Andrica's conjecture

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The function An for the first 100 primes.
The function An for the first 100 primes.
The function An for the first 200 primes.
The function An for the first 200 primes.
The function An for the first 500 primes.
The function An for the first 500 primes.

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 pn is the nth prime number. If gn = pn + 1 − pn denotes the nth prime gap, then Andrica's conjecture can also be rewritten as

g_n < 2\sqrt{p_n} + 1.

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[edit] 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+1}} - \sqrt{p_n} is plotted in the figures opposite. The high-water marks for An occur for n = 1, 2, and 4, with A4\approx 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.

[edit] Generalizations

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

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

where pn 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 \approx 0.567148 ... (sequence A038458 in OEIS), known as the Smarandache constant, which occurs for n = 30. [3]

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

p _ {n+1} ^ x - p_ n ^ x < 1 for x < xmin.

[edit] See also

[edit] 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

[edit] External links