Andrica's conjecture

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%2B1}} - \sqrt{p_n} < 1

holds for all n, where p_n is the nth prime number. If g_n = p_{n%2B1} - p_n denotes the nth prime gap, then Andrica's conjecture can also be rewritten as

g_n < 2\sqrt{p_n} %2B 1.

Contents

Empirical evidence

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for n up to 1.3002 x 1016.[2]

The discrete function A_n = \sqrt{p_{n%2B1}}-\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

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

p _ {n%2B1} ^ x - p_ n ^ x = 1,

where p_n is the nth prime and n can be any positive integer.

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.[3]

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

p _ {n%2B1} ^ x - p_ n ^ x < 1 for x < x_{\min}.

See also

References and notes

  1. ^ D. Andrica, "Note on a conjecture in prime number theory." Studia Univ. Babes-Bolyai Math. 31 (1986), no. 4, 44–48.
  2. ^ Prime Numbers: The Most Mysterious Figures in Math, John Wiley & Sons, Inc., 2005, p.13.
  3. ^ M. L. Perez. Five Smarandache Conjectures on Primes

External links