A Fortunate number, named after
Reo Fortune
Reo Franklin Fortune (27 March 1903 – 25 November 1979) was a New Zealand-born social anthropologist. Originally trained as a psychologist, Fortune was a student of some of the major theorists of British and American social anthropology i ...
, is the smallest integer ''m'' > 1 such that, for a given positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''n'', ''p''
''n''# + ''m'' is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, where the
primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...
''p''
''n''# is the product of the first ''n'' prime numbers.
For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes (2, 3, 5, 7, 11, 13 and 17), which is 510510. Adding 2 to that gives another even number, while adding 3 would give another multiple of 3. One would similarly rule out the integers up to 18. Adding 19, however, gives 510529, which is prime. Hence 19 is a Fortunate number. The Fortunate number for ''p''
''n''# is always above ''p''
''n'' and all its divisors are larger than ''p''
''n''. This is because ''p''
''n''#, and thus ''p''
''n''# + ''m'', is divisible by the
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s of ''m'' not larger than ''p''
''n''.
The Fortunate numbers for the first primorials are:
:
3,
5,
7,
13,
23,
17,
19, 23,
37,
61,
67, 61,
71,
47,
107,
59, 61,
109, etc. .
The Fortunate numbers sorted in numerical order with duplicates removed:
: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, ... .
Fortune conjectured that no Fortunate number is
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials ...
(''Fortune's conjecture'').
A Fortunate prime is a Fortunate number which is also a prime number. , all the known Fortunate numbers are prime. A composite Fortunate number, if one exists, would have to be greater than or equal to ''p''
''n+1''2.
References
* Chris Caldwell
"The Prime Glossary: Fortunate number"at the
Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin.
The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...
.
*
{{Classes of natural numbers
Integer sequences
Prime numbers