Thabit number
Encyclopedia
In number theory
, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3·2n−1 for a non-negative integer n. The first few Thabit numbers are:
The binary representation of the Thabit number 3·2n−1 is n+2 digits long, consisting of "10" followed by n 1s.
The first few Thabit numbers that are prime
(also known as 321 primes):
, the known n values which give prime Thabit numbers are:
The primes for n≥234760 were found by the distributed computing
project 321 search. The largest of these, 3·24235414−1, has 1274988 digits and was found by Dylan Bennett in April 2008. The former record was 3·23136255−1 with 944108 digits, found by Paul Underwood in March 2007.
is also prime, a pair of amicable number
s can be calculated as follows:
So, for example, n=2 gives the Thabit number 11, and n=1 gives the Thabit number 5, and our third term is 71. Then, 22=4, multiplied by 5 and 11 results in 220
, whose divisors add up to 284
, and 4 times 71 is 284, whose divisors add up to 220.
The only known n satisfying these conditions are 2, 4 and 7, corresponding to the Thabit numbers 11, 47 and 383.
The 9th Century astronomer
Thābit ibn Qurra
is credited as the first to study these numbers and their relation to amicable numbers.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a Thabit number, Thâbit ibn Kurrah number, or 321 number is an integer of the form 3·2n−1 for a non-negative integer n. The first few Thabit numbers are:
- 2, 5, 1111 (number)11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
, 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...
, 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...
, 9595 (number)95 is the natural number following 94 and preceding 96.-In mathematics:Ninety-five is the thirtieth distinct semiprime and the fifth of the form...
, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ...
The binary representation of the Thabit number 3·2n−1 is n+2 digits long, consisting of "10" followed by n 1s.
The first few Thabit numbers that are prime
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...
(also known as 321 primes):
- 2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ...
, the known n values which give prime Thabit numbers are:
- 0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734, 3136255, 4235414
The primes for n≥234760 were found by the distributed computing
Distributed computing
Distributed computing is a field of computer science that studies distributed systems. A distributed system consists of multiple autonomous computers that communicate through a computer network. The computers interact with each other in order to achieve a common goal...
project 321 search. The largest of these, 3·24235414−1, has 1274988 digits and was found by Dylan Bennett in April 2008. The former record was 3·23136255−1 with 944108 digits, found by Paul Underwood in March 2007.
Amicable numbers
When both n and n-1 yield prime Thabit numbers, and is also prime, a pair of amicable numberAmicable number
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
s can be calculated as follows:
-
and 
So, for example, n=2 gives the Thabit number 11, and n=1 gives the Thabit number 5, and our third term is 71. Then, 22=4, multiplied by 5 and 11 results in 220
220 (number)
220 is the natural number following 219 and preceding 221.-In mathematics:It is a composite number, with its divisors being 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, making it an amicable number with 284...
, whose divisors add up to 284
284 (number)
Two hundred eighty-four is the natural number following 283 and preceding 285.Its divisors are 1, 2, 4, 71, and 142, adding up to 220, in turn, the divisors of 220 add up to 284, making the two a pair of amicable numbers....
, and 4 times 71 is 284, whose divisors add up to 220.
The only known n satisfying these conditions are 2, 4 and 7, corresponding to the Thabit numbers 11, 47 and 383.
The 9th Century astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
Thābit ibn Qurra
Thabit ibn Qurra
' was a mathematician, physician, astronomer and translator of the Islamic Golden Age.Ibn Qurra made important discoveries in algebra, geometry and astronomy...
is credited as the first to study these numbers and their relation to amicable numbers.