The Diophantus Problem: How Old Was the Father of Algebra?

A post about the Demochares puzzle from the Greek Anthology earned 50 reactions, so the series continues. Today I want to tell you about what is arguably the most famous and meaningful problem of antiquity — the Diophantus problem.

Who Was Diophantus, the Father of Algebra?

You have almost certainly heard of Diophantus. He lived in the third century in Alexandria. He is often called the “father of algebra” — he was the first to solve equations systematically and introduced the earliest prototype of algebraic notation.

His principal work, Arithmetica, directly inspired Fermat to formulate his famous Last Theorem. It was in the margins of Diophantus’s Arithmetica that Fermat wrote his legendary claim — that xⁿ + yⁿ = zⁿ has no integer solutions for n > 2 — while famously noting that the margin was too small to contain the proof.

The Diophantus Problem and Its Legacy in Modern Mathematics

Diophantine equations — those in which we seek integer solutions — lie at the heart of some of the hardest open problems in number theory: from Hilbert’s tenth problem to the abc conjecture and the seventh Millennium Prize Problem (the Birch and Swinnerton-Dyer conjecture).

What is striking is that, like so many great thinkers of antiquity, almost everything we know about Diophantus comes from later references and indirect evidence. The only “biographical” source describing his life is the very puzzle at the center of this post — epigram number 126 from the Greek Anthology.

The Only Biography of Diophantus: Epigram 126

Whether this epigram is historically accurate, no one knows for certain. And yet it is precisely on this poem that all existing estimates of Diophantus’s lifespan are based.

Here is the puzzle, rendered close to the original:

God granted him one sixth of his life as a boy;
and one twelfth more as a youth, while his cheeks were growing their first down.
Then one seventh before he took a wife.
Five years after his marriage, a son was born.
Alas, the late-born and beloved child of this wise man
lived only half his father’s life before cold fate took him.
Diophantus consoled himself in the science of numbers for four more years,
and then reached the end of his journey.

Can You Solve the Diophantus Problem?

I am sure you can work it out. Share how you approached it in the comments.

Answer: 84 years