Andrica's conjecture

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(a) The function A_n for the first 100 primes.
(b) The function A_n for the first 200 primes.
(c) The function A_n 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 A_n 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

\sqrt{p_{n+1}} - \sqrt{p_n} < 1

holds for all n, where p_n is the nth prime number. If g_n = p_{n+1} - p_n denotes the nth prime gap, then Andrica's conjecture can also be rewritten as

g_n < 2\sqrt{p_n} + 1.

Empirical evidence[edit]

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for n up to 1.3002 x 1016.[2] Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended to past 1.4 x 1018.

The discrete function A_n = \sqrt{p_{n+1}}-\sqrt{p_n} is plotted in the figures opposite. The high-water marks for A_n 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:

 p _ {n+1} ^ x - p_ n ^ x = 1,

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

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

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

 p _ {n+1} ^ x - p_ n ^ x < 1 for x < x_{\min}.

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]