Diophantine Equations

Around 1800 BCE, a Babylonian student carved numbers into a clay tablet called Plimpton 322[^1].

It contained 15 perfect Pythagorean triples—sets of three whole numbers where:

Examples:

• (3, 4, 5) → 9 + 16 = 25 ✓
• (119, 120, 169)
• And 13 others

This is a Diophantine equation—a puzzle where you only want whole number answers.

Diophantus of Alexandria[^2] wrote Arithmetica, the first systematic treatment of these equations.

Called the “Father of Algebra”

His innovation:

• Focused on finding solutions systematically
• Not just verifying them
• Introduced early algebraic notation
• Still thought geometrically

Original puzzles:

1. x² + y² = z² (Pythagorean triples)
2. n(n+1)/2 = m² (Triangular numbers that are also squares)
3. x² - y² = n (Difference of two squares)
4. x + y = a², x - y = b² (Systems with multiple square constraints)

Founded by Alexander the Great in 331 BCE, Alexandria became the intellectual capital of the ancient world.

Why it mattered:

• Home to the Great Library of Alexandria (largest library in the ancient world)
• The Mouseion (Museum) - the first research institution
• Attracted the greatest minds: Euclid, Archimedes, Eratosthenes, and Diophantus

Location:

• Mediterranean coast of Egypt
• Strategic position between Europe, Africa, and Asia
• Perfect hub for trade and ideas

For 600+ years, Alexandria was where mathematics, astronomy, medicine, and philosophy flourished.

Umar ibn al-Khattab (Omar) was the second Caliph of Islam, ruling from 634-644 CE.

What’s a Caliph? A Caliph (خليفة, “successor”) is the political and religious leader of the Islamic community after Prophet Muhammad.

The first Caliph was Abu Bakr (632-634 CE), Muhammad’s close companion.

Conquest of Alexandria: Omar’s general Amr ibn al-As conquered Alexandria in 641-642 CE, bringing Egypt into the Islamic empire.

The Library Myth: A popular legend claims Omar ordered the Great Library burned. This is a myth invented 500+ years later—no contemporary sources support it. The Library had likely already declined by then.

From Alexandria to Spain: Not long after taking Alexandria (641-642 CE), the Islamic empire continued expanding westward…

Islamic Spain (Al-Andalus): 711-1492 CE

• Islamic forces invaded Iberia in 711 CE (just 70 years after Alexandria)
• Islamic rule lasted 781 years in Iberia
• 1492: Reconquista complete - Granada falls to Catholic Monarchs
• Spain becomes Catholic, but Arabic deeply influenced Spanish language

The Novel (1510): Just 18 years after Islamic Spain ended, Spanish author Garci Rodríguez de Montalvo wrote Las Sergas de Esplandián, featuring a mythical island called “California” ruled by Queen Calafia.

The Connection: “Calafia” likely derives from “Caliph” (خليفة) - the Islamic ruler title, still fresh in Spanish memory.

How California Got Its Name: When Spanish explorers reached the Baja California peninsula in the 1530s-1540s, they named it after the mythical island from the novel!

Legacy: Hundreds of Spanish words come from Arabic - algebra, alcohol, admiral, guitar, orange…

Diophantine equations aren’t just math games—they show up everywhere:

Real-world constraints:

• Commerce: Can’t buy 2.5 chickens or sell 3.7 tickets
• Astronomy: Calendars need whole days aligned with moon cycles
• Music: Harmonious intervals use integer ratios (Perfect fifth: 3:2, Major third: 5:4)
• Physics: Quantum energy levels are discrete integers
• Planets: Orbital resonances at near-integer ratios (Jupiter’s moons: 1:2:4 Laplace resonance)

When reality demands whole numbers, you need Diophantine equations.

The Problem: Ancient astronomers needed to synchronize different celestial cycles:

• Solar year: ~365.25 days
• Lunar month: ~29.5 days
• Planetary periods: Mars (687 days), Jupiter (4,333 days), Saturn (10,759 days)

The Challenge: When do these cycles align? When will the same moon phase occur on the same date?

CRT Solution: The Chinese used the Remainder Theorem to predict:

• Eclipse cycles
• Planetary conjunctions
• Calendar intercalations (when to add leap months)

Example: The Metonic cycle (19 years = 235 lunar months) was discovered independently in Babylon, Greece, and China using similar remainder-based reasoning!

Modern equivalent: Your phone’s calendar handles time zones, leap years, and daylight saving - all remainder arithmetic!

Chinese Zodiac carvings, Kushida Shrine

Crystal lattices are periodic structures in 3D space—atoms arranged in repeating patterns.

The CRT connection:

• Atoms repeat in different directions with different periods
• X-direction: period = a (lattice constant)
• Y-direction: period = b
• Z-direction: period = c

Given diffraction data from multiple lattice planes, we need to find positions that satisfy:

x ≡ r₁ (mod a)
y ≡ r₂ (mod b)
z ≡ r₃ (mod c)

CRT solves this! Just like ancient calendar synchronization, modern physicists use CRT to:

• Determine atomic positions from X-ray diffraction patterns
• Index crystal planes (Miller indices)
• Reconstruct 3D structures from 2D measurements

PrScO₃ crystal (100M× magnification)

Form: x² - Ny² = 1

Who solved it:

• Brahmagupta (7th century): composition law
• Bhāskara II (12th century): chakravala method

Why it matters: Provides rational approximations to irrational square roots!

Example for √2: Solve x² - 2y² = 1

• (3, 2) → 3/2 = 1.5
• (17, 12) → 17/12 ≈ 1.4167
• (577, 408) → 577/408 ≈ 1.414215…

Fun fact: John Pell (17th c.) had nothing to do with it. Euler mistakenly attributed it to him!

Bhāskara II

Pierre de Fermat[^3] was reading Diophantus when he wrote in the margin:

“It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers… I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

Fermat’s Last Theorem:

xⁿ + yⁿ = zⁿ has NO integer solutions when n > 2

What we know:

• n = 2: Infinitely many solutions
• n = 3, 4, 5, …: Zero solutions

The problem: No one could find that “truly marvelous proof” that the margin of Diophantus’s book was supposedly just a tiny bit too small to contain — for 358 years

Pierre de Fermat

June 1993, Isaac Newton Institute, Cambridge (UK)

Andrew Wiles[^4] announced three lectures: “Modular Forms, Elliptic Curves, and Galois Representations”

To most mathematicians, this sounded like a normal number theory topic.

Days 1 and 2: The audience of professional mathematicians listened as Wiles presented deep, technical arguments connecting elliptic curves and modular forms—difficult but not unprecedented.

Day 3, end of final lecture: Wiles calmly wrote on the blackboard:

xⁿ + yⁿ = zⁿ has no integer solutions for n > 2, FLT

Q.E.D.

The room fell completely silent… then erupted in applause and astonishment.

He had just proven Fermat’s Last Theorem—unsolved for 356 years.

Andrew Wiles

An elliptic curve is a Diophantine equation of the form:

y² = x³ + ax + b

Why Wiles needed them:

• To prove Fermat’s Last Theorem, Wiles connected it to the Modularity Theorem for elliptic curves
• He proved: every elliptic curve is modular
• This indirectly proved Fermat’s equation has no solutions for n > 2

Modern impact:

• Cryptography: ECC (Elliptic Curve Cryptography) powers Bitcoin, Signal, and modern HTTPS
• Millennium Prize: Birch and Swinnerton-Dyer Conjecture ($1M unsolved problem about elliptic curves)
• Theoretical Physics: String theory and mirror symmetry

Various elliptic curves y² = x³ + ax + b

August 8, 1900, Paris

At the International Congress of Mathematicians, David Hilbert[^5] presented a visionary lecture that would shape mathematics for the next century.

Hilbert’s 23 Problems:

The impact: This list became the roadmap for 20th-century mathematics. We will come back to this list in later topics.

David Hilbert

Einstein’s Field Equations:

Hilbert’s Action (Lagrangian formulation):

The difference:

• Einstein: Physical intuition (F = ma style Newtonian Mechanics)
• Hilbert: Mathematical formalism (Lagrangian mechanics)

Timeline:

• Hilbert submitted: Nov 20, 1915
• Einstein published: Nov 25, 1915
• Both arrived independently

Hilbert’s 10th Problem asked:

“Given any Diophantine equation… devise a process to determine whether the equation is solvable in rational integers.”

Can we write an algorithm that determines if ANY Diophantine equation has integer solutions?

Example:

• Input: 3x² + 7xy - 5y³ = 42
• Output: YES or NO
• Question: Can a program answer this for every equation?

The 70-year quest: Mathematicians worked for decades trying to find such an algorithm. In 1970, Yuri Matiyasevich[^6] (building on Davis, Putnam, Robinson) proved that no such algorithm exists. Some Diophantine equations are undecidable. The halting problem can be encoded as a Diophantine equation.

Impact: Linked number theory to logic, computability, and the philosophy of mathematics

Connects to:

• Gödel’s Incompleteness Theorems
• Turing’s Halting Problem
• Church-Turing thesis

Yuri Matiyasevich