Interprime

Average of two consecutive odd primes

In mathematics, an interprime is the average of two consecutive odd primes.[1] For example, 9 is an interprime because it is the average of 7 and 11. The first interprimes are:

4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, ... (sequence A024675 in the OEIS)

Interprimes cannot be prime themselves (otherwise the primes would not have been consecutive).[1]

Since there are infinitely many primes, there are also infinitely many interprimes. The largest known interprime as of 2018[update] may be the 388342-digit n = 2996863034895 · 21290000, where n + 1 is the largest known twin prime.[2]

See also

  • Prime gap
  • Twin primes
  • Cousin prime
  • Sexy prime
  • Balanced prime – a prime number with equal-sized prime gaps above and below it

References

  1. ^ a b Weisstein, Eric W. "Interprime". mathworld.wolfram.com. Retrieved 2020-08-10.
  2. ^ Caldwell, Chris K. "The Top Twenty: Twin Primes". The Prime Pages. Retrieved 27 February 2017.
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Prime number classes
By formula
  • Fermat (22n + 1)
  • Mersenne (2p − 1)
  • Double Mersenne (22p−1 − 1)
  • Wagstaff (2p + 1)/3
  • Proth (k·2n + 1)
  • Factorial (n! ± 1)
  • Primorial (pn# ± 1)
  • Euclid (pn# + 1)
  • Pythagorean (4n + 1)
  • Pierpont (2m·3n + 1)
  • Quartan (x4 + y4)
  • Solinas (2m ± 2n ± 1)
  • Cullen (n·2n + 1)
  • Woodall (n·2n − 1)
  • Cuban (x3 − y3)/(x − y)
  • Leyland (xy + yx)
  • Thabit (3·2n − 1)
  • Williams ((b−1)·bn − 1)
  • Mills (⌊A3n⌋)
By integer sequence
By property
Base-dependentPatterns
  • Twin (p, p + 2)
  • Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
  • Triplet (p, p + 2 or p + 4, p + 6)
  • Quadruplet (p, p + 2, p + 6, p + 8)
  • k-tuple
  • Cousin (p, p + 4)
  • Sexy (p, p + 6)
  • Chen
  • Sophie Germain/Safe (p, 2p + 1)
  • Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
  • Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
  • Balanced (consecutive p − n, p, p + n)
By sizeComplex numbersComposite numbersRelated topicsFirst 61 primes
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