Package de.tilman_neumann.jml.primes.bounds

Class NthPrimeUpperBounds

java.lang.Object

de.tilman_neumann.jml.primes.bounds.NthPrimeUpperBounds

public class NthPrimeUpperBounds
extends Object

Bounds for the n.th prime p(n).

Author:

Tilman Neumann

Method Summary

Modifier and Type	Method	Description
static long	Axler2013(long n)	Axler 2013 page viii Korollar G for n >= 8009824.
static long	combinedUpperBound(long n)	Computes an upper bound for p(n) given n.
static long	Dusart1999(long n)	Dusart 1999 page 14 for n >= 39017.
static long	Dusart2010p7(long n)	Dusart 2010 page 7, Lemma 6.5: Holds for n >= 178974.
static long	Dusart2010p8(long n)	Dusart 2010 page 8, Proposition 6.6: Holds for n >= 688383.
static long	Robin1983(long n)	Robin 1983 "for n >= 7022": My tests say it works for n >= 8601.
static long	RobinJacobsen(long n)	Robin 1983 modified by D.Jacobsen for n = 6...39016 _only_.
static long	RosserSchoenfeld01(long n)	Rosser/Schoenfeld 1961 for n >= 6.
static long	RosserSchoenfeld02(long n)	Rosser/Schoenfeld 1961 for n >= 20.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

combinedUpperBound

public static long combinedUpperBound(long n)

Computes an upper bound for p(n) given n.

Parameters:

n -

Returns:

upper bound for n.th prime

RosserSchoenfeld01

public static long RosserSchoenfeld01(long n)

Rosser/Schoenfeld 1961 for n >= 6.

Parameters:

n -

Returns:

upper bound for n.th prime

RosserSchoenfeld02

public static long RosserSchoenfeld02(long n)

Rosser/Schoenfeld 1961 for n >= 20.

Parameters:

n -

Returns:

upper bound for n.th prime

RobinJacobsen

public static long RobinJacobsen(long n)

Robin 1983 modified by D.Jacobsen for n = 6...39016 _only_.

Parameters:

n -

Returns:

upper bound for n.th prime

Robin1983

public static long Robin1983(long n)

Robin 1983 "for n >= 7022": My tests say it works for n >= 8601.

Parameters:

n -

Returns:

upper bound for n.th prime

Dusart1999

public static long Dusart1999(long n)

Dusart 1999 page 14 for n >= 39017.

Parameters:

n -

Returns:

upper bound for n.th prime

Dusart2010p7

public static long Dusart2010p7(long n)

Dusart 2010 page 7, Lemma 6.5: Holds for n >= 178974.

Parameters:

n -

Returns:

upper bound for n.th prime

See Also:

"https://arxiv.org/PS_cache/arxiv/pdf/1002/1002.0442v1.pdf"

Dusart2010p8

public static long Dusart2010p8(long n)

Dusart 2010 page 8, Proposition 6.6: Holds for n >= 688383.

Parameters:

n -

Returns:

upper bound for n.th prime

See Also:

"https://arxiv.org/PS_cache/arxiv/pdf/1002/1002.0442v1.pdf"

Axler2013

public static long Axler2013(long n)

Axler 2013 page viii Korollar G for n >= 8009824.

Parameters:

n -

Returns:

upper bound for n.th prime