Andrica's conjecture

(a) The function

A_n

for the first 100 primes.

(b) The function

A_n

for the first 200 primes.

A_n

for the first 500 primes.

Graphical proof for Andrica's conjecture for the first (a)100, (b)200 and (c)500 prime numbers. The function

A_n

is always less than 1.

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 $p_n$ is the nth prime number. If $g_n = p_{n+1} - p_n$ denotes the nth prime gap, then Andrica's conjecture can also be rewritten as

g_n < 2\sqrt{p_n} + 1.

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] Using a table of maximal gaps and the above gap inequality, the confirmation value can be extended exhaustively to 4 x 10¹⁸.

The discrete function $A_n = \sqrt{p_{n+1}}-\sqrt{p_n}$ 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+1} ^ x - p_ n ^ x = 1,

where $p_n$ is the nth prime and x can be any positive number.

The largest possible solution x is easily seen to occur for $n=1$ , when x_max=1. The smallest solution x is conjectured to be x_min ≈ 0.567148... (sequence A038458 in OEIS) which occurs for n = 30.

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

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

for

x < x_{\min}.

References and notes

↑ Andrica, D. (1986). "Note on a conjecture in prime number theory". Studia Univ. Babes-Bolyai Math. 31 (4): 44–48. ISSN 0252-1938. Zbl 0623.10030.
↑ Prime Numbers: The Most Mysterious Figures in Math, John Wiley & Sons, Inc., 2005, p.13.

Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 978-0-387-20860-2. Zbl 1058.11001.

External links

Prime number conjectures

Hardy–Littlewood 1st 2nd Agoh–Giuga Andrica's Artin's Bateman–Horn Brocard's Bunyakovsky Chinese hypothesis Cramér's Dickson's Elliott–Halberstam Firoozbakht's Gilbreath's Grimm's Landau's problems Goldbach's weak Legendre's Twin prime Legendre's constant Lemoine's Mersenne Oppermann's Polignac's Pólya Redmond–Sun Schinzel's hypothesis H Waring's prime number

Andrica's conjecture

Empirical evidence

Generalizations

See also

References and notes

External links