Number Theory

Pythagoras's theorem, Euclid's way

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. — Euclid, Elements, Proposition I.47 Long before the Greeks, the surveyors of Egypt, who re-marked the fields after each Nile flood, were called rope-stretchers. The story goes that they used a rope with 12 equally spaced knots: stretched into a triangle with sides 3, 4 and 5, it gave a right angle. The Babylonians knew much more: a clay tablet from around 1800 BC, Plimpton 322, lists right triangles with whole-number sides, some of them huge, such as 3,367, 3,456 and 4,825.

October 12, 2026 Read

Fermat's little theorem: powers and remainders

Take the number 2 and raise it to larger and larger powers. Divide each result by 7 and note the remainder: Power $2^1$ $2^2$ $2^3$ $2^4$ $2^5$ $2^6$ Value 2 4 8 16 32 64 Remainder mod 7 2 4 1 2 4 1 At $2^6$ the remainder is 1. Now try 3 and the prime 5: $3^4=81$, which leaves remainder 1 when divided by 5. And 10 and the prime 13: $10^{12}=1{,}000{,}000{,}000{,}000$, which also leaves remainder 1 when divided by 13.

October 7, 2026 Read

Four squares are enough

Take any positive whole number and try to write it as a sum of perfect squares, $0, 1, 4, 9, 16, 25, \ldots$, using as few terms as possible: $$\begin{aligned} 5&=4+1,\\ 6&=4+1+1,\\ 7&=4+1+1+1,\\ 30&=25+4+1,\\ 2026&=2025+1=45^2+1^2. \end{aligned}$$Some numbers are squares, others need two, others three. The number 7 needs four: there is no other way. Could some number need five? In 1770 Lagrange proved that no number does: Every positive whole number is a sum of four squares of integers.

October 14, 2026 Read

The Sophie Germain identity: factoring what looks prime

Some numbers look prime and are not. 629, for example, is not even, does not end in 5 and is not a multiple of 3. But $629=5^4+4$, and that form gives it away: it factors by an identity that bears Sophie Germain’s name. The identity For any numbers $a$ and $b$, $$\begin{aligned}a^4+4b^4={}&\left(a^2+2b^2+2ab\right)\\&\times\left(a^2+2b^2-2ab\right).\end{aligned}$$The trick for finding it is completing the square, as in the quadratic formula. The sum $a^4+4b^4$ is almost the square of $a^2+2b^2$; only the middle term, $4a^2b^2$, is missing. Adding and subtracting that term,

October 13, 2026 Read

Algebra done with squares and rectangles

If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments. — Euclid, Elements, Proposition II.4 The Greeks had no letters for unknowns, no plus or minus signs, no equals sign. When they wanted to speak of $a\cdot b$, they spoke of a rectangle with sides $a$ and $b$; when they wanted $a^2$, they spoke of a square with side $a$. The words “square” and “cube” that we use for powers come from this. Book II of the Elements is made entirely in this way: it consists of identities of algebra, stated and proved as facts about areas. Historians call it geometric algebra.

October 12, 2026 Read

Fermat numbers: a conjecture and a counterexample

Mathematicians have always dreamed of a formula that produces only prime numbers. In 1640 Fermat thought he had found one. Five primes in a row Fermat studied numbers of the form $$F_n=2^{2^n}+1.$$The exponent is a power of 2: $1,2,4,8,16,\ldots$ Computing the first few: $$F_0=3,\quad F_1=5,\quad F_2=17,\quad F_3=257,\quad F_4=65{,}537.$$All five are prime. Checking $65{,}537$ by hand, trying divisors up to its square root, was already hard work. Fermat wrote to Frénicle de Bessy that he was convinced all numbers of this form were prime, although he admitted he had no proof.

October 7, 2026 Read

Shuffling the roots: Lagrange's theorem

In 1770 algebra had a problem that had been stuck for more than two centuries. There were formulas for solving equations of degree 2, the quadratic formula, and also of degrees 3 and 4, published by Cardano in 1545. For degree 5, nobody could find anything. The best mathematicians had tried, and every method that worked up to degree 4 jammed at 5. Lagrange did not try yet another method. He asked a different question: why do the old methods work?

October 14, 2026 Read

Chladni figures and the prize for vibrating plates

Sprinkle sand on a metal plate held at its centre and draw a violin bow across the edge. The plate begins to sing, and the sand, as if it had a will of its own, flees some places and gathers in others, drawing lines, crosses, stars and symmetrical curves. Change the note, and the drawing changes. These are Chladni figures. How a Chladni plate is excited: the plate is held at the centre and a violin bow is drawn across the edge. Engraving from an 1879 book.Unknown artist, 1879 · Wikimedia Commons · public domain Sound made visible The German Ernst Chladni published these figures in 1787, in a book called Discoveries on the Theory of Sound. The qualitative explanation is simple. When the plate vibrates, almost all its points move up and down and throw the sand away. But some lines stay still: these are the nodal lines. The sand slides there and stays. The drawing is a map of the places where the plate does not move.

October 13, 2026 Read

Euclid's algorithm: the greatest common divisor

The Euclidean algorithm is the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day. — Donald Knuth, The Art of Computer Programming, vol. 2 (1969) You have two rods, one 252 cm long and the other 105 cm, and you want the longest ruler that measures both exactly, a whole number of times. In today’s language you want the greatest common divisor, $\gcd(252,105)$. You could list all the divisors of each number and look for the largest common one, but with big numbers that is impractical. In Book VII of the Elements, Euclid teaches a much faster way.

October 12, 2026 Read

Counting the prime numbers

The prime numbers seem to appear at random: 2, 3, 5, 7, 11, 13, then a gap until 17, 19, 23, another until 29. But seen from afar, they obey a rule. Call $\pi(x)$ the number of primes less than or equal to $x$. While still a teenager, Gauss noticed, looking at tables, that near a number $x$ the density of primes is about $\frac{1}{\ln x}$, and proposed the estimate $$\pi(x) \approx \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t},$$the logarithmic integral. Nobody knew how to prove that it was good, nor how to explain the deviations.

October 31, 2026 Read

The derivative without infinitesimals and the mean value theorem

At the end of the eighteenth century, calculus worked very well and nobody could properly explain why. Leibniz’s $dx$ were “infinitely small” quantities: sometimes they were treated as if they were different from zero, so that one could divide by them, and right afterwards as if they were zero, so that one could discard them. Bishop Berkeley had mocked this in 1734, calling them “ghosts of departed quantities”. Lagrange was bothered by it. In 1797 he published a book whose subtitle is a whole programme:

October 14, 2026 Read

Mean curvature: how much a surface bends

To write the equation of vibrating plates, Sophie Germain had to answer a question that sounds simple: how much is a surface bent at a point? Her answer, mean curvature, is still used today to study soap bubbles, cell membranes and aircraft skins. The curvature of a curve Let us start with a line. A straight line does not bend, and we say its curvature is zero. A circle bends equally at every point, and the smaller it is, the more it bends. That is why the curvature of a circle of radius $R$ is defined as

October 13, 2026 Read