A prime number is an integer greater than $1$ that has exactly two positive divisors — itself and $1$. Every other integer greater than $1$ is composite — it can be written as a product of two smaller integers. The first ten primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$, and there are infinitely many. Primes are the indivisible building blocks of every other integer, and they sit at the heart of modern cryptography.
Quick Reference
Field | Value |
|---|---|
Definition | An integer $> 1$ with exactly two positive divisors: $1$ and itself |
Symbol | No standard symbol; the set is sometimes denoted $\mathbb{P}$ |
First 10 primes | $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ |
Number of primes | Infinitely many (proved by Euclid, c. 300 BCE) |
Used in | Number theory, RSA cryptography, hashing, error-correcting codes |
What is a Prime Number?
A prime number has exactly two divisors: $1$ and itself. The number $7$ is prime because the only positive integers that divide it cleanly are $1$ and $7$. The number $9$ is not prime because $3$ also divides it — so $9$ has three divisors ($1, 3, 9$).
The number $1$ is not prime, by convention. If $1$ were allowed, the unique prime factorisation of every other integer would break — so mathematicians defined $1$ out of the set. The number $2$ is the only even prime; every other even number is divisible by $2$, so it has at least three divisors and cannot be prime.
Prime Numbers List — 1 to 100
There are exactly $25$ primes from $1$ to $100$:
$$2, 3, 5, 7, 11, 13, 17, 19, 23, 29,$$ $$31, 37, 41, 43, 47, 53, 59, 61, 67, 71,$$ $$73, 79, 83, 89, 97$$
The primes thin out as numbers grow. There are $25$ primes below $100$, but only $21$ between $100$ and $200$, and only $16$ between $200$ and $300$. The thinning is captured by the Prime Number Theorem: the count of primes up to $n$ is approximately $\frac{n}{\ln n}$.
How to Identify a Prime — The Sieve of Eratosthenes
The cleanest method to find all primes up to a limit $N$ is the sieve developed by Eratosthenes of Cyrene around $200$ BCE:
List all integers from $2$ to $N$.
Circle $2$ (the first prime). Cross out every multiple of $2$ that is greater than $2$.
Move to the next un-crossed number ($3$). Circle it. Cross out every multiple of $3$ greater than $3$.
Continue with the next un-crossed number, repeating.
Stop when you reach $\sqrt{N}$. Every remaining un-crossed number is prime.
The sieve gives the primes $1$ to $100$ in a few minutes by hand, and it is the foundation of every modern prime-testing algorithm.
Why Prime Numbers Matter
The primes are the building blocks of arithmetic. Euclid's Fundamental Theorem of Arithmetic states that every integer greater than $1$ can be written as a product of primes in exactly one way (up to ordering). The number $60 = 2 \times 2 \times 3 \times 5$, and no other product of primes equals $60$. This unique factorisation is what makes primes the atoms of integer arithmetic.
In modern cryptography, primes do the heavy lifting. RSA encryption — the system that protects bank transactions, email, and most internet communication — relies on the fact that multiplying two large primes is easy, but factoring the product back into those primes is computationally hard. Two primes with $300$ digits each multiply in a fraction of a second; their product would take more than the age of the universe to factor on the fastest known computer.
Cryptography is where prime numbers stopped being a curiosity of pure math and became infrastructure.
[MATHEMATICIANS & HISTORY CALLOUT]
Title: Euclid's proof that primes never run out
Mathematician: Euclid of Alexandria (c. 300 BCE, Greece)
Date and place: Alexandria, c. 300 BCE
The story: In Elements Book IX, Proposition 20, Euclid proved that there are infinitely many primes — using a single short argument that has stood for $2{,}300$ years. He assumed there were only finitely many primes, multiplied them all together, added $1$, and showed that the result either was a new prime or had a prime factor not in the original list. Either way, the assumption was wrong; there must be more primes than any finite list. The proof is one of the cleanest in mathematics, requires no machinery, and is still taught in number-theory courses worldwide.
Why it matters: The infinitude of primes is the foundation of every modern cryptographic system. RSA encryption works because primes are abundant — large enough primes are always available, and there is no finite list of "the primes worth knowing about." Euclid's proof, written before the wheel was widely deployed for transport, secures every credit-card transaction today.
The story did not stop with Euclid.
Pierre de Fermat (1607–1665, France) introduced the small theorem now named after him, which states that for any prime $p$ and any integer $a$ not divisible by $p$, $a^{p-1} \equiv 1 \pmod{p}$.
Bernhard Riemann (1826–1866, Germany) connected the distribution of primes to the zeros of an analytic function — the Riemann Hypothesis, still unsolved, is widely considered the most important open problem in mathematics.
Worked Examples Prime Numbers
Example 1: Test whether $97$ is prime (the wrong path first)
The instinct is to divide $97$ by every integer from $2$ to $96$. That works but is wasteful. Stop and use a better rule.
You only need to test divisors up to $\sqrt{97} \approx 9.85$ — so divisors $2, 3, 5, 7$ (skipping composites). If none of these divides $97$, the number is prime.
$97 \div 2$: $97$ is odd, not divisible.
$97 \div 3$: digit sum $9 + 7 = 16$, not divisible by $3$.
$97 \div 5$: $97$ does not end in $0$ or $5$, not divisible.
$97 \div 7$: $7 \times 13 = 91$, $7 \times 14 = 98$. Not divisible.
Final answer: $97$ is prime.
Example 2: Prime factorise $84$
$$84 = 2 \times 42 = 2 \times 2 \times 21 = 2 \times 2 \times 3 \times 7$$
Or written with exponents: $84 = 2^2 \times 3 \times 7$.
Final answer: $84 = 2^2 \times 3 \times 7$.
Example 3: Find the next prime after $31$
Test $32$: even, not prime. Test $33 = 3 \times 11$, not prime. Test $34$: even, not prime. Test $35 = 5 \times 7$, not prime.
Test $36$: even, not prime. Test $37$: not divisible by $2, 3, 5$, and $\sqrt{37} \approx 6.08$, so we need only check those. $37$ is prime.
Final answer: The next prime after $31$ is $37$.
Common Mistakes of Prime Number
Mistake 1: Treating $1$ as a prime number.
Where it slips in: The student reasons that $1$'s only divisors are $1$ and itself, so $1$ should be prime.
Don't do this: Include $1$ in any prime list.
The correct way: By convention, primes are integers strictly greater than $1$. Excluding $1$ preserves the unique-factorisation property.
Mistake 2: Treating $2$ as composite because it is even.
Where it slips in: The student remembers "even numbers are not prime" and applies the rule to $2$.
Don't do this: Skip $2$ when listing primes.
The correct way: $2$ is the only even prime — its only divisors are $1$ and $2$. Every other even number has $2$ as a third divisor, but $2$ itself does not have any "other" even divisor.
Mistake 3: Stopping the divisor test too early.
Where it slips in: Testing divisors $2, 3, 5$ on, say, $221$, finding none divide it, and declaring $221$ prime — but $221 = 13 \times 17$.
Don't do this: Stop at small primes.
The correct way: Test all primes up to $\sqrt{N}$. For $N = 221$, $\sqrt{221} \approx 14.87$, so test $2, 3, 5, 7, 11, 13$. The number $13$ catches the factorisation.
Mistake 4: Confusing prime with odd.
Where it slips in: The student assumes every odd number is prime.
Don't do this: Treat odd as a sufficient condition.
The correct way: Many odd numbers are composite — $9 = 3^2$, $15 = 3 \times 5$, $21 = 3 \times 7$. Odd is necessary (after $2$) but not sufficient.
The real-world version of Mistake 3 is the RSA challenge problems. RSA Laboratories published large numbers and offered cash for anyone who could factor them. Several were factored by distributed computing networks running for months. The fact that some are still unfactored after decades is exactly why RSA encryption is still in use — the divisor test simply does not finish in any reasonable time at the sizes used in real cryptography.
Try These Next
Now try this: list all primes between $100$ and $120$ using the Sieve of Eratosthenes (you only need to test divisors up to $\sqrt{120} \approx 10.95$, so primes $2, 3, 5, 7$). If you get stuck, return to the sieve description above.
If your child is comfortable with primes up to $100$, the natural next step is prime factorisation of larger numbers and modular arithmetic — both lead directly to RSA encryption. At Bhanzu, trainers connect prime numbers to where they actually do useful work, so the topic earns its place in the curriculum.
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