The numbers Euclid proved never run out, 2,300 years ago.
12 splits into 2×6, 3×4, 2×2×3. So which numbers cannot be split at all?
The atoms of multiplication — primes.
Experiment
Hands-on experiment
Predict first — among 2 to 100, how many are prime (unsplittable by multiplication)?
Run the sieve — cross out multiples in turn
A prime's multiples are not prime. Cross out multiples of 2, 3, 5, 7 — what survives?
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Split matter and you reach atoms; split a natural number by multiplication and you reach numbers that split no further. These 'atoms of multiplication' are the primes.
The key fact is uniqueness: 60 is 2²×3×5 and no other combination of primes will ever produce it (the Fundamental Theorem of Arithmetic). This uniqueness lets us dissect the structure of any number completely.
Then the 20th century found an unexpected use. Multiplying two large primes takes an instant; splitting the product back takes supercomputers billions of years — that asymmetry became the heart of internet encryption (RSA).
Insight
Insights from the video
“Primes are the periodic table of the natural numbers.”
As chemists understand matter by decomposing it into elements, mathematicians understand numbers by decomposing them into primes. Divisors, multiples, GCDs — all readable from one factorization.
“Your card payment is guarded by prime numbers.”
Behind the https padlock is the product of two 600-digit primes. A pure-math property — 'hard to split' — became security itself.
Misconception
Common misconceptions
1 is a prime too.
If 1 counted as prime, 12 = 2²×3 = 1×2²×3 = 1²×2²×3… — infinitely many decompositions, destroying uniqueness. Excluding 1 isn't an arbitrary exception; it's the choice that protects unique factorization.
Primes must eventually run out as numbers grow.
There are infinitely many primes — Euclid proved it 2,300 years ago. They thin out but never stop, and multimillion-digit primes are still being discovered today.
Formula
Writing it as math
What the sieve filtered and the machine confirmed, written in mathematical language.
Definition of prime
A number that multiplication cannot split further. 2, 3, 5, 7, 11, … — the survivors of the sieve.
Fundamental Theorem of Arithmetic
Every natural number decomposes into primes, and up to order there is exactly one way — why the machine always returned the same answer.
Infinitude of primes (Euclid)
Assume finitely many primes, multiply them all and add 1: the result is divisible by none of them — contradiction. The primes never end.
In Real Life
Where you meet it in real life
Internet encryption (RSA)
Public-key cryptography stands on the asymmetry: you can publish the product of two large primes, yet no one can recover the primes. It locks card payments, messengers, and online banking.
The 13- and 17-year cicadas
North American cicadas emerge on 13- and 17-year cycles — primes. Evolution chose them to minimize overlap with predators' 2-, 3-, and 4-year cycles.
Hash tables and random numbers
Hash functions and pseudorandom generators favor modular arithmetic with prime-sized moduli — collisions and repeating patterns are minimized.
Gear design
Give two meshing gears coprime tooth counts (GCD 1) and the same pair of teeth meet as rarely as possible — wear spreads evenly.
Try Yourself
Test yourself
Q1Is 91 prime?Show answer ▾
No — 91 = 7×13. It survives 2, 3, and 5, so it looks prime, but 7 catches it. Since √91 ≈ 9.5 you only had to check up to 7 — exactly why the sieve stopped at multiples of 7.
Q2Sieving primes up to 100, why is it enough to cross out multiples of 7 and stop?Show answer ▾
Every composite n has a prime factor ≤ √n. A composite up to 100 has a prime factor ≤ √100 = 10, i.e., one of 2, 3, 5, 7. Sieve those four and everything left is prime.
Q3Can factorization count divisors of a big number? How many divisors does 720 have?Show answer ▾
720 = 2⁴×3²×5, so it has (4+1)(2+1)(1+1) = 30 divisors. You're choosing how many times to use each prime (0 to aᵢ) — counting made possible by unique factorization.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Irrational Numbers
After seeing the gaps in the number line, the inner structure of the naturals looks new.
← Irrational Numbers labLeads to next
Powers
The language of factorization, p^a — writing 'how many times the same atom was multiplied' is exactly what powers are.
Go to the Powers lab →Related
Labs worth exploring together
Related lab
Powers
The notation p^a in factorizations IS the power — the language for counting atoms.
Go to the Powers lab →Related lab
Counting Cases
Counting divisors is a counting problem: 'how many times shall I use each prime?'
Go to the Counting Cases lab →