Fortunate number
Encyclopedia
A Fortunate number, named after Reo Fortune
, for a given positive integer
n is the smallest integer m > 1 such that pn# + m is a prime number
, where the primorial
pn# 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 pn# is always above pn. This is because pn#, and thus pn# + m, is divisible by the prime factor
s of m for m = 2 to pn.
The Fortunate numbers for the first primorials are:
The Fortunate numbers sorted in numerical order with duplicates removed:
Reo Fortune conjectured that no Fortunate number is composite
. A Fortunate prime is a Fortunate number which is also a prime number. , all the known Fortunate numbers are also Fortunate primes.
Reo Fortune
Reo Franklin Fortune was a New Zealand social anthropologist. Originally trained as a psychologist, Fortune was a lecturer in social anthropology at the Cambridge University, and a specialist in Melanesian language and culture. He was married to Margaret Mead, with whom he undertook field studies...
, for a given positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
n is the smallest integer m > 1 such that pn# + m is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, where the primorial
Primorial
In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied...
pn# 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 pn# is always above pn. This is because pn#, and thus pn# + m, is divisible by the prime factor
Prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
s of m for m = 2 to pn.
The Fortunate numbers for the first primorials are:
- 3, 5, 7, 1313 (number)13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
, 2323 (number)23 is the natural number following 22 and preceding 24.- In mathematics :Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. Twenty-three is also the fifth factorial prime, the third Woodall prime...
, 1717 (number)17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...
, 1919 (number)19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...
, 23, 3737 (number)37 is the natural number following 36 and preceding 38.-In mathematics:It is a prime number, the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form...
, 6161 (number)61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....
, 6767 (number)67 is the natural number following 66 and preceding 68. It is an odd number.-In mathematics:Sixty-seven is the 19th prime number , an irregular prime, a lucky prime, the sum of five consecutive primes , and a Heegner number.Since 18! + 1 is divisible by 67 but 67 is not one more than a multiple of...
, 61, 7171 (number)71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....
, 4747 (number)47 is the natural number following 46 and preceding 48.-In mathematics:Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, and the sixth Lucas prime. Forty-seven is a highly cototient number...
, 107107 (number)107 is the natural number following 106 and preceding 108.-In mathematics:One hundred [and] seven is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime....
, 5959 (number)59 is the natural number following 58 and preceding 60.-In mathematics:Fifty-nine is the 17th smallest prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary...
, 61, 109109 (number)109 is the natural number following 108 and preceding 110.-In mathematics:One hundred [and] nine is the 29th prime number, and also a Chen prime. The previous prime is 107, making them both twin primes...
, 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 .
Reo Fortune conjectured that no Fortunate number is composite
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
. A Fortunate prime is a Fortunate number which is also a prime number. , all the known Fortunate numbers are also Fortunate primes.