Catalytic reforming: Difference between revisions

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In [[number theory]], a '''Thabit number''', '''Thâbit ibn Kurrah number''', or '''321 number''' is an integer of the form 3·2<sup>''n''</sup>−1 for a [[non-negative integer]] ''n''.
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The first few Thabit numbers are:
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:[[2 (number)|2]], [[5 (number)|5]], [[11 (number)|11]], [[23 (number)|23]], [[47 (number)|47]], [[95 (number)|95]], 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, ... {{OEIS|id=A055010}}
 
The 9th Century [[Iraq]]i [[Muslim]] [[mathematician]], [[physician]], [[astronomer]] and [[translator]] [[Thābit ibn Qurra]] is credited as the first to study these numbers and their relation to [[amicable number]]s.<ref name="Rashed">{{cite book|last=Rashed|first=Roshdi|title=The development of Arabic mathematics: between arithmetic and algebra.|publisher=Kluwer Academic Publishers|location=Dordrecht, Boston, London|year=1994|volume=156|isbn=0-7923-2565-6|page=277}}</ref>
 
== Properties ==
The binary representation of the Thabit number 3·2<sup>''n''</sup>−1 is ''n''+2 digits long, consisting of "10" followed by ''n'' 1s.
 
The first few Thabit numbers that are [[prime number|prime]] (also known as '''321 primes'''):
 
:2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, ... {{OEIS|id=A007505}}
 
{{As of|2008|4}}, the known ''n'' values which give prime Thabit numbers are:<ref>[http://www.mersenneforum.org/321search/How%20many%20digits%20these%20primes%20have.html]</ref><ref>[http://primes.utm.edu/primes/page.php?id=84769]</ref>
 
:0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 47, 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 {{OEIS|id=A002235}}
 
The primes for ''n''≥234760 were found by the [[distributed computing]] project '''321 search'''.<ref>[http://www.mersenneforum.org/321search/The%20status%20of%20the%20search.html]</ref> The largest of these, 3·2<sup>4235414</sup>−1, has 1274988 digits and was found by Dylan Bennett in April 2008. The former record was 3·2<sup>3136255</sup>−1 with 944108 digits, found by Paul Underwood in March 2007.
 
In 2008, [[Primegrid]] has taken over the search for Thabit primes.<ref>http://primes.utm.edu/bios/page.php?id=479</ref> It is still searching and has already found Thabit primes for the following ''n'':<ref>http://primes.utm.edu/primes/lists/short.txt</ref>
:4235414, 6090515
 
It is also searching for primes of the form 3·2<sup>''n''</sup>+1. Such primes occur for the following ''n'':
:2291610, 5082306, 7033641
 
==Connection with amicable numbers==
 
When both ''n'' and ''n''-1 yield prime Thabit numbers, and <math>9 \cdot 2^{2n - 1} - 1</math> is also prime, a pair of amicable numbers can be calculated as follows:
: <math>2^n(3 \cdot 2^{n - 1} - 1)(3 \cdot 2^n - 1)</math> and <math>2^n(9 \cdot 2^{2n - 1} - 1).</math>
 
For example, ''n'' = 2 gives the Thabit number 11, and ''n'' = 1 gives the Thabit number 5, and our third term is 71. Then, 2<sup>2</sup>=4, multiplied by 5 and 11 results in [[220 (number)|220]], whose divisors add up to [[284 (number)|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.
 
== References ==
{{reflist}}
*{{MathWorld|title=Thâbit ibn Kurrah Number|urlname=ThabitibnKurrahNumber}}
 
{{Prime number classes|state=collapsed}}
{{Classes of natural numbers}}
 
[[Category:Integer sequences]]

Latest revision as of 11:40, 4 December 2014

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