463 is an integer that lies between 462 and 464. It is an odd number and, importantly, a prime: it has no positive divisors other than 1 and itself. As a prime greater than 3, it falls into common arithmetic residue classes and participates in several small prime clusters with its neighbors.
Mathematical properties
463 is a prime number congruent to 3 modulo 4 (463 ≡ 3 (mod 4)). Because it is a prime of the form 4k+3, it cannot be expressed as a sum of two integer squares. It is congruent to 1 modulo 6 (463 ≡ 1 (mod 6)), as with many primes larger than 3. 463 forms a twin-prime pair with 461 (difference 2) and lies near 467, so it appears in a small cluster of primes (461, 463, 467).
Representations and notations
Common representations of 463 include these forms:
- Roman numerals: CDLXIII
- Binary: 111001111 (base 2)
- Hexadecimal: 1CF (base 16)
- Decimal digit sum: 4 + 6 + 3 = 13
As an integer label it behaves like any prime in modular arithmetic, quadratic residues, and primality-based constructions used in number theory and cryptography, although 463 itself is too small for modern cryptographic key sizes.
Occurrence and uses
Numbers such as 463 appear in indexing, cataloguing, model numbers, and year designations. When 463 is used as a year number it may refer to events in the year 463 CE or 463 BCE; those years are part of broader historical periods (late antiquity and classical antiquity respectively). As a simple prime, 463 is mainly of interest in elementary number theory, recreational mathematics, and in lists of prime numbers.
Notable facts and distinctions
463 is not a Sophie Germain prime (2p+1 = 927 is composite) and is not a safe prime. Its prime status makes it a basic example in discussions of prime congruence classes, small prime clusters, and properties that distinguish primes of the form 4k+1 from those of the form 4k+3.