Fortunate number

The Fortunate number for a given positive natural number is defined as the difference between (= product of the first prime numbers) and the smallest prime number that is at least 2 greater than . ${\displaystyle f_{n}}$ ${\displaystyle n}$ ${\displaystyle P_{n}}$ ${\displaystyle n}$ ${\displaystyle P_{n}}$

They are named after Reo Franklin Fortune who studied them.

${\displaystyle f_{n}=\min {\Bigl \{}m>1\;{\Big |}\,\prod _{i=1}^{n}p_{i}+m\in \mathbb {P} {\Bigr \}}}$.

The first 50 Fortunate numbers are:

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, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, ... (Follow A005235 in OEIS )

The sequence of the Fortunate numbers is sorted and without repetitions :

3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397, 401, 409, 419, 421, 439, 443, ... (Follow A046066 in OEIS )

example

Calculating the 8th Fortunate number : The product of the first 8 prime numbers is . The next prime number greater by at least 2 is . This prime number is larger than the prime number product . So is . ${\displaystyle f_{8}}$
${\displaystyle P_{8}=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19=9699690}$ ${\displaystyle p=9699713}$ ${\displaystyle p-P_{8}=9699713-9699690=23}$ ${\displaystyle P_{8}}$ ${\displaystyle f_{8}=23}$

Fortunate prime numbers

A Fortunate number that is also prime is called a Fortunate prime number . So far, all known Fortunate numbers are prime numbers.

Reo Fortune assumed that all Fortunate numbers are prime, a still unsolved problem ( Fortune's conjecture , English Fortune's conjecture ).

Less fortunate numbers

In a corresponding way, Paul Carpenter also defines the less-fortunate numbers (or lesser fortunate numbers ) as

${\displaystyle l_{n}=\min {\Bigl \{}m>1\;{\Big |}\,\prod _{i=1}^{n}p_{i}-m\in \mathbb {P} {\Bigr \}}}$.

They are therefore defined as the difference between (= product of the first prime numbers) and the largest prime number, which is at least 2 smaller than . For these numbers, too, it is not known whether they are all prime. ${\displaystyle P_{n}}$ ${\displaystyle n}$ ${\displaystyle P_{n}}$

Examples

• The less-fortunate number is not defined because is and therefore no prime number exists that is at least 2 smaller than . ${\displaystyle l_{1}}$ ${\displaystyle P_{1}=2}$ ${\displaystyle P_{1}}$
• Calculation of the less fortunate number : ${\displaystyle l_{9}}$

The product of the first 9 prime numbers is . The next lower prime is . The prime number product is larger than the prime number . So is . ${\displaystyle P_{9}=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23=223092870}$ ${\displaystyle p=223092827}$ ${\displaystyle P_{9}-p=223092870-223092827=43}$ ${\displaystyle p}$ ${\displaystyle l_{9}=43}$

• The first 50 Less-fortunate numbers are (starting with ): ${\displaystyle l_{2}=3,l_{3}=7,\ldots }$

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, 223, 293, 593, 293, 457, 227, 311, 373, 257, ... ( Follow A055211 in OEIS )

• The sequence of the Less-fortunate numbers is sorted and without repetitions :

3, 7, 11, 13, 17, 23, 29, 41, 43, 47, 59, 67, 73, 83, 89, 97, 101, 107, 113, 127, 131, 151, 173, 179, 191, 197, 223, 227, 233, 239, 251, 257, 263, 281, 293, 307, 311, 313, 317, 331, 347, 367, 373, 379, 389, 431, 433, 439, 443, 449, ...

property

• The first 1000 less fortunate numbers are prime numbers.

guess

• It is assumed that all Less-fortunate-numbers are prime numbers.

See also

• Narcissistic number
• Munchausen number

literature

• Richard Kenneth Guy : Unsolved problems in number theory . 2nd Edition. Springer , 1994, ISBN 0-387-94289-0 , pp. 7-8 .

Web links

• Eric W. Weisstein : Fortunate Prime . In: MathWorld (English).
• Problems & Puzzles: Problems. Problem 4. Fortune's Conjecture. primepuzzles.net, accessed April 20, 2020 . 

Individual evidence

1. ^ Fortunate number . In: The Prime Glossary . Retrieved April 19, 2008.
2. ^ Richard Kenneth Guy : Unsolved problems in number theory. Springer , 1994, pp. 7–8 , accessed December 23, 2018 . 
3. ↑ ^{a b} Comments on OEIS A055211