c. 200 – 284 AD
The Father of Algebra — whose Arithmetica launched the study of integer and rational solutions to polynomial equations, inspiring Fermat's Last Theorem
"His boyhood lasted 1/6 of his life; his beard grew after 1/12 more; after 1/7 more he married; 5 years later a son was born; the son lived half the father's life and died 4 years before him."
Solution: Let x = lifespan. x/6 + x/12 + x/7 + 5 + x/2 + 4 = x. Solving: x = 84 years.
His masterwork, originally 13 books. Six survived in Greek; four more were rediscovered in Arabic translation in 1968. Contains 189 problems seeking rational or integer solutions to equations.
A partially surviving treatise investigating numbers expressible as sums of figurate numbers, connecting to the Pythagorean tradition of number shapes.
Diophantus refers to a separate collection of "Porisms" (corollaries) in the Arithmetica, but this work has not survived. Some results can be inferred from his references.
Introduced abbreviations for unknowns and powers: ς for the unknown, Δυ for x², Kυ for x³. This was a revolutionary step between rhetorical and symbolic algebra.
A Diophantine equation is a polynomial equation where only integer (or rational) solutions are sought. This transforms algebra from a continuous to a discrete problem.
"To divide a given square into two squares." — i.e., given a², find rational x, y such that x² + y² = a².
This exact problem — "Arithmetica II.8" — is the one beside which Pierre de Fermat wrote his famous marginal note in 1637:
"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers... I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."
— Fermat, marginal note on Arithmetica II.8This launched Fermat's Last Theorem, unsolved for 358 years until Andrew Wiles proved it in 1995.
Before Diophantus, algebraic problems were stated entirely in words (rhetorical algebra). He introduced a syncopated notation — abbreviations for the unknown and its powers.
Diophantus developed systematic techniques for finding rational solutions, including:
For example, to solve x² + y² = a², set y = mx − a, reducing to a linear equation in x.
"Find two square numbers whose difference is a given number" (say 60).
Let the numbers be (x+3)² and (x−3)². Then (x+3)² − (x−3)² = 12x = 60, so x = 5.
Answer: 64 and 4 ✓
Diophantus's genius was choosing the right parametrization. By writing unknowns as (x+a) and (x−a), he forced the desired structure to emerge. This anticipates modern algebraic techniques by over a millennium.
Diophantus observed that no integer of the form 4n + 3 can be expressed as a sum of two squares. This result, properly proved by Euler 1500 years later, opened the theory of quadratic forms.
(a²+b²)(c²+d²) = (ac+bd)² + (ad−bc)². The product of two sums of squares is itself a sum of squares. This is a precursor to Gaussian integers and the Brahmagupta-Fibonacci identity.
Effectively parametrized all Pythagorean triples: a = m²−n², b = 2mn, c = m²+n². Though this was known to the Babylonians, Diophantus gave it a systematic algebraic treatment.
Noted conditions under which problems have no solutions — e.g., that no sum of two cubes can be a cube (a special case of Fermat's Last Theorem, not proved until modern times).
Pose the problem in terms of finding numbers with given properties
Choose a clever form for the unknowns with one free parameter
Substitute to get a single equation in one unknown
Find the rational solution and verify
Diophantus typically found one solution, not all solutions. He didn't develop a general theory of solvability, but rather showed extraordinary ingenuity problem-by-problem. Each problem in the Arithmetica has a unique trick.
He rejected negative numbers entirely, calling equations with negative solutions "absurd." He also worked with rationals, not just integers — the restriction to integers is a modern convention named after him.
The central mystery of Diophantus is himself. He is the most influential mathematician of late antiquity, yet we know almost nothing about his life.
When Claude-Gaspard Bachet published a new Latin translation of Arithmetica in 1621, Pierre de Fermat obtained a copy and filled its margins with 48 observations. These marginal notes launched modern number theory. The most famous — Fermat's Last Theorem — took 358 years to resolve.
Hypatia of Alexandria (c. 360–415 AD) wrote a commentary on the Arithmetica, now lost. She may have been the last ancient scholar to work seriously with Diophantus's ideas before the Arabic revival.
The entire field of algebraic number theory — studying integer solutions to polynomial equations — descends directly from Diophantus. Fermat, Euler, Gauss, Kummer, and Wiles all worked in this tradition.
Modern arithmetic geometry studies rational points on algebraic varieties — the natural generalization of Diophantine problems to higher dimensions. The Birch and Swinnerton-Dyer conjecture (a Millennium Prize problem) is Diophantine in spirit.
In 1900, Hilbert asked: "Is there a general algorithm to decide solvability of Diophantine equations?" In 1970, Matiyasevich proved the answer is NO — the problem is undecidable.
Modern public-key cryptography (RSA, elliptic curve cryptography) relies fundamentally on the difficulty of certain Diophantine problems — finding integer solutions to specific equations.
RSA security rests on the difficulty of factoring large numbers — a Diophantine problem at its core.
Reed-Solomon codes used in CDs, DVDs, and deep-space communication are built on polynomial arithmetic over finite fields.
Operations research uses integer linear programming — finding integer solutions to systems of linear inequalities — for logistics, scheduling, and resource allocation.
Lattice point geometry underlies X-ray crystallography, where discrete Bragg reflections correspond to integer solutions of diffraction conditions.
Elliptic curve cryptography — finding rational points on elliptic curves — secures Bitcoin and most modern digital currencies.
Pythagorean tuning and just intonation require finding rational approximations to irrational frequency ratios — a Diophantine approximation problem.
Thomas Heath (1910). The classic English-language study with translation and detailed mathematical commentary.
Isabella Bashmakova (1997). An accessible modern account tracing the influence from antiquity to Fermat and beyond.
Simon Singh (1997). The gripping story of Fermat's Last Theorem, starting from Diophantus's margin to Wiles's proof.
Titu Andreescu, Dorin Andrica, Ion Cucurezeanu (2010). Problem-solving approach to modern Diophantine methods.
André Weil (1984). From Hammurapi to Legendre — a masterful historical survey by one of the 20th century's greatest number theorists.
Jean Christianidis & Jeffrey Oaks (2022). A new scholarly edition with fresh analysis of Diophantus's methods and notations.
"I have discovered a truly marvellous demonstration of this proposition, which this margin is too narrow to contain."
— Pierre de Fermat, marginalia on Diophantus's Arithmetica II.8 (1637)Diophantus of Alexandria · c. 200–284 AD · Father of Algebra