Firuzbeht hypothesis

Spacing function between prime numbers

The Firuzbaht ^[1] ^[2] hypothesis is a hypothesis about the distribution of prime numbers . The hypothesis is named after the Iranian mathematician Farida Firuzbeht from the university in Isfahan, who expressed it in 1982.

Content

Hypothesis Approval

The hypothesis states that $p_{n}^{1/n}$ ${\ displaystyle p_ {n} ^ {1 / n}}$ (Where $p_{n}$ ${\ displaystyle p_ {n}}$ - n- th prime number) is a strictly decreasing function of n , i.e.

{\sqrt[{n+1}]{p_{n+1}}}<{\sqrt[{n}]{p_{n}}}

{\ displaystyle {\ sqrt [{n + 1}] {p_ {n + 1}}} <{\ sqrt [{n}] {p_ {n}}}}

for all

n\geq 1.

{\ displaystyle n \ geq 1.}

Equivalent to:

p_{n+1}<p_{n}^{1+{\frac {1}{n}}}

{\ displaystyle p_ {n + 1} <p_ {n} ^ {1 + {\ frac {1} {n}}}}

for all

n\geq 1,

{\ displaystyle n \ geq 1,}

see sequences A182134 , A246782 .

Confirmation of hypothesis

Using the table of maximum intervals , Farida Firuzbeht tested her hypothesis to 4,444⋅10 ¹² ^[2] . With the extended max interval table, the hypothesis was tested for all primes up to $10^{19}$ ${\ displaystyle 10 ^ {19}}$ ^[3] ^[4] .

Relationship with other hypotheses

If the hypothesis is true, then the function of the intervals between prime numbers $g_{n}=p_{n+1}-p_{n}$ ${\ displaystyle g_ {n} = p_ {n + 1} -p_ {n}}$ must satisfy the inequality ^[5] :

g_{n}<(\log p_{n})^{2}-\log p_{n}

{\ displaystyle g_ {n} <(\ log p_ {n}) ^ {2} - \ log p_ {n}}

for all

n>4.

{\ displaystyle n> 4.}

Moreover ^[6] ,

g_{n}<(\log p_{n})^{2}-\log p_{n}-1

{\ displaystyle g_ {n} <(\ log p_ {n}) ^ {2} - \ log p_ {n} -1}

for all

n>9,

{\ displaystyle n> 9,}

see also sequence A111943 . The hypothesis is among the strongest hypotheses about upper bounds for the intervals between prime numbers; it is even somewhat stronger than Cramer and Shanks hypotheses ^[4] . The hypothesis implies the strong form of Cramer's hypothesis , and therefore it is incompatible with the heuristics of Granville, Pints, ^[7] ^[8] ^[9] and Meier ^[10] ^[11] , in which

g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}

{\ displaystyle g_ {n}> {\ frac {2- \ varepsilon} {e ^ {\ gamma}}} (\ log p_ {n}) ^ {2}}

meets infinitely many times for any $\varepsilon >0,$ ${\ displaystyle \ varepsilon> 0,}$ Where $\gamma$ ${\ displaystyle \ gamma}$ means Euler's constant - Mascheroni .

Two related hypotheses (see comments on the sequence A182514 )

\left({\frac {\log(p_{n+1})}{\log(p_{n})}}\right)^{n}<e,

{\ displaystyle \ left ({\ frac {\ log (p_ {n + 1})} {{log (p_ {n})}} \ right) ^ {n} <e,}

which is somewhat weaker and

\left({\frac {p_{n+1}}{p_{n}}}\right)^{n}<n\log(n)

{\ displaystyle \ left ({\ frac {p_ {n + 1}} {p_ {n}}} \ right) ^ {n} <n \ log (n)}

for all

n>5,

{\ displaystyle n> 5,}

which is stronger.

Notes

↑ Ribenboim, 2004 , p. 185.
↑ ¹ ² Rivera, 2012 .
↑ Gaps between consecutive primes
↑ ¹ ² Kourbatov, 2018 .
↑ Sinha, 2010 , p. 1–10.
↑ Kourbatov, 2015 .
↑ Granville, 1995 , p. 12–28.
↑ Granville, 1995 , p. 388–399.
↑ Pintz, 2007 , p. 232–471.
↑ Adleman, McCurley, 1994 , p. 291–322.
↑ Maier, 1985 , p. 221-225.

Literature

Paulo Ribenboim. The Little Book of Prizes Second Edition. - 2004. - ISBN 0-387-20169-6 .
Carlos Rivera. Conjecture 30. The Firoozbakht Conjecture . - 2012.
Granville A. Harald Cramér and the distribution of prime numbers // Scandinavian Actuarial Journal. - 1995. - Vol. one.
Andrew Granville. Unexpected irregularities in the distribution of prime numbers // Proceedings of the International Congress of Mathematicians. - 1995. - Vol. one.
Janos Pintz. Cramér vs. Cramér: On Cramér's probabilistic model for primes // Funct. Approx. Comment. Math .. - 2007. - V. 37 , no. 2
Leonard Adleman , Kevin McCurley. Theoretical Complexity, II. Algorithmic number theory (Ithaca, NY, 1994) ,. - Berlin: Springer, 1994. - T. 877. - (Lecture Notes in Comput. Sci.).
Helmut Maier. Primes in short intervals // The Michigan Mathematical Journal. - 1985. - V. 32 , no. 2 - ISSN 0026-2285 . - DOI : 10.1307 / mmj / 1029003189 .
Alexei Kourbatov. Prime Gaps: Firoozbakht Conjecture . - 2018.
Nilotpal Kanti Sinha. Cramer's conjecture. - 2010. - arXiv : 1010.1399 .
Alexei Kourbatov. Upper bounds for the gaps related to Firoozbakht's conjecture // Journal of Integer Sequences. - 2015. - Vol. 18. - arXiv : 1506.03042 .

Links

Hans Riesel. Prime Numbers and Computer Methods for Factorization, Second Edition. - Birkhauser, 1985. - ISBN 3-7643-3291-3 .

[_f33f380745996816-1] Ribenboim, 2004 , p. 185.

[_1de8c8164b0a6db3-2] ¹ ² Rivera, 2012 .

[3] Gaps between consecutive primes

[_7b3fd28c6367edcb-4] ¹ ² Kourbatov, 2018 .

[_b3dd1c50abb9ff4e-5] Sinha, 2010 , p. 1–10.

[_88e700f6cc5b81f2-6] Kourbatov, 2015 .

[_2488ae7fb811eee3-7] Granville, 1995 , p. 12–28.

[_9210c92eb3ade57e-8] Granville, 1995 , p. 388–399.

[_95e81e5a2dc8b2ab-9] Pintz, 2007 , p. 232–471.

[_077c422c517de4f4-10] Adleman, McCurley, 1994 , p. 291–322.

[_36dfc7905ad7c121-11] Maier, 1985 , p. 221-225.

Firuzbeht hypothesis

Content

Hypothesis Approval

Confirmation of hypothesis

Relationship with other hypotheses

See also

Notes

Literature

Links

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