List of Prime Numbers: the Prime Integers in Any Range
There are 25 prime integers from 1 to 100, the first 2 and the last 97, and 168 from 1 to 1,000. Any range up to 100,000 numbers wide is listed with its count, twin primes and widest gap.
Primes in range
- Primes from 1 to 100
- 25 primes
- The primes
- 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
- Smallest and largest
- 2 and 97
- Share of the range that is prime
- 25 of 100 numbers, 25%
- Prime number theorem estimate About x / ln(x) primes lie below x. The true count, 25, runs ahead of the estimate, and the ratio of the two tends to 1 as x grows.
- 100 / ln(100) = 22
- Twin primes (p and p + 2 both prime)
- 8 pairs: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)
- Largest gap between neighbors
- 8, from 89 to 97
- Sum of the primes
- 1,060
| Numbers | How many primes | Primes |
|---|---|---|
| 1 to 10 | 4 | 2, 3, 5, 7 |
| 11 to 20 | 4 | 11, 13, 17, 19 |
| 21 to 30 | 2 | 23, 29 |
| 31 to 40 | 2 | 31, 37 |
| 41 to 50 | 3 | 41, 43, 47 |
| 51 to 60 | 2 | 53, 59 |
| 61 to 70 | 2 | 61, 67 |
| 71 to 80 | 3 | 71, 73, 79 |
| 81 to 90 | 2 | 83, 89 |
| 91 to 100 | 1 | 97 |
- Found with a sieve of Eratosthenes: cross out every multiple of each prime up to the square root of the range's end, and what is left is prime.
- Ranges of up to 100,000 numbers, anywhere below a trillion.
How the Prime Factorization Calculator works
A prime number is a whole number above 1 whose only divisors are 1 and itself. The primes up to 100 are 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.
The sieve of Eratosthenes finds every prime in a range at once. Write out the numbers from 2 to 100. Keep 2 and cross out every later multiple of 2. The next number left standing, 3, is prime: keep it and cross out its multiples. Do the same for 5 and 7. The next survivor is 11, and 11 × 11 = 121 is past 100, so everything still standing is prime. A composite number up to 100 always has a prime factor no bigger than 10, which is why four rounds clear the whole list.
Each round can start at the prime's square. By the time you reach 7, its multiples 14, 21, 28, 35 and 42 are already gone, crossed out by 2, 3 and 5, so the first new one is 49.
How many primes there are. There are 4 primes below 10, 25 below 100, 168 below 1,000, 1,229 below 10,000 and 9,592 below 100,000. They thin out at a predictable rate: the prime number theorem says roughly x / ln(x) of the numbers up to x are prime. For 100,000 that estimate is 8,686 against the true 9,592, and the relative error keeps shrinking as x grows. A quarter of the numbers up to 100 are prime, and under a tenth of those up to 100,000.
Twin primes and gaps. Twin primes are pairs two apart, such as 41 and 43. There are 8 pairs below 100 and 1,224 below 100,000; whether they go on forever is still unproved. The gaps between neighboring primes widen, unevenly: the widest below 100 is 8, from 89 to 97, and the widest below 100,000 is 72, from 31,397 to 31,469.
Prime Factorization Calculator FAQ
What are the prime numbers from 1 to 50?
There are 15: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. Their sum is 328.
How many prime numbers are there between 100 and 200?
21, from 101 to 199. Seven of those pairs are twin primes: 101 and 103, 107 and 109, 137 and 139, 149 and 151, 179 and 181, 191 and 193, and 197 and 199.
Is 1 a prime number?
No. A prime has exactly two divisors and 1 has one, so every list of primes starts at 2, the only even prime.
Do the prime numbers ever run out?
No. Multiply any finite list of primes together and add 1: the result leaves a remainder of 1 when divided by every prime on the list, so its own prime factors are missing from it. Euclid gave that proof around 300 BC.