additiveprime

Additiveprime is a term used in number theory to denote a prime number that can be written as the sum of two prime numbers. Because every odd prime greater than 2 is odd, while the sum of two odd primes is even, the only possible representation of an odd prime as a sum of two primes is 2 plus another prime. Therefore, an odd prime p is additiveprime precisely when p − 2 is also prime. In other words, additiveprimes are exactly the larger members of twin prime pairs (p, p−2). The number 2 is not additiveprime since it cannot be expressed as a sum of two primes.

Examples and form: the initial additiveprimes are 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, and so on. Each of these corresponds to a twin prime pair (p−2, p). For instance, 5 corresponds to (3,5), and 139 corresponds to (137,139).

Properties and significance: the set of additiveprimes is closely tied to the twin prime conjecture. If there are infinitely many twin prime pairs, there would be infinitely many additiveprimes. Conversely, a proof that only finitely many twin primes exist would imply only finitely many additiveprimes. Beyond this, additiveprimes are used to illustrate basic parity considerations in prime representations and to explore the distribution of primes in twin prime patterns.

See also: twin primes, Goldbach’s conjecture.