What numbers are the edges of a right triangle?

Abstract

In this chapter we study numbers that appear as the side lengths of primitive right triangles. We use rings of Gaussian integers to prove our main theorems. We give a quick review of the basic properties of the ring of Gaussian integers. For a more thorough exposition we refer the reader to the classical text by Sierpinski [46] or Conrad [69].

Appendices

Exercises

• 5.1 Write the following numbers as a product of irreducibles of $\mathbb Z[i]$:

  • a. 56;

  • b. $4 + 6i$;

  • c. $3 + 5i$;

  • d. $9 +i$;

  • e. $7 +24i$.

• 5.2 Compute $\gcd (6-17i, 18+i)$.

• 5.3 Solve the equation $x+y+z=xyz=1$ in Gaussian integers.

• 5.4 Determine all irreducible elements with norm less than 100.

• 5.5 Devise a test to decide whether $x+iy$ is the square of a Gaussian integer.

• 5.6 Determine all Gaussian integers which are the sum of two squares of Gaussian integers.

• 5.7 Show that a Gaussian integer $x + iy$ is a sum of the squares of three Gaussian integers if and only if y is even.

• 5.8 What can you say about right triangles with integral sides such that the legs differ by 1? What if the difference is a fixed number d?

• 5.9 What can you say about right triangles with integral sides such that the sum of the legs is a fixed number s?

• 5.10 What can you say about a right triangle with integral sides such that the perimeter and the hypotenuse are squares?

• 5.11 Write 45305 as a sum of two squares.

• 5.12 For a natural number n, show that if the equation $n = x^2 + y^2$, $x, y > 0$, $2 \mid x$, has more than one solution, then n is not prime.

• 5.13 Find a formula for the number of primitive right triangles with a leg equal to a number n in terms of the divisors of n.

• 5.14 Prove the following result of Gauss [16, page 172]: Every hypotenuse composed of k distinct primes belongs to

  $$ \left[ \frac{k}{1}\right] + 2 \left[ \frac{k}{2}\right] + 2^2 \left[ \frac{k}{3}\right] + \dots + 2^{k-1}\left[ \frac{k}{k}\right] $$

  different right triangles. Of these triangles, $2^{k-1}$ are primitive.

• 5.15 ($\maltese$) Determine if 31897485916040 is a sum of two squares. If it is, determine in how many ways.

Notes

1.1 Primes of special forms

The problem of deciding which polynomials produce prime numbers goes back centuries. Euler made the famously wrong claim that the polynomial $f(x) = x^2 - x + 41$ has the property that f(n) is a prime number for every integer n. The values $f(0), f(1), f(2), f(3), \dots , f(40)$ are all prime, though f(41) is clearly not. We will see in Exercise 6.2 that there are no non-constant polynomials f(x) such that f(n) is a prime number for every integral value of n. Despite this rather disappointing statement, one could still ask whether there are polynomials that produce infinitely many primes. The answer is a definite yes. For example, every odd prime is either congruent to 1 modulo 4 or congruent to 3 modulo 4. This means that at least one of the polynomials $4x+1$ or $4x+3$ produces infinitely many primes. We will see in Chapter 6 that in fact both of these polynomials capture infinitely many primes.

For a general polynomial of degree 1, one can effectively decide whether the polynomial produces infinitely many primes. Suppose $f(x) = ax + b$ with $a, b \in \mathbb Z$. If $\gcd (a, b) = d> 1$, then the polynomial has no chance of producing infinitely many primes. It turns out that this is the only obstruction. The following is a celebrated theorem of Dirichlet [2, Theorem 7.3]:

Theorem 5.11.

If a, b are integers with $b>0$, and $\gcd (a, b) =1$, then the arithmetic progression

$$ a, a+b, a+2b, a + 3b, a+4b, \dots $$

contains infinitely many prime numbers.

Unfortunately, we do not know an algebraic/elementary proof of this fact. The standard proofs of Dirichlet’s Theorem use complex analysis and, though not terribly hard, are beyond the scope of this small volume. We give several examples of this theorem in Chapter 6. We also present the proof of an important special case in Exercise 6.22.

For polynomials of degree larger than 1 the situation is considerably more complicated. For example, in 1912, Landau conjectured that the polynomial $f(x) = x^2 + 1$ produces infinitely many primes. At the time of this writing it is still not known if Landau’s Conjecture is true. The best result in this direction is due to Henryk Iwaniec who in 1978 proved that there are infinitely many integers n such that $n^2 + 1$ is the product of at most two prime numbers.

If we consider quadratic polynomials in more than one variable, then the situation is better understood. Theorem 5.7 gives a linear necessary and sufficient condition for the representability of a prime by a quadratic expression—namely, that an odd prime p is representable by the quadratic form $x^2 + y^2$ if and only if p is of the form $4k+1$, implying that there are infinitely many primes of the form $x^2 + y^2$. There are other results of similar nature for representability of prime numbers by polynomials of the form $x^2 + ny^2$ dating back to, at least, Fermat and Euler. For example, a prime is of the form $x^2 + 2y^2$ for integers x, y if and only if $p \equiv 1, 3 \ \mathrm {mod}\ 8$. See Cox [14] for an in-depth study of primes that are representable by quadratic forms in two variables.

1.2 Algebraic number theory

We understand the phrase algebraic number theory in two different, but related, ways. The first one is algebraic number/theory, as in number theory done using algebraic methods, and the second one is algebraic/number theory as in the theory of algebraic numbers. In terms of the first interpretation, Chinese Remainder Theorem 2.24 is really a statement about ideals in a general ring; Euler’s Theorem 2.31 is a special case of Lagrange’s theorem in finite group theory; Lemma 2.49 is a consequence of the statement that every finite subgroup of a field is cyclic. What we did in this chapter with Gaussian integers is part of the second interpretation, and as we saw in this chapter we used our results on Gaussian integers to prove a statement about ordinary integers. Another example is our results from Appendix B which we will use in our proof of the Law of Quadratic Reciprocity in Chapter 7.

An important result in this chapter is Theorem 5.10 which establishes unique factorization in Gaussian integers. Unfortunately, this uniqueness of factorization fails in general number rings. A famous example is the ring $\mathbb Z[\sqrt{-5}] = \{ x + y \sqrt{-5} \mid x, y \in \mathbb Z\}$. We have $6 = 2 \cdot 3 = (1 + \sqrt{-5})\cdot (1- \sqrt{-5})$, and it is not hard to see that $2, 3, 1+ \sqrt{-5}, 1- \sqrt{-5}$ are all irreducible elements. It was Richard Dedekind who discovered that the fix for the failure of unique factorization in this and other number rings was to utilize ideals. Let us briefly explain Dedekind’s ideas in a slightly more modern language than was available to him. We will use the notion of an algebraic integer defined in Appendix B. We define a number field to be a field K obtained by adjoining a finite number of algebraic integers to $\mathbb Q$. Define the ring of integers $\mathscr {O}_K$ of K to be set of all algebraic integers contained in K. Theorem B.4 shows that $\mathscr {O}_K$ is a ring. Dedekind showed that every ideal of $\mathscr {O}_K$ is a product of prime ideals of $\mathscr {O}_K$ in an essentially unique fashion. Since every ideal of $\mathbb Z$ and $\mathbb Z[i]$ is principal, Dedekind’s result implies the unique factorization theorems of these rings.

Algebraic number theory was brought to new heights in the hands of David Hilbert and Emil Artin who early in 20th century found spectacular generalizations of the Law of Quadratic Reciprocity, known as Reciprocity Laws. These laws were further generalized by Shimura and Taniyama, who also discovered new connections to the theory of elliptic curves and modular forms. The most general reciprocity laws were conjectured by Robert Langlands in the 60s and 70s. Even though these conjectures remain largely open, they have inspired much progress in the last few decades.

For an elementary introduction to algebraic number theory, see [50]. Samuel [43] is a timeless classic. Murty and Esmonde’s book [37] is a much recommended problem-solving-based approach to algebraic number theory. More advanced readers already familiar with basic algebraic number theory, abstract algebra, and measure theory are encouraged to read Weil’s Basic Number Theory [56]. This book is far from basic, but in the words of Norbert Schappacher, if you learn number theory from this book, you will never forget it. Mazur [86] is an excellent expository article explaining the connections between modular forms and Diophantine equations. The book [17] is an account of the history of class field theory. Gelbart [75] is a not-so-elementary introduction to the Langlands program.