Is There a Biggest Prime Number?

A prime number is a whole number bigger than 1 whose only factors are 1 and itself: 7 can only be split as 1 × 7. A whole number bigger than 1 that has other factors is composite: 12 = 3 × 4. The number 1 is neither (you will see why below).

To find every prime up to 100, you don’t need to test each number one by one. A trick from ancient Greece, the sieve of Eratosthenes, does it by crossing out:

1. Start with 2, the smallest prime. Cross out every multiple of 2 after it: 4, 6, 8, …
2. The smallest number not crossed out is 3. It must be prime, because no smaller number divides it. Cross out its multiples.
3. Keep going: each time, the smallest number left is the next prime, and you cross out its multiples.

You only needed four primes, 2, 3, 5 and 7. Why could you stop at 7, with 11 still waiting?

Any composite number can be split into two factors, a small one and a big one, like 91 = 7 × 13. The smaller factor can’t be bigger than 10, because 11 × 11 = 121 is already past 100. So every composite number up to 100 has a factor of 10 or less, and that factor is 2, 3, 5 or 7 or is built from them (4, 6, 8, 9, 10). Either way it was crossed out by 2, 3, 5 or 7. What is left is prime.

Counting them

Here is how many primes there are in each hundred up to 1,000:

Numbers	1–100	101–200	201–300	301–400	401–500	501–600	601–700	701–800	801–900	901–1,000
Primes	25	21	16	16	17	14	16	14	15	14

They get fewer, but not smoothly: 401–500 has more than 301–400. Primes don’t follow a simple rule that tells you where the next one is. Now count much farther:

Up to	10	100	1,000	10,000	1 lakh	10 lakh	1 crore
Primes	4	25	168	1,229	9,592	78,498	664,579
About one number in	2.5	4	6	8	10	13	15

Up to 100, one number in four is prime. Up to 1 crore, only one in fifteen.

Close to a big number, the pattern is even neater. Near a 1 followed by some zeros, about one number in 2.3 × (the number of zeros) is prime. Near 1 crore (7 zeros), one in about 16; near 1 kharab (11 zeros), one in about 25. Counting 10,000 numbers after each start in our program: 1 in 16.3 after 1 crore, 1 in 20.5 after 1 arab (9 zeros, the rule says 20.7), 1 in 25.4 after 1 kharab. The great German mathematician Carl Friedrich Gauss noticed this thinning when he was 15 or 16, counting primes in blocks. Proving it took another century, until 1896, and the proof is far beyond school maths. But the question in this topic needs nothing like that.

Primes matter because every whole number is made of them. Keep splitting a composite number until you can’t split any more, and you are left with primes only:

360 = 36 × 10 = (6 × 6) × (2 × 5) = 2 × 3 × 2 × 3 × 2 × 5 = 2³ × 3² × 5

Start differently, 360 = 8 × 45 = (2 × 2 × 2) × (3 × 3 × 5), and you get exactly the same primes: three 2s, two 3s and one 5. However you split it, the primes at the end are always the same. This is the fundamental theorem of arithmetic: every whole number bigger than 1 is a prime or a product of primes, in just one way (apart from the order you write them in). That is why the primes are called the building blocks of the whole numbers, the way atoms build every substance.

It also explains why 1 is not counted as a prime. If it were, 6 = 2 × 3 = 1 × 2 × 3 = 1 × 1 × 2 × 3 … would have endless “different” factorizations, and the theorem would need an exception.

Prime factorizations make HCF and LCM, from Grade 7, easy to see. 84 = 2 × 2 × 3 × 7 and 90 = 2 × 3 × 3 × 5. They share one 2 and one 3, so their HCF is 2 × 3 = 6. The LCM takes each prime as often as either number needs it: 2 × 2 × 3 × 3 × 5 × 7 = 1,260.

Suppose Ishan is right, and there is a last prime. Then all the primes could be written on one list. Let’s test a list. Say someone claims the only primes are 2, 3 and 5.

Multiply them all, and add 1:

2 × 3 × 5 + 1 = 31

Now try to divide 31 by the primes on the list. 30 is a whole number of 2s, of 3s and of 5s, so 31, one more, is not: dividing by any of them leaves 1 over.

So none of 2, 3 or 5 divides 31. But every number bigger than 1 is built from primes. So 31 is either a prime itself, or built from primes that are not on the list. Either way, the list was not complete. (Here 31 is itself prime.)

Try it with other lists. Choose any primes you like.

The same thing happens with any list, however long. Call the list’s primes p, q, r, … and their product P. Then P + 1 leaves 1 over when divided by any of them, so it is either prime or built from primes not on the list. No list of primes can ever be complete. There are infinitely many primes, and there is no biggest one. This is Euclid’s theorem.

Mathematicians usually tell the argument like this:

1. Suppose the primes do run out, so that a list holds every one of them.
2. Multiply them all and add 1. The new number leaves 1 over when divided by every prime on the list.
3. But the new number is bigger than 1, so some prime divides it, and that prime can’t be on the list.
4. That contradicts step 1, where the list held every prime. So step 1 was false: the primes never run out.

This way of proving something, by supposing the opposite and showing that it leads to something impossible, is called proof by contradiction. Euclid himself put it a little more directly: give me any list of primes, and I will show you a prime that isn’t on it. Both say the same thing.

If there are infinitely many primes, can there still be long stretches with none at all? Yes, as long as you like. Here is a stretch of five:

1 × 2 × 3 × 4 × 5 × 6 = 720.

• 722 = 720 + 2: 2 divides 720 and 2, so 2 divides 722.
• 723 = 720 + 3: 3 divides both, so 3 divides 723.
• 724, 725, 726 the same way, divided by 4, 5 and 6.

Five composite numbers in a row: 722, 723, 724, 725 and 726. Want 100 in a row? Multiply 1 × 2 × 3 × … × 101, and add 2, 3, 4, …, 101 to it in turn: each sum is divided by the number you added. The product is a huge number, with 160 digits, but the argument works just as well. The first real stretch of over 100 composites starts much lower: between the primes 370,261 and 370,373 there are 111 composite numbers in a row.

Some primes are as close as two odd numbers can be: 3 and 5, 11 and 13, 29 and 31, 71 and 73. These are called twin primes. Do they too go on forever? Nobody knows. It is one of the oldest open questions in mathematics, simple enough to say to a Grade 6 student, and still unsolved in 2026.

If primes go on forever, perhaps there is a formula that makes them. In 1772 the Swiss mathematician Leonhard Euler wrote down a famous one: take any whole number n and work out

n × n + n + 41

For n = 0 it gives 41, a prime. For n = 1, 43; n = 2, 47; n = 3, 53; n = 10, 151. All prime. Our program checked every n from 0 to 39: forty primes in a row.

Mathematicians have proved that no formula of this kind (a polynomial: powers of n multiplied by whole numbers, added together) gives only primes, unless it gives the same number every time. Nobody has found a simple formula that makes every prime, and none is expected. The primes have to be hunted.

• What Don’t Mathematicians Know About Prime Numbers? (Grades 9–12) Twin primes, Goldbach’s guess and other questions as easy to ask as Euclid’s, still unanswered.
• Why Do the Divisibility Tests Work? How to tell if 3, 9 or 11 divides a number from its digits alone: quick checks for the factors of a number you are splitting into primes.
• Is a Pattern That Works Ten Times Always True? More patterns like Euler’s formula, that hold for a long time and then break.
• Can You Write a Message Only a Friend Can Read? Secret codes, with remainders.
• How Can Two Strangers Agree on a Secret in Public? (Grades 9–12) How primes keep your messages safe on the internet.
• Is There Room at a Full Hotel with Endless Rooms? What “infinitely many” really means.