Andrica's conjecture (named afte
Dorin Andrica
is a conjecture regarding the

gaps Gaps is a member of the Montana group of Patience games, where the goal is to arrange all the cards in suit from Deuce (a Two card) to King. Other solitaire games in this family include Spaces, Addiction, Vacancies, Clown Solitaire, Paganini, ...

between

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s. The conjecture states that the inequality :

\sqrt - \sqrt < 1

holds for all

n

, where

p_n

is the ''n''th prime number. If

g_n = p_ - p_n

denotes the ''n''th

prime gap A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ W ...

, then Andrica's conjecture can also be rewritten as :

g_n < 2\sqrt + 1.

Empirical evidence

Imran Ghory has used data on the largest prime gaps to confirm the conjecture for

n

up to 1.3002 × 10¹⁶.''Prime Numbers: The Most Mysterious Figures in Math'', John Wiley & Sons, Inc., 2005, p. 13. Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 × 10¹⁸. The discrete function

A_n = \sqrt-\sqrt

is plotted in the figures opposite. The high-water marks for

A_n

occur for ''n'' = 1, 2, and 4, with ''A''₄ ≈ 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 _  ^ x - p_ n ^ x = 1,

where

p_n

is the ''n''th prime and ''x'' can be any positive number. The largest possible solution for ''x'' is easily seen to occur for ''n''=1, when ''x''_max = 1. The smallest solution for ''x'' is conjectured to be ''x''_min ≈ 0.567148... which occurs for ''n'' = 30. This conjecture has also been stated as an

inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

, the generalized Andrica conjecture: :

p _  ^ x - p_ n ^ x < 1

for

x < x_.

References and notes

External links

''Andrica's Conjecture''
at

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* {{Prime number conjectures Conjectures about prime numbers Unsolved problems in number theory

Empirical evidence

Generalizations

See also

References and notes

External links