Diophantus Biography

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World of Mathematics on Diophantus

Little is known about the life of the Greek mathematician Diophantus. However, his work led to one of the greatest mathematical challenges of all time, Fermat's last theorem.

Diophantus was born and lived in Alexandria, now in Egypt, which was at the time a great center of culture and learning in the Greek world. We have no record of the date of his birth or death, but we do have two pieces of evidence regarding when and how long he lived. One is a letter written in the 11th century, that tells that the bishop of Laodicea, Anatolius, dedicated his work on Egyptian computation to Diophantus, who was a good friend; since we know that Anatolius was bishop around 270 a.d., it would make sense that Diophantus was his contemporary.

A bit more sleuthing is required to learn a few details of Diophantus's life, which were left behind in a mathematical puzzle called "Diophantus's riddle." If it is to be believed, Diophantus married at age 33, had a son who died at age 42, and the mathematician himself died four years later, at age 84.

Where little is known about his personal life, Diophantus's work in mathematics is documented in his Arithmetika, which was supposed to have consisted of 13 books, but only six have survived. (It is possible that the six books were the only ones completed.) His works show that he may have studied with Babylonian teachers, for he was influenced by both Greek and Babylonian practices.

Diophantus devised an early form of algebra. He worked with equations that are solved in terms of integers. Some of his equations had no single solution; such equations, with more than one or even an infinite number of solutions, are called Diophantine, or indeterminate, equations today. Although they can't be solved down to one answer, they can be generalized to fit a number of solutions.

Diophantus used a symbol to indicate the unknown quantities in his equations, which was a major innovation. He was also the first Greek mathematician to treat fractions as numbers.

Before Diophantus, most Greek mathematics were concerned with practical problems drawn from everyday concerns, such as agriculture and finance; the math was either computational or geometric. However, by devising indeterminate equations, Diophantus opened the door to number theory.

The eighth problem of the second book of Arithmetika led to a puzzle that stumped generations of mathematicians. When looking at the problem in 1637, Pierre de Fermat wrote: "On the other hand it is impossible to separate a cube into the sum of two cubes . . . or any power except a square into the sum of two powers with the same exponent. I have discovered a truly marvelous proof of this, which however the margin is not large enough to contain." For over 300 years, Fermat's last theorem went unsolved. It was a riddle that Diophantus surely would have appreciated greatly.