Integral solutions to the Pythagorean Equation

Abstract

In this chapter we present two different methods to find the solutions of the Pythagorean Equation, one algebraic and one geometric. We then apply the geometric method to find solutions of some other equations.

Appendices

Exercises

• 3.1 For an integer D, find the integral solutions of Pell’s Equation (3.4).

• 3.2 Find the rational solutions to $x^2 - y^2=1$ by writing $x-y = m/n$ and $x+y = n/m$.

• 3.3 Find every integral solution of the equation

  $$ a^2 + b^2 + c^2 = d^2. $$

• 3.4 Prove that the only integral solution to the equation $x^2 + y^2 + z^2 = 2 xyz$ is $x=y=z=0$.

• 3.5 Find all the rational solutions of $x^2 + y^2 = z^2 + t^2$.

• 3.6 Show that for all natural numbers n, the equation $x^2 - y^2 = n^3$ is solvable in integers x, y. Determine the number of solutions if n is odd.

• 3.7 Show that the equation

  $$ x^2 + (x+1)^2 + (x+2)^2 + (x+3)^2 + (x+4)^2 = y^2 $$

  has no solutions in integers $x, y \in \mathbb Z$.

• 3.8 Find all the solutions of the equation

  $$ 3(x^2 + y^2) + 2xy = 664 $$

  in integers x, y.

• 3.9 Show that for every $t \in \mathbb Z$ the triple

  $$ (x, y, z) = (9t^4, 1-9t^3, 3t - 9t^4) $$

  satisfies

  $$ x^3 + y^3 + z^3 = 1. $$

  Also verify that for each $t \in \mathbb Z$

  $$ (x, y , z) = (1 + 6 t^3 , 1-6t^3, -6t^3) $$

  is a solution of the equation $x^3 + y^3 + z^3 = 2$. Show that the equation $x^3 + y^3 + z^3 = 4$ has no solutions in $\mathbb Z$. It is in general not known how to solve equations of the form $x^3 + y^3 + z^3 =n$ with $x, y, z \in \mathbb Z$.

• 3.10 Find all integral right triangles whose hypotenuse is a square.

• 3.11 Find all right triangles one of whose legs is a square.

• 3.12 Find all primitive right triangles with square perimeter.

• 3.13 Show that for every $n \in \mathbb N$, there are at least n distinct primitive right triangles which share a leg.

• 3.14 Show that for every $n \in \mathbb N$, there are at least n distinct primitive right triangles which share their hypotenuse.

• 3.15 Find all integral right triangles whose side lengths form an arithmetic progression.

• 3.16 Show that for every n there are n points in the plane, not all of which are on a straight line, such that the distance between every two of them is an integer. How about infinitely many points?

• 3.17 Show that for every Pythagorean triple (u, v, w) we have

  $$ (uv)^4 + (vw)^4 + (wu)^4 = (w^4-u^2v^2)^2. $$

  Conclude that the equation

  $$ x^4 + y^4 + z^4 = t^2 $$

  has infinitely many solutions in integers x, y, z, t such that $\gcd (x, y, z) = 1$.

• 3.18 Solve the system of Diophantine equations

  $$ {\left\{ \begin{array}{ll} x^2 + t = u^2, \\ x^2 - t = v^2. \end{array}\right. } $$

• 3.19 Verify that the points (1, 0) and (0, 2) satisfy the equation

  $$ y^2 = x^3 - 5x + 4. $$

  Use the geometric method of this chapter to find more solutions.

• 3.20 Verify that the point $(-3, 9)$ satisfies the equation $y^2 = x^3 - 36 x$. Use this point to produce more solutions.

• 3.21 Use infinite descent to show that there is no rational number $\gamma$ such that $\gamma ^2 = 2$.

• 3.22 Show that there are no non-zero integral solutions to the following equations:

  • a. $2x^4 - 2y^4 = z^2$;

  • b. $x^4 + 2 y^4 =z^2$;

  • c. $x^4 - y^4 = 2z^2$;

  • d. $8 x^4 - y^4 =z^2$.

• 3.23 Show that the only solutions to $x^4 + y^4 = 2z^2$ in integers are $z = \pm x^2$ and $|y|=|x|$.

• 3.24 ($\maltese$) Find the number of solutions (x, y, z) in integers of the equation $x^2 - 5 y^2 = z^2$ with $|x|, |y|, |z| < 1000$.

• 3.25 ($\maltese$) Find 25 pairs of integers (x, y) such that $x^2 - 2 y^2 = 1$. You might want to use Equation (3.7) of the Notes.

• 3.26 ($\maltese$) Find ten pairs of rational numbers (x, y) such that $y^2 = x^3 + 3$.

Notes

1.1 Pell’s Equation

Traditionally, Pell’s Equation is Equation (3.3) with the extra assumption that x, y are integers. The equation

$$ x^2 - Dy^2 = -1, $$

too, is called Pell’s Equation. Calling any of these equations Pell’s Equation is a famous mischaracterization by Euler. Historically these equations were of interest to mathematicians for hundreds of years before Euler and his contemporaries; see, for example, [27, Ch. 2]. This last reference states that in 628 the great Indian mathematician Brahmagupta (598–670 CE) discovered the identity

$$ (a^2 - D b^2)(p^2 - D q^2) = (ap + D bq)^2 - D (aq + bp)^2. $$

An immediate consequence of this fact is the remarkable statement that if Pell’s Equation $x^2 - D y^2 = \pm 1$ has a non-trivial integral solution, i.e., one where $y \ne 0$, it will have infinitely many integral solutions. In fact, let $(x_1, y_1)$ be the solution of the equation

$$ x^2 - D y^2 = \pm 1, $$

with $x_1, y_1 >0$, and $x_1$ the smallest possible. We call $(x_1, y_1)$ the fundamental solution. Then, there are two possibilities:

1. If $x_1^2 - D y_1^2 = +1$, then the equation $x^2 - D y^2 = -1$ has no solutions. Furthermore, every solution of the equation $x^2 - D y^2 = +1$ is of the form $(\pm x_N, \pm y_N)$ with

$$\begin{aligned} x_N + \sqrt{D} y_N = (x_1 + \sqrt{D} y_1)^N \end{aligned}$$

(3.7)

for some $N \in \mathbb Z$.

2. If $x_1^2 - D y_1^2 = -1$, then the equation $x^2 - D y^2 = -1$ has solutions $(\pm x_N, \pm y_N)$ determined by Equation (3.7) with $N \in \mathbb Z$ odd. The solutions of $x^2 - D y^2 = +1$ are the pairs $(\pm x_N, \pm y_N)$ with $N \in \mathbb Z$ even.

For example when $D=2$, the fundamental solution to $x^2 - 2 y^2 = \pm 1$ is (1, 1) which satisfies $1^2 - 2 \cdot 1^2 = -1$. If $N=2$, we compute

$$ (1+ \sqrt{2})^2 = 3 + 2 \sqrt{2}, $$

and it is clear that (3, 2) satisfies $3^2 - 2. 2^2 =+1$. If $N=3$,

$$ (1+\sqrt{2})^3 = 7 + 5 \sqrt{2}, $$

and $7^2 - 2 \cdot 5^2 = -1$.

Because of these observations, finding the solutions of Pell’s Equation reduces to the search for the fundamental solution. Note that even though the fundamental solution $(x_1,y_1)$ is the smallest solution of the equation, it does not have to be small in any reasonable sense. For example, the smallest solution of $x^2 - 61 y^2 =1$ is $(x, y) = (1766319049, 226153980)$. The most effective way to write down the fundamental solution is via continued fractions. This method was originally discovered by the Indian mathematicians Jayadeva (c. 950–$\sim$ 1000 CE) and Bhaskara (c. 1114 –1185 CE) who completed Brahmagupta’s method, though they gave no formal proof of this. The formal proof was provided by Lagrange in the 18th century. For a complete history of this subject we refer the reader to Weil’s book [57]. For details of this method, see [27, Ch. 3] or [33, Ch. 7], especially §7.6.3.

1.2 Elliptic curves

The cubic curves considered in §3.3 are called elliptic curves. These are some of the most important objects in all of mathematics, and they have been the subject of intense research for a few hundred years. The genesis of the adjective in the name of these curves goes back to 17th and 18th centuries. Let us briefly explain the connection; see [92] for details and references.

Consider the ellipse with the equation

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, $$

with $a > b$. It is an easy integration exercise to show that the area of the ellipse is equal to $\pi a b$. Now suppose we want to compute the perimeter of the ellipse.

Fig. 3.4

Ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

A parametrization for the ellipse is given by

$$ {\left\{ \begin{array}{ll} x = a \sin t\\ y = b \cos t \end{array}\right. } \quad 0 \le t \le 2 \pi . $$

By the arc length formula, itself an application of the Pythagorean Theorem, the perimeter $\ell$ of the ellipse is equal to

$$ \ell = \int _{0}^{2 \pi } \sqrt{ \left( \frac{dx}{dt}\right) ^2 + \left( \frac{dy}{dt}\right) ^2} \, dt $$

$$ = 4 \int _0^{\pi /2} \sqrt{a^2 \cos ^2 t + b^2 \sin ^2 t} \, dt $$

$$ = 4 a \int _0^{\pi /2} \sqrt{1-k^2 \sin ^2 t} \, dt, $$

with $k^2 = 1-b^2/a^2$. A change of variables with $u = \sin t$ gives

$$\begin{aligned} \ell = 4 a \int _0^1 \frac{\sqrt{1 - k^2 u^2}}{\sqrt{1-u^2}} \, du. \end{aligned}$$

(3.8)

This is a special value of an elliptic integral of second kind. In general, elliptic integrals of the second kind are defined as follows: For $0 \le w \le 1$ we define

$$ E(w) = \int _0^w \frac{\sqrt{1 - k^2 u^2}}{\sqrt{1-u^2}} \, du. $$

Elliptic integrals are in general not expressible in terms of elementary functions. Because of their many applications in mathematical physics these types of integrals attracted a lot of attention starting in the 18th century. It was Abel in the 19th century who realized that the correct object of study is the inverse of the function E. The motivation for this point of view is the $\sin ^{-1}$ integral: We know

$$ \sin ^{-1}w = \int _0^w \frac{du}{\sqrt{1-u^2}}, $$

but the more natural function to work with is the inverse function of $\sin ^{-1}$, the ubiquitous sine. Going back to Equation (3.8), we make one more change of variable $z=1-k^2 u^2$ to obtain

$$ \ell = 2a \int _{\lambda }^1 \frac{z}{\sqrt{z(1-z)(z-\lambda )}} \, dz, $$

with $\lambda = 1-k^2$. Upon setting $z = q+\frac{1+\lambda }{3}$ the integral transforms to

$$ \ell = 2a \int _{\frac{2\lambda -1}{3}}^{\frac{2-\lambda }{3}} \frac{q + \frac{1 + \lambda }{3}}{\sqrt{-q^3 + \frac{1}{3} (\lambda ^2 + \lambda -1) q + \frac{1}{27}(2 \lambda ^3 + 3 \lambda ^2 - 3\lambda - 2)}} \, d q. $$

Finally (!), set $q = - \root 3 \of {4} v$ to get

$$ \ell = 2\root 3 \of {4} a \int ^{-\frac{2\lambda -1}{3\root 4 \of {3}}}_{-\frac{2-\lambda }{3\root 3 \of {4}}} \frac{- \root 3 \of {4} v + \frac{1 + \lambda }{3}}{\sqrt{4 v^3 - \frac{\root 3 \of {4}}{3} (\lambda ^2 + \lambda -1) v + \frac{1}{27}(2 \lambda ^3 + 3 \lambda ^2 - 3\lambda - 2)}} \, d v. $$

Let

$$ g_2 = \frac{\root 3 \of {4}}{3} (\lambda ^2 + \lambda -1), $$

and

$$ g_3 = - \frac{1}{27}(2 \lambda ^3 + 3 \lambda ^2 - 3\lambda - 2). $$

Karl Weierstrass defined a function $\wp (u)$ with the property that

$$ u = \int _{\wp (u)}^\infty \frac{dv}{\sqrt{4 z^3 - g_2 z - g_3}}. $$

So clearly $\ell$ is related to the function $\wp$. A remarkable property of the $\wp$-function is that it satisfies the functional equation

$$ \wp '(z)^2 = 4 \wp (z)^3 - g_2 \wp (z) - g_3, $$

i.e., the point $(\wp (z), \wp '(z))$ lies on the curve

$$\begin{aligned} y^2 = 4 x^3 - g_2 x - g_3. \end{aligned}$$

(3.9)

In fact, the points $(\wp (z), \wp '(z))$ give a full parametrization for the points with complex coordinates on the curve. Furthermore,

$$ \wp (u+v) = - \wp (u) - \wp (v) + \frac{1}{4} \left( \frac{\wp '(u) - \wp '(v)}{\wp (u) - \wp (v) }\right) ^2, $$

and

$$ \wp (-u) = \wp (u), \quad \wp '(-u) = - \wp '(u). $$

These formulae have an interesting interpretation for the points on the curve. We define a group law $\oplus$ on the set of points of the curve as follows: For a point A on the curve, define $-A$ to be the reflection of A with respect to the x-axis; for three points A, B, C, we say $A \oplus B = C$ if A, B, and $-C$ are colinear; and O, the identity point, is the point at infinity in the direction of the y axis, i.e. $A \oplus (-A)$ for any point A.

The work we did in §3.3 shows that if $g_2, g_3 \in \mathbb Q$, then the $\oplus$ of any two points with coordinates in $\mathbb Q$ will be again a point with coordinates in $\mathbb Q$. Clearly, also, for a point A with rational coordinates, $-A$ will have rational coordinates. This means that the collection of points on the curve with rational coordinates forms a group. It is a truly surprising fact that, by a theorem of Mordell, this group is finitely generated. We refer the reader to [48, Ch. 3] or [47, Ch. VIII] for details.

1.3 Fermat’s Last Theorem

Fermat’s Last Theorem is an esoteric statement with no applications as such, but despite its obscurity it has given rise to an enormous amount of mathematics. Edwards [19] presents algebraic number theory as it was originally motivated by false proofs of Fermat’s Last Theorem. “The Proof”, a NOVA documentary [114] on Wiles’ work, is an excellent account of the last steps toward the proof. Charles Mozzochi’s endearing photo essay “The Fermat Diary” [36] is a photo album of all those whose works contributed to the proof of the theorem in the last fifty years. Finally, even though it is written for experts, Sir Andrew Wiles’ introduction to his masterful paper in the Annals of Mathematics [110] is a delight to read.

1.4 The abc Conjecture

The abc Conjecture is an easy to state conjecture with many surprising consequences in number theory. The conjecture was formulated by D. W. Masser and J. Oesterlé in the 80’s. This is the statement:

Conjecture 3.12

(The abc Conjecture). If $\varepsilon > 0$, then the number of triples (a, b, c) of coprime natural numbers such that $c = a + b$ and

$$ c > \left( \prod _{p \mid abc} p\right) ^{1 + \varepsilon } $$

is finite.

The conjecture could also be formulated as follows: For every $\varepsilon > 0$, there is a constant $\kappa _\varepsilon >0$ such that for every triple (a, b, c) of coprime natural numbers satisfying $c = a + b$ we have

$$ c \le \kappa _\varepsilon \left( \prod _{p \mid abc} p\right) ^{1 + \varepsilon }. $$

To see a quick application, let us apply the abc Conjecture to Fermat’s Last Theorem. Suppose we have three coprime natural numbers x, y, z such that $x^n + y^n = z^n$. If $\varepsilon >0$ is given, then applying the abc Conjecture with $a = x^n$, $b = y^n$, and $c = z^n$ shows that with the exception of finitely many choices of x, y, z we have

$$ z^n \le \left( \prod _{p \mid x^n y^n z^n} p\right) ^{1 + \varepsilon } $$

Next, $p \mid x^n y^n z^n$ if and only if $p \mid xyz$. So we have

$$ \prod _{p \mid x^n y^n z^n} p = \prod _{p \mid x y z} p. $$

Now we observe that if n is a natural number, $\prod _{p \mid n} p \le n$. Using this observation we have

$$ \prod _{p \mid x y z} p \le xyz < z^3. $$

In the last step we have used the fact that $x < z$ and $y < z$. Putting everything together, we conclude that except for finitely many choices of x, y, z we have

$$ z^n < (z^3)^{1+\varepsilon } = z^{3(1 + \varepsilon )}. $$

This implies that $n < 3(1+\varepsilon )$. Since the choice of $\varepsilon$ is arbitrary, this means $n \le 3$. What we have proved is the following:

Corollary 3.13

(Assuming abc Conjecture). For each $n > 3$, Fermat’s equation $x^n + y^n = z^n$ has at most finitely many solutions in coprime natural numbers x, y, z.

The statement of the abc Conjecture is ineffective. This means that for a fixed $\varepsilon >0$ the conjecture does not provide any estimate for the number or the size of triples (a, b, c) satisfying the conditions of the conjecture. There are several explicit versions of the abc Conjecture in literature. Here we state one of these explicit conjectures which is due to Alan Baker [63].

To state Baker’s abc Conjecture we need some notation. For a natural number n, we set $\mathrm {rad}\,(n)$ to be the product of the prime divisors of n, i.e.,

$$ \mathrm {rad}\,(n) = \prod _{p \mid n} p. $$

For example, $\mathrm {rad}\,(1)=1$, $\mathrm {rad}\,(12) = 2 \times 3 = 6$ and $\mathrm {rad}\,(25) = 5$. We also let $\omega (n) = \sum _{p \mid n} 1$, i.e., the number of prime divisors of n. With this definition we have $\omega (1) = 0$, $\omega (12) = 2$, $\omega (25) =1$. Using this notation, the original abc Conjecture asserts that for $\varepsilon > 0$, there is $\kappa _\varepsilon > 0$ such that for a triple (a, b, c) of coprime natural numbers, we have

$$ c < \kappa _\varepsilon (\mathrm {rad}\,(abc))^{1 + \varepsilon }. $$

Conjecture 3.14

(Baker’s abc Conjecture). Let (a, b, c) be a triple of coprime natural numbers such that $c = a + b$. Let $N = \mathrm {rad}\,(abc)$ and $r = \omega (N)$. Then

$$ c < \frac{6}{5} N \frac{(\log N)^r}{r!}. $$

We leave it to the reader to verify that Baker’s abc Conjecture in fact implies the abc Conjecture. The papers by Granville and Tucker [77] and Waldschmidt [107] outline various applications of the abc Conjecture. In April of 2012, Shinichi Mochizuki of Kyoto University announced a proof of the abc Conjecture occupying hundreds of pages. At the time of this writing it is still not known if Mochizuki’s proof is correct, and for that reason the abc Conjecture is still considered open.