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Andrica's conjecture

Conjecture about gaps between prime numbers From Wikipedia, the free encyclopedia

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

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(a) The function for the first 100 primes.
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(b) The function for the first 200 primes.
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(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. It is conjectured that the function is always less than 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


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Empirical evidence

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for up to 1.3002×1016.[2] Using a more recent table of maximal gaps, the confirmation value can be extended exhaustively to 2×1019 > 264.

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.

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Generalizations

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Value of x in the generalized Andrica's conjecture for the first 100 primes, with the conjectured value of xmin labeled.

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 for x is easily seen to occur for n=1, when xmax = 1. The smallest solution for 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
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See also

References and notes

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