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In number theory, Firoozbakht’s conjecture (or the Firoozbakht conjecture^[1]^[2]) is a conjecture about the distribution of prime numbers which states that $p_{n}^{1/n}\,$ (where $p_{n}\,$ is the nth prime) is a strictly decreasing function of n, i.e.,

p_{n+1}^{1/(n+1)}<p_{n}^{1/n}{\text{ for all }}n\geq 1.

Equivalently: $p_{n+1}<p_{n}^{1+1/n}{\text{ for all }}n\geq 1,$ see OEIS: A182134.

Another equivalent statement is $\left({\frac {p_{n+1}}{p_{n}}}\right)^{n}<p_{n}{\text{ for all }}n\geq 1,$ see OEIS: A205827 and OEIS: A182514.

The conjecture is named after Farideh Firoozbakht, from the University of Isfahan, who stated it in 1982. If this conjecture is true, then the prime gap function $g_{n}=p_{n+1}-p_{n}$ satisfies $g_{n}<(\log p_{n})^{2}-\log p_{n}{\text{ for all }}n>4,$ which is sharper than (hence implies) Cramér's conjecture $g_{n}=O((\log p_{n})^{2}).$ Using a table of maximal gaps, the inequality can be verified for all primes below 4×10¹⁸.^[3]

The conjecture is believed to be false, as it contradicts the Cramér–Granville heuristic.^[4]^[5]^[6] Pintz refers to this as MCM, the modified Cramér model.^[7]

Notes

^ Paulo Ribenboim 2004, p. 185
^ Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.
^ Gaps between consecutive primes
^ Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28.
^ Granville, Andrew. "Consequences of Legendre's conjecture"..
^ Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: pp. 388–399 {{citation}}: |pages= has extra text (help).
^ Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): pp. 232–471 {{citation}}: |pages= has extra text (help)

References

Ribenboim, Paulo (2004). The Little Book of Bigger Primes Second Edition. Springer-Verlag. ISBN 0-387-20169-6. {{cite book}}: Invalid |ref=harv (help)
I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th Ed., John Wiley & Sons, Inc. NY, 1991. See p. 408.

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

[2] Rivera, Carlos. "Conjecture 30. The Firoozbakht Conjecture". Retrieved 22 August 2012.

[3] Gaps between consecutive primes

[4] Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28.

[5] Granville, Andrew. "Consequences of Legendre's conjecture"..

[6] Granville, Andrew (1995), "Unexpected irregularities in the distribution of prime numbers" (PDF), Proceedings of the International Congress of Mathematicians, 1: pp. 388–399 {{citation}}: |pages= has extra text (help).

[7] Pintz, János (2007), "Cramér vs. Cramér: On Cramér's probabilistic model for primes", Funct. Approx. Comment. Math., 37 (2): pp. 232–471 {{citation}}: |pages= has extra text (help)

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 The conjecture is named after Farideh Firoozbakht, from the [[University of Isfahan]], who stated it in 1982. If this conjecture is true, then the [[prime gap]] function <math>g_n = p_{n+1} - p_n </math> satisfies <math> g_n < (\log p_n)^2 - \log p_n \text{ for all } n > 4,</math> which is sharper than (hence implies) [[Cramér's conjecture]] <math> g_n = O((\log p_n)^2).</math> Using a table of [[Prime gaps#Numerical results|maximal gaps]], the inequality can be verified for all primes below 4{{e|18}}.<ref>[http://www.ieeta.pt/~tos/gaps.html Gaps between consecutive primes]</ref>
+The conjecture is believed to be false, as it contradicts the [[Cramér's conjecture#Cramér–Granville conjecture|Cramér–Granville heuristic]].<ref>{{citation |last=Granville |first=A. |title=Harald Cramér and the distribution of prime numbers |journal=Scandinavian Actuarial Journal |volume=1 |issue= |year=1995 |pages=12–28 |url=http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf }}.</ref><ref>{{cite web|last=Granville|first=Andrew|title=Consequences of Legendre's conjecture|url=http://mathoverflow.net/a/122939/6043}}.</ref><ref>{{citation |last=Granville |first=Andrew |title=Unexpected irregularities in the distribution of prime numbers |journal=Proceedings of the International Congress of Mathematicians |volume=1 |year=1995 |pages=pp. 388–399 |url=http://www.dms.umontreal.ca/~andrew/PDF/icm.pdf }}.</ref> Pintz refers to this as MCM, the modified Cramér model.<ref>{{citation|last=Pintz|first=János|title=Cramér vs. Cramér: On Cramér's probabilistic model for primes|journal=Funct. Approx. Comment. Math.|volume=37|issue=2|year=2007|pages=pp. 232–471|url=http://projecteuclid.org/euclid.facm/1229619660 }}</ref>
 ==See also==

v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers