Fibonacci, Leonardo Pisano

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Fibonacci, Leonardo Pisano

Italian Number Theorist 1175–1240

Leonardo Pisano Fibonacci (c. 1175–c. 1240) is considered by many to be the greatest number theorist of the Middle Ages. The following sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987, 1597, 2584, 4181, …

defined by F₁ = 1, F₂ = 1, and for n ≥ 3, F_n = F_n–1 + F_n–2 is called the Fibonacci sequence in his honor. The Fibonacci sequence evolved from the following problem in Fibonacci's book Liber abbaci.

A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The answer is F₁₂ = 377.

It is worth noting that Fibonacci did not name the Fibonacci sequence; the sequence was given the name by the nineteenth-century French mathematician, Edouard Lucas. Lucas also found many important applications of the Fibonacci sequence.*

*Today, the Fibonacci Association publishes a journal, The Fibonacci Quarterly, whose primary focus is to promote interest in Fibonacci and related numbers.

Fibonacci made many other contributions to mathematics. He is credited with introducing the Hindu-Arabic numerals to Europe. This is the positional number system based on the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and a decimal point.

In Liber quadratorum ("The Book of Squares"), Fibonacci described a method to construct Pythagorean triples . If he wanted to find two squares whose sum was a square, he took any odd square as one of the two squares. He then found the other square by adding all the odd numbers from 1 up to but not including the odd square. For example, if he took 9 as one of the two squares, the other square is obtained by adding all the odd numbers up to 9—that is, 1, 3, 5, and 7, whose sum is 16, a square. And 9 + 16 = 25, another square. Also, in this book Fibonacci proved that there are no positive integers m and n, such that m ² + n ² and m ² − n ² are both squares.