Information AboutRepunit |
| CATEGORIES ABOUT REPUNIT | |
| prime numbers | |
| base-dependent integer sequences | |
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DEFINITION The repunits are defined mathematically as : Thus, the number ''R''''n'' consists of ''n'' copies of the digit 1. The sequence of repunits starts in OEIS ). REPUNIT PRIMES Historically, the definition of repunits was motivated by recreational mathematicians looking for Prime Factors of such numbers. It is easy to show that if ''n'' is divisible by ''a'', then ''R''''n'' is divisible by ''R''''a''. For example, 9 is divisible by 3, and indeed ''R''9 is divisible by ''R''3—in fact, 111111111 = 111 · 1001001. Thus, for ''R''''n'' to be prime ''n'' must necessarily be prime. But it is not sufficient for ''n'' to be prime; for example, ''R''3 = 111 = 3 · 37 is not prime. Except for this case of ''R''3, ''p'' can only divide ''R''''n'' if ''p = 2kn + 1'' for some ''k''. Repunit primes turn out to be rare. ''R''''n'' is prime for ''n'' = 2, 19, 23, 317, 1031,... (sequence in OEIS). ''R''49081 and ''R''86453 are Probably Prime . It has been conjectured that there are infinitely many repunit primes . GENERALIZATIONS Professional mathematicians used to consider repunits an arbitrary concept, arguing that it depends on the use of Decimal Numerals . But the arbitrariness can be removed or reduced by generalizing the idea to base-''b'' repunits: : In fact, the base-2 repunits are the well-respected Mersenne Number s ''M''''n'' = 2''n'' − 1. The Cunningham Project endeavours to document the integer factorizations of (among other numbers) the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12. The prime repunits are a subset of the Permutable Prime s, i.e., primes that remain prime after any Permutation of their digits. It is easy to [http://www.caliban.org.uk/pmwiki/pmwiki.php?n=Blogs.RichardRothwell.RepUnits prove] that given n, such that n is not exactly divisible by 2 or p, there exists a repunit in base 2p that is a multiple of n. SEE ALSO EXTERNAL LINKS Web sites
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