Level 1
The Atoms of Arithmetic
What primes are, and the strange way they hide.
Pick a whole number — say 60. You can break it into smaller pieces that multiply back together: 60 = 6 × 10. Those pieces break further: 6 = 2 × 3, and 10 = 2 × 5. But now you are stuck. 2, 3 and 5 refuse to split. They are primes: whole numbers greater than 1 whose only divisors are 1 and themselves.
Numbers that do split, like 60, are called composite. And here is the first deep fact of arithmetic: every composite number shatters into primes in exactly one way. Not one way per method — one way, full stop. Try it below: however you choose to split, the gold leaves at the bottom of the tree are always the same.
60 = 22 × 3 × 5
Split any composite number into two smaller factors and keep going; the process always bottoms out in primes (gold), and — remarkably — in the same primes no matter how you split. That uniqueness is the fundamental theorem of arithmetic.
Mathematicians call this the fundamental theorem of arithmetic: every whole number greater than 1 is a product of primes, uniquely up to reordering. It is why primes deserve the name atoms of arithmetic. Chemistry has its periodic table; multiplication has the primes. Everything else is a molecule.
Catching primes with a sieve
So how do you findthe atoms? The oldest method still works beautifully. Around 240 BCE, Eratosthenes of Cyrene — chief librarian at Alexandria, the man who first measured the Earth — noticed you don’t need to test numbers one by one. You can eliminate the non-primes wholesale: keep 2, cross out every multiple of 2; keep 3, cross out its multiples; and so on. Whatever survives is prime. Step through it yourself:
Keep the first number standing, cross out all its multiples; repeat. Notice the first new crossing for each prime p happens at p² — everything smaller was already removed — so once p² leaves the grid, every survivor is prime.
Watch the survivors as the sieve runs. They thin out — between 1 and 100 there are 25 primes, but between 301 and 400 only 16 — yet they keep coming, and they arrive in no pattern anyone can see. 89 and 97 are prime; nothing between them is. Why? The sieve gives no hint. It produces the primes without explaining them.
Do they ever run out?
The thinning raises an uncomfortable question: maybe primes simply stop. Maybe past some enormous number, everything is composite. Around 300 BCE, Euclid of Alexandria settled this with one of the most admired arguments in all of mathematics — a proof you can hold in your head for the rest of your life.
- EuclidSuppose you had a completelist of every prime that exists. Humour me — let’s say it’s {2, 3, 5} and nothing else. (Use the control below to make your list longer.)
An interactive rendering of Proposition 20, Book IX of Euclid's Elements (c. 300 BCE) — arguably the first great proof in number theory.
So the primes are infinite. The supply of atoms never runs dry — and notice the proof never names a single new prime. It conjures one out of pure logic, sight unseen. That is what a proof is: a guarantee covering infinitely many cases at once, something no amount of checking could ever deliver. Keep that distinction in mind; it returns with force in Level 5.
Chaos, seen from a distance
Infinitely many primes, then — but scattered how? In 1963, the physicist Stanisław Ulam, doodling through a dull conference talk, wrote the integers in a spiral and circled the primes. What he saw has been unsettling people ever since.
Write 1 in the middle, spiral the integers outward, light up the primes. Nobody arranged those diagonal streaks — they emerge. Diagonals correspond to quadratic formulas like n² + n + 41, some of which are unusually rich in primes. The streaks whisper that primes have structure. What structure? That question is this whole site.
Locally, primes look random: there is no formula that cheaply hands you the next one, and adjacent primes can be 2 apart or 200 apart. Yet zoom out and those ghostly diagonals appear — order seeping through the static. This is the central tension of our entire journey, so let’s say it plainly:
Primes look random up close, but lawful from afar. The Riemann Hypothesis is the precise statement of how lawful they are.
Go deeperPrimes guard your secrets: cryptography★★★
Multiplying two 300-digit primes takes a computer microseconds. Reversing the process — recovering the primes from their 600-digit product — would take the best known classical algorithms longer than the age of the universe. RSA encryption, which still protects much of the world’s commerce, rests exactly on that asymmetry: the public key contains the product, the private key the primes. The atoms of arithmetic became, twenty-three centuries after Euclid, load-bearing infrastructure.
Go deeperTwin primes: the conjecture next door★★★
Pairs like (3, 5), (11, 13), (101, 103) — primes only 2 apart — keep appearing as far as anyone has looked. The twin prime conjecturesays they never stop. It is still unproven, but in 2013 Yitang Zhang stunned the field by proving that some gap below 70,000,000 occurs infinitely often — the first finite bound ever. A massive collaborative effort (the Polymath project, with James Maynard’s methods) drove the bound down to 246, where it stands. From 246 to 2 is the remaining distance — a fair picture of how number theory progresses: enormous effort, partial light.
Go deeperHow large can the gaps get?★★★
Arbitrarily large. The run of numbers contains no primes at all: is divisible by for each , so choosing huge manufactures a prime-free desert as long as you like. Yet primes also cluster: gaps of 2 plausibly recur forever (the twin prime conjecture). Both extremes coexist in one sequence — randomness-like behaviour that nonetheless, as we will see in Level 2, obeys exact statistical law.
What you now know
Primes are the indivisible atoms from which every whole number is built, in exactly one way. The sieve finds them mechanically; Euclid’s proof guarantees they never run out; and the Ulam spiral shows their scatter is not pure noise — faint structure hides in it. The obvious next move is to stop staring at individual primes and start counting them in bulk. That simple shift of perspective — from atoms to statistics — is where the law begins to show.