Fortunate numbers

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Reo F. Fortune defined the $\scriptstyle n\,$ th Fortunate number as the smallest prime greater than $\scriptstyle {p_{n}}\#+1,\,n\,\geq \,1,\,$ where $\scriptstyle {p_{n}}\#,\,$ the $\scriptstyle n\,$ th primorial number, denotes the product of the primes up to $\scriptstyle p_{n}\,$.

Paul Carpenter defined the $\scriptstyle n\,$ th lesser Fortunate number as the greatest prime less than $\scriptstyle {p_{n}}\#-1,\,n\,\geq \,2,\,$ where $\scriptstyle {p_{n}}\#,\,$ the $\scriptstyle n\,$ th primorial number, denotes the product of the primes up to $\scriptstyle p_{n}\,$.

Conjectures

Fortunate numbers conjecture

Reo F. Fortune conjectured that Fortunate numbers are always prime, i.e. if $\scriptstyle q\,$ is the smallest prime greater than $\scriptstyle {p_{n}}\#+1\,$, then $\scriptstyle m\,=\,q-{p_{n}}\#,\,n\,\geq \,1,\,$ is prime.

For the $\scriptstyle n\,$ th Fortunate number to be composite, it has to be at least the square of $\scriptstyle p_{\{n+1\}}\,$ (the $\scriptstyle (n+1)\,$ thprime). Since $\scriptstyle p_{\{n+1\}}\,\sim \,(n+1)\log(n+1)\,\sim \,n\log n\,$ (Cf. prime number theorem) this gives $\scriptstyle (p_{\{n+1\}})^{2}\,\sim \,(n\log n)^{2}\,$.

The Cramér-Granville conjecture claims that $\scriptstyle p_{n+1}-p_{n}\,<\,M(\log p_{n})^{2},\,$ for some $\scriptstyle M\,>\,1,\,$ where $\scriptstyle p_{0}\,=\,2\,$ and $\scriptstyle p_{n}\,$ is the $\scriptstyle n\,$ th odd prime.

Thus we have

$q-{p_{n}}\#<M(\log({p_{n}}\#))^{2},\,M\,>\,1\,$.

Since

$\lim _{n\to \infty }({p_{n}}\#)^{\left({\frac {1}{p_{n}}}\right)}=e,\,$

we have

$\lim _{n\to \infty }{\frac {\log({p_{n}}\#)}{p_{n}}}=1,\,$

and thus

$\lim _{n\to \infty }\left\{{\frac {q-{p_{n}}\#}{(p_{n})^{2}}}<M\left({\frac {\log({p_{n}}\#)}{p_{n}}}\right)^{2}=M\right\},\,M\,>\,1\,$.

Now

$q-{p_{n}}\#\sim M(p_{n})^{2}\sim M(n\log n)^{2},\,M\,>\,1,\,$

is greater than

$(p_{\{n+1\}})^{2}\sim (n\log n)^{2},\,$

which seems to put the Fortunate numbers conjecture at risk, but this depends on how large $\scriptstyle M\,$ is! Might this be a case of the strong law of small numbers?

Note that if $\scriptstyle q\,=\,p_{n}\#+1\,$ is prime (Cf. quasi-primorial primes), then $\scriptstyle m\,=\,q-p_{n}\#\,=\,1\,$ is noncomposite (although it is not prime, it is a unit).

Lesser Fortunate numbers conjecture

Paul Carpenter conjectured that lesser Fortunate numbers are always prime, i.e. if $\scriptstyle q\,$ is the greatest prime less than $\scriptstyle {p_{n}}\#-1\,$, then $\scriptstyle m\,=\,{p_{n}}\#-q,\,n\,\geq \,2,\,$ is prime.

For the $\scriptstyle n\,$ th lesser Fortunate number to be composite, it has to be at least the square of $\scriptstyle p_{\{n+1\}}\,$ (the $\scriptstyle (n+1)\,$ thprime). Since $\scriptstyle p_{\{n+1\}}\,\sim \,(n+1)\log(n+1)\,\sim \,n\log n\,$ (Cf. prime number theorem) this gives $\scriptstyle (p_{\{n+1\}})^{2}\,\sim \,(n\log n)^{2}\,$.

The Cramér-Granville conjecture claims that $\scriptstyle p_{n+1}-p_{n}\,<\,M(\log p_{n})^{2},\,M\,>\,1,\,$ where $\scriptstyle p_{0}\,=\,2\,$ and $\scriptstyle p_{n}\,$ is the $\scriptstyle n\,$ th odd prime. Thus we have $\scriptstyle {p_{n}}\#-q\,<\,M(\log q)^{2},\,M\,>\,1\,$.

Note that if $\scriptstyle q\,=\,p_{n}\#-1\,$ is prime (Cf. almost-primorial primes), then $\scriptstyle m\,=\,p_{n}\#-q\,=\,1\,$ is noncomposite (although it is not prime, it is a unit).

Sequences

A005235 Fortunate numbers: least $\scriptstyle m\,>\,1\,$ such that $\scriptstyle {p_{n}}\#\,+\,m,\,n\,\geq \,1,\,$ is prime, where $\scriptstyle {p_{n}}\#,\,$ the $\scriptstyle n\,$ th primorial number, denotes the product of the primes up to $\scriptstyle p_{n}\,$.

{3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, ...}

A055211 Lesser Fortunate numbers: least $\scriptstyle m\,>\,1\,$ such that $\scriptstyle {p_{n}}\#\,-\,m,\,n\,\geq \,2,\,$ is prime, where $\scriptstyle {p_{n}}\#,\,$ the $\scriptstyle n\,$ th primorial number, denotes the product of the primes up to $\scriptstyle p_{n}\,$.

{3, 7, 11, 13, 17, 29, 23, 43, 41, 73, 59, 47, 89, 67, 73, 107, 89, 101, 127, 97, 83, 89, 97, 251, 131, 113, 151, 263, 251, 223, 179, 389, 281, 151, 197, 173, 239, 233, 191, 223, ...}