**Fibonacci Problem:**

Leonardo of Pisa (famous as Fibonacci) (1173) wrote a book “Liber Abaci” (1202), wherein he introduced Hindu-Arabic numerals in Europe. In 1225, Frederick II declared him as the greatest mathematician in Europe when he posed the following problem to defeat his opponents.

**Question:**

Determine the rational numbers x, y and z to satisfy the following equations:

and .

**Solution:**

Definition: Euler defined a congruent number to be a rational number that is the area of a right-angled triangle, which has rational sides. With p, q, and r as a Pythagorean triplet such that , then is a congruent number.

It can be shown that square of a rational number cannot be a congruent number. In other words, there is no right-angled triangle with rational sides, which has an area as 1, or 4, or , and so on.

Characteristics of a congruent number: A positive rational number n is a congruent number, if and only if there exists a rational number u such that and are the squares of rational numbers. (Thus, the puzzle will be solved if we can show that 5 is a congruent number and we can determine the rational number ). First, let us prove the characterisitic mentioned above.

Necessity: Suppose n is a congruent number. Then, for some rational number p, q, and r, we have and . In that case,

and n are rational numbers and we have

and similarly,

.

Setting , we get and are squares of rational numbers.

Sufficiency:

Suppose n and u are rational numbers such that and are rational, when

and

and 2n are rational numbers satisfying is a rational square and also , a rational number which is a congruent number.

So, we see that the Pythagorean triplets can lead our search for a congruent number. Sometimes a Pythagorean triplet can lead to more than one congruent number as can be seen with . This set obviously gives 180 as a congruent number. But, as , we can also consider a rational Pythagorean triplet , which gives a congruent number 5 (we were searching for this congruent number in this puzzle!). We also determine the corresponding .

The puzzle/problem is now solved with , which gives , and .

One can further show that if we take three rational squares in AP, , and , and , with their product defined as a rational square and n as a congruent number, then , is a rational point on the elliptic curve .

**Reference:**

1) Popular Problems and Puzzles in Mathematics: Asok Kumar Mallik, IISc Press, Foundation Books, Amazon India link:

2) Use the internet, or just Wikipedia to explore more information on Fibonacci Numbers, Golden Section, Golden Angle, Golden Rectangle and Golden spiral. *You will be overjoyed to find relationships amongst all the mentioned “stuff”. *

Cheers,

Nalin Pithwa