Overview
Eighty-three is the integer after 82 and before 84. It is an odd natural number and is prime: its only positive divisors are 1 and 83. Because of this, 83 appears in many classifications and lists in elementary number theory and in practical contexts such as chemistry and computing.
Mathematical properties
As a prime, 83 has several notable secondary attributes. It is the 23rd prime number, so it is a super-prime (a prime whose index in the sequence of primes is itself prime). It is also a safe prime because (83−1)/2 = 41 is prime, and it is a Sophie Germain prime since 2×83+1 = 167 is prime. In addition, 83 qualifies as a Chen prime because 83+2 = 85 is a product of two primes (a semiprime).
- Prime index: 23 (prime)
- Binary: 1010011
- Hexadecimal: 0x53
- Decimal digits sum: 8 + 3 = 11
Representations and patterns
In positional notation 83 is written as "83" in base ten, 1010011 in binary and 53 in hexadecimal. In the ASCII character set, decimal 83 corresponds to the uppercase letter S. The number is an Eisenstein prime with no imaginary part and a Gaussian prime, because of its congruence classes modulo 3 and 4 respectively. It can be expressed as the sum of three consecutive primes: 23 + 29 + 31 = 83.
History, nomenclature and context
The English name "eighty-three" follows the standard compound form for two-digit numbers. As a year number, "83" can refer to 83 BCE/BC or AD 83, and in modern times to years such as 1983; these usages are common in historical and cultural references. For background on prime numbers and classifications, see general references prime classifications and introductions to number theory at elementary number theory.
Uses and notable facts
In chemistry, the atomic number 83 identifies the element bismuth, a heavy post-transition metal with distinctive physical and chemical properties; for information about the element see bismuth (element 83). In computing and encoding, the value 83 has explicit meanings such as the ASCII code for 'S' (see character encodings). The number also appears in various identification systems, model numbers, and years. For additional numeric data and enumerations that include 83, consult numeric tables and databases at numerical references.
Distinctions
Among two-digit primes, 83 stands out for being both a safe prime and a Sophie Germain prime. It is not a twin prime, since neither 81 nor 85 is prime, but it does have relatively small prime gaps with nearby primes (79 and 89). Its appearance across mathematical classifications and physical indexing systems makes it a frequently encountered small prime in both theory and applications.