Yes, 4,001 is a prime number. Its only factors are 1 and 4,001.
A number is prime if no smaller prime up to its square root divides it. The square root of 4,001 is about 63.25, so it is enough to test the primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61.
| Divide by | Remainder | Divides evenly? |
|---|---|---|
| 2 | 1 | No |
| 3 | 2 | No |
| 5 | 1 | No |
| 7 | 4 | No |
| 11 | 8 | No |
| 13 | 10 | No |
| 17 | 6 | No |
| 19 | 11 | No |
| 23 | 22 | No |
| 29 | 28 | No |
| 31 | 2 | No |
| 37 | 5 | No |
| 41 | 24 | No |
| 43 | 2 | No |
| 47 | 6 | No |
| 53 | 26 | No |
| 59 | 48 | No |
| 61 | 36 | No |
None of them divides 4,001 evenly, so 4,001 is prime.
| Number | Prime? |
|---|---|
| 2,027 | Yes |
| 2,029 | Yes |
| 2,999 | Yes |
| 3,001 | Yes |
| 5,003 | Yes |
| 7,919 | Yes |
| 9,973 | Yes |
| 10,007 | Yes |
Another number? Use the prime number checker for any value, or see the list of prime numbers.