Andrica's conjecture (named afte
Dorin Andrica is a
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
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
:
holds for all
, where
is the ''n''th prime number. If
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
:
Empirical evidence
Imran Ghory has used data on the largest prime gaps to confirm the conjecture for
up to 1.3002 × 10
16.
[''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 × 10
18.
The discrete function
is plotted in the figures opposite. The high-water marks for
occur for ''n'' = 1, 2, and 4, with ''A''
4 ≈ 0.670873..., with no larger value among the first 10
5 primes. Since the Andrica function decreases
asymptotically
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
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:
:
where
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:
:
for
See also
*
Cramér's conjecture In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and t ...
*
Legendre's conjecture
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. The conjecture is one of Landau's problems (1912) on prime numbers; , the conjecture has neither be ...
*
Firoozbakht's conjecture
In number theory, Firoozbakht's conjecture (or the Firoozbakht conjecture) is a conjecture about the distribution of prime numbers. It is named after the Iranian mathematician Farideh Firoozbakht who stated it first in 1982.
The conjecture states ...
References and notes
*
External links
''Andrica's Conjecture''at
PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ...
''Generalized Andrica conjecture''at
PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ...
*
{{Prime number conjectures
Conjectures about prime numbers
Unsolved problems in number theory