What integers are areas of right triangles?

Abstract

In this chapter we study the set of integers that are the area of a right triangle with integer sides.

Appendices

Exercises

• 4.1 Determine all right triangles with integral sides such that the perimeter and the area are equal.

• 4.2 Show that two right triangles with equal hypotenuse and area are congruent.

• 4.3 Show that for every $n \in \mathbb N$, there are n distinct integral right triangles with the same area.

• 4.4 A Heronian triangle is a triangle with rational sides whose area is a rational number. Show that triangles with side lengths (13, 14, 15) and (65, 119, 180) are Heronian.

• 4.5 Show that there are infinitely many isosceles Heronian triangles.

• 4.6 Let ABC and ACD be right triangles with rational sides which share a side AC as in Figure 4.1. Show that the triangle ABD is Heronian. Conversely, suppose ABD is a Heronian triangle with $\angle BAD$ the largest angle of the triangle. Draw the altitude AC and show that the triangles ABC and ACD are right triangles with rational sides.

• 4.7 Show $15 \not \in \mathscr {S}$.

• 4.8 Show that a square-free natural number n is a congruent number if and only if there is a rational number x such that $x^2 - n$ and $x^2 + n$ are squares of rational numbers.

• 4.9 Show that 2 and 3 are not congruent numbers.

• 4.10 Prove Theorem 4.5 by direct computation.

• 4.11 Show that in Theorem 4.5, for $n \in \mathbb N$, x, y are positive rational numbers, if and only if a, b, c are positive rational numbers.

• 4.12 Show that the only solutions of $y^2 = x^3 - n^2 x$ with $xy=0$ are (0, 0) and $(\pm n, 0)$.

• 4.13 Find three rational right triangles with area 6.

• 4.14 ($\maltese$) Find fifty congruent numbers.

• 4.15 ($\maltese$) Find ten rational right triangles with area 30.

• 4.16 ($\maltese$) Use Tunnell’s Theorem 4.8 from the Notes to find all congruent numbers less than 100.

Fig. 4.1

The diagram for Problem

Notes

1.1 The history of congruent numbers

Like many other concepts in elementary number theory, the standard reference for the history of congruent numbers is Dickson’s classic book [16], especially Chapter XVI. The definition that Dickson uses is different from ours. He defines a congruence number to be a natural number n if there is a rational number x such that $x^2-n$ and $x^2+n$ are squares of rational numbers; that this definition is equivalent to our definition is Exercise 4.8. Let us mention here that if S is the area of the right triangle with sides a, b, c, with c the hypotenuse, then

$$ c^2 \pm 4 S = c^2 \pm 2 ab = a^2 + b^2 \pm 2ab = (a\pm b)^2. $$

This means, we have a three term arithmetic progression

$$ \left( \frac{c}{2}\right) ^2 - S,\quad \left( \frac{c}{2}\right) ^2, \quad \left( \frac{c}{2}\right) ^2 + S $$

consisting of rational squares. This is perhaps the reason for the name congruent. Dickson mentions that in tenth century an Iranian mathematician and this author’s fellow townsman Mohammad Ben Hossein Karaji (953–1029) stated that the problem of determining congruent numbers was the “principal object of the theory of rational right triangles.” Dickson [16, Ch. XVI] is a wonderful review of work by various mathematicians on the problem of characterizing congruent numbers over the millennium up to its publication. For a modern treatment of this subject we refer the reader to [30, Ch. 1].

1.2 Tunnell’s theorem

The theory of rational points on cubic curves, the theory of elliptic curves, is a rich active area of research with connections to many parts of modern mathematics [47]. In the last three decades many results about congruent numbers have been obtained that use methods and techniques involving elliptic curves. It appears that Stephens’s very short paper [97] was the first paper that made the connection to elliptic curves explicit. Tunnell’s paper [105] pushed the theory far. Among other results, Tunnell proved the following surprising theorem:

Theorem 4.8

(Tunnell). Define a formal power series in the variable q by

$$ g = q\prod _{n=1}^\infty (1-q^{8n})(1-q^{16n}), $$

and for each $t \in \mathbb N$ set $\theta _t = \sum _{n \in \mathbb Z} q^{tn^2}$. Define integers a(n) and b(n) via the identities

$$ g \theta _2 = \sum _{n=1}^\infty a(n) q^n, $$

and

$$ g \theta _4 = \sum _{n=1}^\infty b(n) q^n. $$

Then, we have

• If $a(n) \ne 0$, then n is not a congruent number;

• If $b(n) \ne 0$, then 2n is not a congruent number.

Conjecturally, both statements in the theorem should be if and only if. The coefficients a(n), b(n) are computable in terms of the number of solutions in integers of equations of the form

$$ Ax^2 + By^2 + Cz^2 = n $$

for $A, B, C \in \mathbb N$. (We advise the reader to do this as an exercise!) Tunnell recovers a number of previously known results from his numerical criterion. For example, he shows a prime p of the form $8k + 3$ is not congruent, as for such primes $a(p) \equiv 2 \ \mathrm {mod}\ 4$, or that if p, q are primes of the form $8k + 5$, then 2pq is not congruent. It is an easy exercise to derive Theorem 4.4 from Theorem 4.8.

At least conjecturally one expects the existence of many congruent numbers. For example, we have the following conjecture which is a consequence of the Birch and Swinnerton-Dyer Conjecutre [47, Conjecture 16.5]:

Conjecture 4.9

([59, 60]). Every positive integer congruent to 5, 6, or 7 modulo 8 is a congruent number.

Recently some impressive results have been obtained in this direction [101, 102]. Smith [95] has proved that at least $55.9\%$ of positive square free integers $n \equiv 5, 6, 7 \ \mathrm {mod}\ 8$ are congruent numbers. In contrast, Smith [96] has proved that congruent numbers are rare among natural numbers $n \equiv 1, 2, 3 \ \mathrm {mod}\ 8$.