to make sure that $\gcd (m, n) =1$). Behnam downloads the pair (n, e) from Azadeh’s public profile, and computes $y := m^e \ \mathrm {mod}\ n$, i.e., the remainder of the division of $m^e$ by n which will be a number between 0 and n. Behnam keeps the message m secret, but sends the message y to Azadeh over some public channel, e.g., Facebook or SMS. Azadeh receives the message y, and deciphers it by computing

$$ y^d \equiv (m^e)^d \equiv m^{1+ u\phi (n)} \equiv (m^{\phi (n)})^u \cdot m \equiv m \mod n, $$

after using Theorem 2.31. On the other hand, Esmat, an evil person, is listening to the conversation happening between Azadeh and Behnam. Esmat downloads the message y. She also knows (n, e) as these are publicly available. However, at present there is no reasonably fast way to get from the data y, (n, e) to m without knowing d, and knowing d requires $\phi (n)$. As noted above computing $\phi (n)$, at the time of this writing, requires knowing the prime factors of n, which Azadeh is keeping secret.

For example, suppose Azadeh picks the prime numbers $p=101$ and $q=113$ (this is just a prototype; in practice the prime numbers are a few hundred digits long). Hence $n= 101\times 113 = 11413$. We have $\phi (n) = (101-1)(113-1) = 11200$. She also picks $e=3$. Note that $\gcd (3, 11200)=1$. Azadeh’s public key is the pair (11413, 3). What Azadeh is not sharing with the public are the prime numbers 101 and 113. She also keeps secret the number d such that $3d \equiv 1 \mod 11200$. Azadeh can easily compute, for example using SageMath, Appendix C, that $d = 7467$ works. Now suppose Behnam wants to transmit a message m with numerical value 77 to Azadeh. Behnam computes $m^e \ \mathrm {mod}\ n$. In this case since $m = 77$, $e = 3$, and $n = 11413$, he computes

$$ 77^{3} \equiv 13 \mod 11413. $$

So Behnam’s message, which he can communicate over a public channel, is $y=13$. Anyone can read this message x, and everyone knows Azadeh’s public key (11413, 3). So the problem that Esmat, the evil person, needs to solve is this: Find m such that $m^3 \equiv 13 \ \mathrm {mod}\ 1 1413$. For Azadeh, this is easy. All she needs to do is compute

$$ 13^{7467} \equiv 77 \mod 11413, $$

which she can easily do using SageMath.

The ElGamal Cryptosystem, named after the Egyptian computer scientist Taher ElGamal, is based on the difficulty of the Discrete Log problem. As mentioned earlier RSA cryptography is based on the idea that it is difficult to go from $(m^e \ \mathrm {mod}\ n , e, n)$ to m. The flip side of this idea is the Discrete Log problem. Let n be a natural number for which we have a primitive root g. Let $1<x<n$ be a natural number that is coprime to n. The Discrete Log problem asks for the determination of an integer $0< l < \phi (n)$ such that $x \equiv g^l \ \mathrm {mod}\ n$.

In the ElGamal Cryptosystem, Azadeh picks a large prime p, a primitive root g modulo p, a random number l, with $1< l < p-1$, and computes $e=g^l \ \mathrm {mod}\ n$. Azadeh’s public key is (p, g, e) which she publishes. She keeps l secret. Behnam wants to send a message m to Azadeh. Benham picks a random integer u, $1< u < p-1$, and computes $x : = g^u \ \mathrm {mod}\ p$, and $y:= m\cdot e^u \ \mathrm {mod}\ p$. Behnam sends the pair (x, y) over a public channel to Azadeh. Azadeh recovers m by computing

$$ y \cdot x^{-l} \equiv m \cdot (g^l)^u \cdot (g^u)^{-l}\equiv m \mod p. $$

We refer the reader to [53], especially Ch. 6 for RSA and Ch. 7 for ElGamal.

1.5 Primitive roots and Artin’s conjecture

The notion of the order of a modulo n made an appearance in Gauss’s book [21, articles 315-317], when he considered the decimal expansion of 1 / p for a prime number p, $p \ne 2, 5$. In this case, the fraction 1 / p is purely periodic and its period is equal to $o_{p}(10)$. In general, we saw in this chapter that if m, n are natural numbers with $\gcd (m, n)=1$, then the base n expansion of 1 / m is purely periodic with minimal period equal to $o_m(n)$. In particular the minimal period is at most equal to $\phi (m)$. In the case where $m=p$ is a prime number, $\phi (p) = p-1$. The following is a natural question: For a natural number n, are there infinitely many prime numbers p such that the base n-expansion of 1 / p has period $p-1$? Note that in this case n will have to be a primitive root modulo p. While the answer to the question is expected to be yes, it is not known for any n, not even $n=10$.

Conjecture 2.60

(Artin 1927). Fix an integer $g \ne -1, 0, 1$ which is not a perfect square. Then there are infinitely many primes p such that g is a primitive root modulo p.

In fact, Artin conjectured an asymptotic formula for $\# \{ p \text { prime} \mid p \le X, o_p(g) = p-1 \}$ of the form $\delta (g) X/\log X$ as $X \rightarrow \infty$, for some constant $\delta (g)>0$. Artin gave a heuristic argument to derive a formula for $\delta (g)$; however, in 1957 Derrick and Emma Lehmer observed that Artin’s predicted formula did not match numerical data. Artin was then able to pinpoint the error in the original heuristic reasoning and corrected the prediction. In 1967 Hooley [81] gave a proof of the predicted asymptotic formula which relied on some version of Riemann’s Hypothesis, not yet proved; see Notes to Chapter 13. See Murty’s expository article [89] for an accessible account of the progress made toward the conjecture up until the time of its publication. For a more up-to-date report on the conjecture and the methods and techniques used in its study, see Moree’s survey [88].