Problem 33 - A History of Mathematics: An Introduction

Question

From the Book of Squares: Find a square number for which the sum of it and its root is a square number and for which the difference of it and its root is similarly a square number. (In modern notation, find \(x, y, z\), such that \(x^{2}+x=z^{2}\) and \(x^{2}-x=y^{2}\). Leonardo began his solution by using the congruous number 24 in solving \(a^{2}+24=b^{2}, a^{2}-24=\) \(c^{2}\); he then divided everything by \(24 .\) )

Step-by-Step Solution

Answer

Question: Find three numbers, x, y, and z that satisfy the equations \(x^2 + x = z^2\) and \(x^2 - x = y^2\). Answer: The three numbers are x = 24, y = 24, and z = √600.

Step 1: Use the congruous number 24 to form two equations

We will begin by using the congruous number 24 in solving the equations \(a^2 + 24 = b^2\) and \(a^2 - 24 = c^2\), as Leonardo did.

Step 2: Divide both equations by 24

To simplify the equations, we will divide both equations by 24: - \((a^2 + 24)/24 = b^2/24\) => \(a^2/24 + 1 = b^2/24\) - \((a^2 - 24)/24 = c^2/24\) => \(a^2/24 - 1 = c^2/24\)

Step 3: Compare these equations to original equations

Now, let's compare the simplified equations we have found with the original two equations: - For \(x^2 + x = z^2\), we can rewrite it as: \(x^2/24 + 1 = z^2/24\) (similar to the first equation above). - For \(x^2 - x = y^2\), we can rewrite it as: \(x^2/24 - 1 = y^2/24\) (similar to the second equation above).

Step 4: Find x

Now our task is to find a number \(x\) that satisfies both these rewritten equations. Since the equations follow the same structure as those we derived using the congruous number 24, we can set \(x=a\).

Step 5: Solve for x

Now, we have an equation in the form \(x^2/24 + 1 = z^2/24\), which is satisfied by our previous equation \(a^2/24 + 1 = b^2/24\). In this case, \(x=a\) and \(z=b\). We can derive that \(z^2 = x^2 + x = a^2 + a\). Since \(a\) is an integer, let's choose \(a = 24\), then we obtain \(x = 24\). Next, we can determine \(z^2\) using the equation \(z^2 = x^2 + x = (24)^2 + 24 = 600\).

Step 6: Find y^2

Now that we have found \(x^2\) and \(z^2\), let's find \(y^2\). We know that \(x^2 - x = y^2\), so we have \(y^2 = 24^2 - 24 = 576\). Thus, \(y = 24\).

Step 7: Verify our results

To verify our results, let's check whether they satisfy the original equations: - For the equation \(x^2 + x = z^2\): \(24^2 + 24 = 600\), which is true. - For the equation \(x^2 - x = y^2\): \(24^2 - 24 = 576\), which is also true. We have found the three numbers: \(x = 24\), \(y = 24\), and \(z = \sqrt{600}\), such that \(x^2 + x = z^2\) and \(x^2 - x = y^2\).

Key Concepts

Congruous Numbers

Congruous numbers are central to this problem. A number is called congruous when it is used to form equations involving sums of squares or differences of squares that yield an integer result. In our exercise, the congruous number is 24, which Leonardo of Pisa originated in this context.
The task involves two key equations:

• First: $a^2 + 24= b^2$
• Second: $a^2 - 24 = c^2$

The purpose of using a congruous number is to simplify finding solutions for these equations. By deriving these equations from an initial problem setup, Leonardo ultimately applies the number 24 to discover integer solutions efficiently.
In modern terms, we relate congruous numbers when seeking integer solutions in the setup of Diophantine equations. Employing a congruous number can make complex integer problem-solving more manageable by ensuring differences and sums of squares result in whole numbers.

Diophantine Equations

Named after the ancient Greek mathematician Diophantus, Diophantine equations are polynomial equations, usually in two or more unknowns, that seek integer solutions. This problem falls under that category. The challenge is to find integers that satisfy specific polynomial expressions.
In this exercise, we have:

• The first equation: $x^2 + x = z^2$
• The second equation: $x^2 - x = y^2$

These equations demand integer solutions for $x$, $y$, and $z$. The difficulty lies in configuring the values so that not only both sides of each equation are integers, but they also represent valid square numbers.
Utilizing congruous numbers assists in solving these equations by systematically aligning them with known patterns. In the end, solving Diophantine equations requires inventive approaches like Leonardo's method, where alternative numbers (such as our 24 congruous number) play a role in simplifying complex integer validations.

Leonardo's Method

Leonardo of Pisa, also known as Fibonacci, designed a method to tackle the problem of finding square numbers that fit specific conditions—a problem fundamentally woven with congruous numbers and integer equations. His method involves breaking down an equation into smaller, more manageable parts.
In this problem:

• He starts by expressing both a sum and a difference of squares equations using the congruous number 24. This translates the problem into finding equivalent simpler equations.
• By dividing through by 24, he recalibrates the equations to make the comparison easier, linking each form back to his base congruous number.

Leonardo's method shines through simplicity. By grounding the problem through known constants and mathematical patterns, he reduces a complex integer problem into a series of steps that can easily be navigated. In essence, his approach of leveraging a congruous number like 24 simplifies the original Diophantine challenge by breaking it into smaller, defined tasks that ultimately unveil the correct integer solutions. This exemplifies how historical approaches continue to inform and underpin contemporary problem-solving strategies in number theory.