401

401 is a natural number that comes after 400 and before 402. It is a prime number—that is, it has no divisors other than 1 and itself. In standard decimal notation, it consists of three digits: four hundred and one.

401 is written as 4·10² + 0·10 + 1. It is an odd number, since it ends in 1. The sum of its digits is 4+0+1 = 5, which shows that it is not divisible by 3 and not divisible by 9.

Mathematical properties

That 401 is prime can be verified by a simple method: the square root of 401 is close to 20, so it is sufficient to test whether it is divisible by primes less than 20, namely 2, 3, 5, 7, 11, 13, 17, and 19. None of them divide 401. The preceding prime is 397, and the next is 409.

401 can be expressed as the sum of two squares: 401 = 20² + 1² = 400 + 1. This is notable because every prime that leaves a remainder of 1 when divided by 4—401 = 4·100 + 1—can be expressed as a sum of two squares in exactly one way. This is a well-known result known as Fermat's theorem on sums of two squares.

401 is not part of a pair of twin primes, because 399 = 3·7·19 and 403 = 13·31 are both composite numbers. When 401 is multiplied by 2 and 1 is added, the result is 803 = 11·73, which means 401 is not a Sophie Germain prime.

Other uses

Outside of mathematics, 401 is used as a numerical code. For example, in the HTTP protocol of the internet, the response code 401 (Unauthorized) is sent when a request requires authentication but the credentials provided are missing or invalid. The number also appears in road designations, traffic sign numbering, and storage systems.