Chapter 6
Other Topics

6.1 Introduction

The theory of numbers is a vast discipline and no single volume can adequately treat of it in all of its phases. A short book can serve only as an introduction; but where the field is so vast such an introduction is much needed. That is the end which the present volume is intended to serve; and it will best accomplish this end if, in addition to the detailed theory already developed, some account is given of the various directions in which the matter might be carried further.

To do even this properly it is necessary to limit the number of subjects considered. Consequently we shall at once lay aside many topics of interest which would find a place in an exhaustive treatise. We shall say nothing, for instance, about the vast domain of algebraic numbers, even though this is one of the most fascinating subjects in the whole field of mathematics. Consequently, we shall not refer to any of the extensive theory connected with the division of the circle into equal parts. Again, we shall leave unmentioned many topics connected with the theory of positive integers; such, for instance, is the frequency of prime numbers in the ordered system of integers—a subject which contains in itself an extensive and elegant theory.

In §§6.2–6.5 we shall speak briefly of each of the following topics: theory of quadratic residues, Galois imaginaries, arithmetic forms, analytical theory of numbers. Each of these alone would require a considerable volume for its proper development. All that we can do is to indicate the nature of the problem in each case and in some cases to give a few of the fundamental results.

In the remaining three sections we shall give a brief introduction to the theory of Diophantine equations, developing some of the more elementary properties of certain special cases. We shall carry this far enough to indicate the nature of the problem connected with the now famous Last Theorem of Fermat. The earlier sections of this chapter are not required as a preliminary to reading this latter part.

6.2 Theory of Quadratic Residues

Let $a$ and $m$ be any two relatively prime integers. In §4.9 we agreed to say that $a$ is a quadratic residue modulo $m$ or a quadratic non-residue modulo $m$ according as the congruence

has or has not a solution. We saw that if $m$ is chosen equal to an odd prime number $p$, then $a$ is a quadratic residue modulo $p$ or a quadratic non-residue modulo $p$ according as

This is known as Euler’s criterion.

It is convenient to employ the Legendre symbol

to denote the quadratic character of $a$ with respect to $p$. This symbol is to have the value $+1$ or the value $-1$ according as $a$ is a quadratic residue modulo $p$ or a quadratic non-residue modulo $p$. We shall now derive some of the fundamental properties of this symbol, understanding always that the numbers in the numerator and the denominator are relatively prime.

From the definition of quadratic residues and non-residues it is obvious that

It is easy to prove in general that

This comes readily from Euler’s criterion. We have to consider the three cases

The method will be sufficiently illustrated by the treatment of the last case. Here we have

Multiplying these two congruences together member by member we have

whence

as was to be proved.

If $m$ is any number prime to $p$ and we write $m$ as the product of factors

where $q\prime ,q\prime \prime ,q\prime \prime \prime ,\dots$ are odd primes, $\alpha$ is zero or a positive integer and $\epsilon$ is $+1$ or $-1$ according as $m$ is positive or negative, we have

as one shows easily by repeated application of relation (2). Obviously,

Hence, it follows from (3) that we can readily determine the quadratic character of $m$ with respect to the odd prime $p$, that is, the value of

provided that we know the value of each of the expressions

where $q$ is an odd prime.

The first of these can be evaluated at once by means of Euler’s criterion; for, we have

and hence

Thus we have the following result: The number $-1$ is a quadratic residue of every prime number of the form $4k+1$ and a quadratic non-residue of every prime number of the form $4k+3$.

The value of the second symbol in (4) is given by the formula

The theorem contained in this equation may be stated in the following words: The number $2$ is a quadratic residue of every prime number of either of the forms $8k+1,8k+7$; it is a quadratic non-residue of every prime number of either of the forms $8k+3,8k+5$.

The proof of this result is not so immediate as that of the preceding one. To evaluate the third expression in (4) is still more difficult. We shall omit the demonstration in both of these cases. For the latter we have the very elegant relation

This equation states the law which connects the quadratic character of $q$ with respect to $p$ with the quadratic character of $p$ with respect to $q$. It is known as the Law of Quadratic Reciprocity. About fifty proofs of it have been given. Its history has been a very interesting one; see Bachmann’s Niedere Zablentheorie, Teil I, pp. 180–318, especially pp. 200–206.

For a further account of this beautiful and interesting subject we refer the reader to Bachmann, loc. cit., and to the memoirs to which this author gives reference.

6.3 Galois Imaginaries

If one is working in the domain of real numbers the equation

has no solution; for there is no real number whose square is $-1$. If, however, one enlarges the “number system” so as to include not only all real numbers but all complex numbers as well, then it is true that every algebraic equation has a root. It is on account of the existence of this theorem for the enlarged domain that much of the general theory of algebra takes the elegant form in which we know it.

The question naturally arises as to whether we can make a similar extension in the case of congruences. The congruence

has no solution, if we employ the term solution in the sense in which we have so far used it. But we may if we choose introduce an imaginary quantity, or mark, $j$ such that

just as in connection with the equation $x^2+1=0$ we would introduce the symbol $i$ having the property expressed by the equation

It is found to be possible to introduce in this way a general set of imaginaries satisfying congruences with prime moduli; and the new quantities or marks have the property of combining according to the laws of algebra.

The quantities so introduced are called Galois imaginaries.

We cannot go into a development of the important theory which is introduced in this way. We shall be content with indicating two directions in which it leads.

In the first place there is the general Galois field theory which is of fundamental importance in the study of certain finite groups. It may be developed from the point of view indicated here. An excellent exposition, along somewhat different lines, is to be found in Dickson’s Linear Groups with an Exposition of the Galois Field Theory.

Again, the whole matter may be looked upon from the geometric point of view. In this way we are led to the general theory of finite geometries, that is, geometries in which there is only a finite number of points. For a development of the ideas which arise here see Veblen and Young’s Projective Geometry and the memoir by Veblen and Bussey in the Transactions of the American Mathematical Society, vol. 7, pp. 241–259.

6.4 Arithmetic Forms

The simplest arithmetic form is $ax+b$ where $a$ and $b$ are fixed integers different from zero and $x$ is a variable integer. By varying $x$ in this case we have the terms of an arithmetic progression. We have already referred to Dirichlet’s celebrated theorem which asserts that the form $ax+b$ has an infinite number of prime values if only $a$ and $b$ are relatively prime. This is an illustration of one type of theorem connected with arithmetic forms in general, namely, those in which it is asserted that numbers of a given form have in addition a given property.

Another type of theorem is illustrated by a result stated in §6.2, provided that we look at that result in the proper way. We saw that the number $2$ is a quadratic residue of every prime of either of the forms $8k+1$ and $8k+7$ and a quadratic non-residue of every prime of either of the forms $8k+3$ and $8k+5$. We may state that result as follows: A given prime number of either of the forms $8k+1$ and $8k+7$ is a divisor of some number of the form $x^2-2$, where $x$ is an integer; no prime number of either of the forms $8k+3$ and $8k+5$ is a divisor of a number of the form $x^2-2$, where $x$ is an integer.

The result just stated is a theorem in a discipline of vast extent, namely, the theory of quadratic forms. Here a large number of questions arise among which are the following: What numbers can be represented in a given form? What is the character of the divisors of a given form? As a special case of the first we have the question as to what numbers can be represented as the sum of three squares. To this category belong also the following two theorems: Every positive integer is the sum of four squares of integers; every prime number of the form $4n+1$ may be represented (and in only one way) as the sum of two squares.

For an extended development of the theory of quadratic forms we refer the reader to Bachmann’s Arithmetik der Quadratischen Formen of which the first part has appeared in a volume of nearly seven hundred pages.

It is clear that one may further extend the theory of arithmetic forms by investigating the properties of those of the third and higher degrees. Naturally the development of this subject has not been carried so far as that of quadratic forms; but there is a considerable number of memoirs devoted to various parts of this extensive field, and especially to the consideration of various special forms.

Probably the most interesting of these special forms are the following:

where $\alpha$ and $\beta$ are relatively prime integers, or, more generally, where $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2-ux+v=0$ where $u$ and $v$ are relatively prime integers. A development of the theory of these forms has been given by the present author in a memoir published in 1913 in the Annals of Mathematics, vol. 13, pp. 30–70.

6.5 Analytical theory of numbers

Let us consider the function

It is clear that we have

where and (for $s$ greater than $0$) is the number of ways in which the positive integer $s$ may be separated into like or distinct summands each of which is a power of $2$.

We have readily

whence

as one readily verifies by equating coefficients of like powers of $x$. From this we have in particular

Thus in (A) we have recurrence relations by means of which we may readily reckon out the values of the number theoretic function . Thus we may determine the number of ways in which a given positive integer $s$ may be represented as a sum of powers of $2$.

We have given this example as an elementary illustration of the analytical theory of numbers, that is, of that part of the theory of numbers in which one employs (as above) the theory of a continuous variable or some analogous theory in order to derive properties of sets of integers. This general subject has been developed in several directions. For a systematic account of it the reader is referred to Bachmann’s Analytische Zahlentheorie.

6.6 Diophantine equations

If is a polynomial in the variables $x,y,z,\dots$ with integral coefficients, then the equation

is called a Diophantine equation when we look at it from the point of view of determining the integers (or the positive integers) $x,y,z,\dots$ which satisfy it. Similarly, if we have several such functions , in number less than the number of variables $x,y,z,\dots$, then the set of equations

is said to be a Diophantine system of equations. Any set of integers $x,y,z,\dots$ which satisfies the equation [system] is said to be a solution of the equation [system].

We may likewise define Diophantine inequalities by replacing the sign of equality above by the sign of inequality. But little has been done toward developing a theory of Diophantine inequalities. Even for Diophantine equations the theory is in a rather fragmentary state.

In the next two sections we shall illustrate the nature of the ideas and the methods of the theory of Diophantine equations by developing some of the results for two important special cases.

6.7 Pythagorean triangles

DEFINITIONS. If three positive integers $x,y,z$ satisfy the relation

they are said to form a Pythagorean triangle or a numerical right triangle; $z$ is called the hypotenuse of the triangle and $x$ and $y$ are called its legs. The area of the triangle is said to be $\frac{1}{2}xy$.

We shall determine the general form of the integers $x$, $y$, $z$, such that equation (1) may be satisfied. Let us denote by $\nu$ the greatest common divisor of $x$ and $y$ in a particular solution of (1). Then $\nu$ is a divisor of $z$ and we may write

Substituting these values in (1) and reducing we have

where $u,v,w$ are obviously prime each to each, since $u$ and $v$ have the greatest common divisor $1$.

Now an odd square is of the form $4k+1$. Hence the sum of two odd squares is divisible by $2$ but not by $4$; and therefore the sum of two odd squares cannot be a square. Hence one of the numbers $u$, $v$ is even. Suppose that $u$ is even and write equation (2) in the form

Every common divisor of $w-v$ and $w+v$ is a divisor of their difference $2v$. Therefore, since $w$ and $v$ are relatively prime, it follows that $2$ is the greatest common divisor of $w-v$ and $w+v$. Then from (3) we see that each of these numbers is twice a square, so that we may write

where $a$ and $b$ are relatively prime integers. From these two equations and equation (3) we have

Since $u$ and $v$ are relatively prime it is evident that one of the numbers $a$, $b$ is even and the other odd.

The forms of $u$, $v$, $w$ given in (4) are necessary in order that (2) may be satisfied. A direct substitution in (2) shows that this equation is indeed satisfied by these values. Hence we have in (4) the general solution of (2) where $u$ is restricted to be even. A similar solution would be obtained if $v$ were restricted to be even. Therefore the general solution of (1) is

where $a$, $b$, $\nu$ are arbitrary integers except that $a$ and $b$ are relatively prime and one of them is even and the other odd.

By means of this general solution of (1) we shall now prove the following theorem:

I. There do not exist integers $m$, $n$, $p$, $q$, all different from zero, such that

It is obvious that an equivalent theorem is the following:

II. There do not exist integers $m$, $n$, $p$, $q$, all different from zero such that

Obviously, we may without loss of generality take $m$, $n$, $p$, $q$ to be positive; and this we do.

The method of proof is to assume the existence of integers satisfying equations (5) and (6) and to show that we are thus led to a contradiction. The argument we give is an illustration of Fermat’s famous method of “infinite descent.”

If any two of the numbers $p$, $q$, $m$, $n$ have a common prime factor $t$, it follows at once from (5) and (6) that all four of them have this factor. For, consider an equation in (5) or in (6) in which these two numbers occur; this equation contains a third number, and it is readily seen that this third number is divisible by $t$. Then from one of the equations containing the fourth number it follows that this fourth number is divisible by $t$. Now let us divide each equation of system (6) through by $t^2$; the resulting system is of the same form as (6). If any two numbers in this resulting system have a common prime factor $t_1$, we may divide through by $t_1^2$; and so on. Hence if a pair of simultaneous equations (6) exists then there exists a pair of equations of the same form in which no two of the numbers $m$, $n$, $p$, $q$ have a common factor other than unity. Let this system of equations be

From the first equation in (7) it follows that $p_1$ and $q_1$ are both even or both odd; and, since they are relatively prime, it follows that they are both odd. Evidently $p_1>q_1$. Then we may write

where $\alpha$ is a positive integer. If we substitute this value of $p_1$ in the first equation of (7), the result may readily be put in the form

Since $q_1$ and $m_1$ have no common prime factor it is easy to see from this equation that $\alpha$ is prime to both $q_1$ and $m_1$, and hence that no two of the numbers $q_1+\alpha ,\alpha ,m_1$ have a common factor.

Now we have seen that if $a$, $b$, $c$ are positive integers no two of which have a common prime factor, while

then there exist relatively prime integers $r$ and $s$, $r>s$, such that

or

Hence from (8) we see that we may write

or

In either case we have

If we substitute in the second equation of (7) and divide by 2 we have

From this equation and the fact that $r$ and $s$ are relatively prime it follows at once that $r$, $s$, $r^2-s^2$ are all square numbers; say,

Now $r-s$ and $r+s$ can have no common factor other than 1 or 2; hence from

we see that either

or

And if it is the latter case which arises, then

Hence, assuming equations of the form (6) we are led either to equations (11) or to equations (12); that is, we are led to new equations of the form with which we started. Let us write the equations thus:

that is, system (13) is identical with that one of systems (11), (12) which actually arises.

Now from (9) and (10) and the relations $p_1=q_1+2\alpha ,r>s$, we see that

Hence $u<p_1$. Also,

Hence $w_1<p_1$. Since $u$ and $w_1$ are both less than $p_1$ it follows that $p_2$ is less than $p_1$. Hence, obviously, $p_2<p$. Moreover, it is clear that all the numbers $p_2,q_2,m_2,n_2$ are different from zero.

From these results we have the following conclusion: If we assume a system of the form (6) we are led to a new system (13) of the same form; and in the new system $p_2$ is less than $p$.

Now if we start with (13) and carry out a similar argument we shall be led to a new system

with the relation $p_3<p_2$, starting from this last system we shall be led to a new one of the same form, with a similar relation of inequality; and so on ad infinitum. But, since there is only a finite number of positive integers less than the given positive integer $p$ this is impossible. We are thus led to a contradiction; whence we conclude at once to the truth of II and likewise of I.

By means of theorems I and II we may readily prove the following theorem:

III. The area of a numerical right triangle is never a square number.

Let the sides and hypotenuse of a numerical right triangle be $u,v,w$, respectively. The area of this triangle is $\frac{1}{2}uv$. If we assume this to be a square number $t^2$ we shall have the following simultaneous Diophantine equations

We shall prove our theorem by showing that the assumption of such a system leads to a contradiction.

If any two of the numbers $u,v,w$ have a common prime factor $p$ then the remaining one also has this factor, as one sees readily from the first equation in (14). From the second equation in (14) it follows that $t$ also has the same factor. Then if we put $u=p{u}_1,v=p{v}_1,w=p{w}_1,t=p{t}_1$, we have

a system of the same form as (14). It is clear that we may start with this new system and proceed in the same manner as before, and so on, until we arrive at a system

where $ū$, $\bar{v}$, $\bar{w}$ are prime each to each.

Now the general solution of the first equation (15) may be written in one of the forms

Then from the second equation in (15) we have

It is easy to see that no two of the numbers $a$, $b$, $a-b$, $a+b$ in the last member of this equation have a common factor; for, if so, $ū$ and $\bar{v}$ would have a common factor, contrary to hypothesis. Hence each of these four numbers is a square. That is, we have equations of the form

whence

But, according to theorem I, no such system of equations can exist. That is, the assumption of equations (14) leads to a contradiction. Hence the theorem follows as stated above.

6.8 The Equation $x^n+y^n=z^n$.

The following theorem, which is commonly known as Fermat’s Last Theorem, was stated without proof by Fermat in the seventeenth century:

If n is an integer greater than 2 there do not exist integers x, y, z, all different from zero, such that

No general proof of this theorem has yet been given. For various special values of $n$ the proof has been found; in particular, for every value of $n$ not greater than 100.

In the study of equation (1) it is convenient to make some preliminary reductions. If there exists any particular solution of (1) there exists also a solution in which $x$, $y$, $z$ are prime each to each, as one may show readily by the method employed in the first part of §6.7. Hence in proving the impossibility of equation (1) it is sufficient to treat only the case in which $x$, $y$, $z$ are prime each to each.

Again, since $n$ is greater than 2 it must contain the factor 4 or an odd prime factor $p$. If $n$ contains the factor $p$ we write $n=mp$, whence we have

If $n$ contains the factor 4 we write $n=4m$, whence we have

From this we see that in order to prove the impossibility of (1) in general it is sufficient to prove it for the special cases when $n$ is 4 and when $n$ is an odd prime $p$. For the latter case the proof has not been found. For the former case we give a proof below. The theorem may be stated as follows:

I. There are no integers $x,y,z$, all different from zero, such that

This is obviously a special case of the more general theorem:

II. There are no integers $p$, $q$, $\alpha$, all different from zero, such that

The latter theorem is readily proved by means of theorem III of §6.7. For, if we assume an equation of the form (2), we have

But, obviously,

Now, from (3) we see that the numerical right triangle determined by (4) has its area equal to the square number $p^2{q}^2{\alpha }^2$. But this is impossible. Hence no equation of the form (2) exists.

1.

Show that the equation ${\alpha }^4+4{\beta }^4={\gamma }^2$ is impossible in integers $\alpha$, $\beta$, $\gamma$ all of which are different from zero.

2.

Show that the system $p^2-q^2=k{m}^2$, $p^2+q^2=k{n}^2$ impossible in integers $p$, $q$, $k$, $m$, $n$, all of which are different from zero.

3*.

Show that neither of the equations $m^4-4n^4=\pm t^2$ is possible in integers $m$, $n$, $t$, all of which are different from zero.

4*.

Prove that the area of a numerical right triangle is not twice a square number.

5*.

Prove that the equation $m^4+n^4={\alpha }^2$ is not possible in integers $m$, $n$, $\alpha$ all of which are different from zero.

6*.

In the numerical right triangle $a^2+b^2=c^2$, not more than one of the numbers $a$, $b$, $c$ is a square.

7.

Prove that the equation $x^{2k}+y^{2k}=z^{2k}$ implies an equation of the form $m^k+n^k=2^{k-2}{t}^k$.

8.

Find the general solution in integers of the equation $x^2+2y^2=t^2$.

9.

Find the general solution in integers of the equation $x^2+y^2=z^4$.

10.

Obtain solutions of each of the following Diophantine equations:

$$\begin{array}{llll}\hfill x^3+y^3+z^3& =2t^3,& \hfill & \\ \hfill x^3+2y^3+3z^3& =t^3,& \hfill & \\ \hfill x^4+y^4+4z^4& =t^4,& \hfill & \\ \hfill x^4+y^4+z^4& =2t^4.& \hfill & \end{array}$$