Andrica's conjecture

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(a) The function for the first 100 primes.
(b) The function for the first 200 primes.
(c) The function for the first 500 primes.
Graphical proof for Andrica's conjecture for the first (a)100, (b)200 and (c)500 prime numbers. The function is always less than 1.

Andrica's conjecture (named after Dorin Andrica) is a conjecture regarding the gaps between prime numbers.[1]

The conjecture states that the inequality

holds for all , where is the nth prime number. If denotes the nth prime gap, then Andrica's conjecture can also be rewritten as

Empirical evidence[edit]

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for up to 1.3002 × 1016.[2] Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 × 1018.

The discrete function is plotted in the figures opposite. The high-water marks for occur for n = 1, 2, and 4, with A4 ≈ 0.670873..., with no larger value among the first 105 primes. Since the Andrica function decreases asymptotically as n increases, a prime gap of ever increasing size is needed to make the difference large as n becomes large. It therefore seems highly likely the conjecture is true, although this has not yet been proven.

Generalizations[edit]

As a generalization of Andrica's conjecture, the following equation has been considered:

where is the nth prime and x can be any positive number.

The largest possible solution x is easily seen to occur for , when xmax = 1. The smallest solution x is conjectured to be xmin ≈ 0.567148... (sequence A038458 in the OEIS) which occurs for n = 30.

This conjecture has also been stated as an inequality, the generalized Andrica conjecture:

for

See also[edit]

References and notes[edit]

  1. ^ Andrica, D. (1986). "Note on a conjecture in prime number theory". Studia Univ. Babes–Bolyai Math. 31 (4): 44–48. ISSN 0252-1938. Zbl 0623.10030. 
  2. ^ Prime Numbers: The Most Mysterious Figures in Math, John Wiley & Sons, Inc., 2005, p. 13.

External links[edit]