Prime Numbers: Checker and Table up to 10,000

What prime numbers are

A prime number is an integer greater than 1 that has exactly two divisors: 1 and itself. 2, 3, 5, 7 and 11 are prime; 4 is not, because it is also divisible by 2. The number 1 is neither prime nor composite, because it has only one divisor, and 2 is the only even prime number: every other even number is divisible by 2. Numbers greater than 1 that are not prime are called composite and can be written as a product of prime numbers in only one way, apart from the order of the factors: this is the prime factorization, guaranteed by the fundamental theorem of arithmetic. There are infinitely many prime numbers, as Euclid already proved.

How to use the calculator

Type an integer and press “Check” or Enter: the calculator tells you whether it is prime and, if it is not, shows its prime factorization, the number of divisors and the smallest prime divisor. You also get the previous and the next prime number. Below there is the table of prime numbers, which you can extend up to 100, 1,000 or 10,000; if the number you checked is prime and falls within the table, it is highlighted.

You can write the number with a dot as the thousands separator, Italian style: 1.250 means one thousand two hundred fifty. A number with a comma, or with a dot that does not separate groups of three digits such as 2.5, is rejected with a message, because prime numbers are integers. The limit is one quadrillion (10¹⁵). The calculation takes place in your browser: no data is sent or saved.

Prime numbers up to 100, 1,000 and 10,000

There are 25 prime numbers less than 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 higher you go, the sparser the primes become: up to 100 one number in four is prime, up to 10,000 about one in eight.

How to tell whether a number is prime

You just need to try dividing it by the numbers up to its square root. The reason is simple: if n = a × b with a ≤ b, then a × a ≤ n, so the smallest factor does not exceed the square root of n. If no number up to the square root divides n, n is prime. For 97 the square root is about 9.8: 2, 3, 5 and 7 are enough, and none of them divides 97, which is therefore prime.

The calculator adds a refinement: after 2 and 3 it only tries numbers of the form 6k − 1 and 6k + 1 (5, 7, 11, 13, 17, 19…), because all the others are multiples of 2 or 3. For a prime number close to the limit of 10¹⁵ about ten million divisions are still needed, so the answer may take a moment.

Calculation example

Here is what the calculator returns for three numbers:

The number of divisors is derived from the factorization: add 1 to each exponent and multiply the results. For 360 = 2³ × 3² × 5 you get (3 + 1) × (2 + 1) × (1 + 1) = 24. 1,000,000,007 is also prime: the previous prime is 999,999,937 and the next is 1,000,000,009.

The sieve of Eratosthenes

To build the table, a very ancient method is used, the sieve of Eratosthenes: write the numbers from 2 onwards, take the first number not crossed out and cross out all its multiples, then repeat with the next remaining number. Those that remain are the prime numbers. Here too it is enough to reach the square root of the limit: for the table up to 10,000 you cross out the multiples of the primes up to 100, and every remaining number is prime.

Limits of the calculator

The cap of one quadrillion is not arbitrary: the browser represents integers exactly only up to 2⁵³ − 1, that is 9,007,199,254,740,991, and beyond that threshold the remainders of divisions would no longer be reliable. The method used is exact but becomes slow with very large numbers, for which different algorithms are needed that are not used here. The calculator shows how many divisors there are, not the list of them. The table stops at 10,000 to stay readable on screen. It is a tool for study and checking: for a class test it is still best to be able to redo the procedure by hand.

Frequently asked questions

How many prime numbers are there from 1 to 100?

There are 25: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. Up to 1,000 there are 168 prime numbers and up to 10,000 there are 1,229.

Why is 1 not a prime number?

Because a prime number must have exactly two distinct divisors, 1 and itself, while 1 has only one. There is also a practical reason: if 1 were prime, prime factorization would no longer be unique, because 6 could be written as 2 × 3, as 1 × 2 × 3 and so on.

How do you break a number down into prime factors?

Divide the number by the smallest prime that divides it and repeat with the quotient, until 1 is left. For 360: divide three times by 2 (180, 90, 45), twice by 3 (15, 5) and once by 5. The result is 360 = 2³ × 3² × 5.

How do you count the divisors of a number?

From the prime factorization: add 1 to each exponent and multiply the results. For 360 = 2³ × 3² × 5 the divisors are (3 + 1) × (2 + 1) × (1 + 1) = 24. A prime number always has 2 divisors.

Related tools: for the greatest common divisor of two or more numbers use the GCD calculator, which finds it with the Euclidean algorithm and shows the prime factorization of each number; the guide to the greatest common divisor explains both the factorization method and Euclid’s method.