Arabic Six Children's Stories

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is a pronic number and the only semiprime to be. [4] It is the first discrete biprime (2 × 3) [5] which makes it the first member of the (2 × q) discrete biprime family, where q is a higher prime. All primes above 3 are of the form 6 n ± 1 for n ≥ 1.

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All integers n {\displaystyle n} that are multiples of 6 are pseudoperfect (all multiples of a pseudoperfect number are pseudoperfect). Six is also the smallest Granville number, or S {\displaystyle {\mathcal {S}}} -perfect number. [8] By Zsigmondy's theorem, if n {\displaystyle n} is a natural number that is not 1 or 6, then there is a prime p {\displaystyle p} such that o r d p ( 2 ) = n {\displaystyle ord_{p}(2)=n} . See A112927 for such p {\displaystyle p} . Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler". [9] Six is a congruent number. [10] The ring of integer of the sixth cyclotomic field Q(ζ 6) , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω 2, where ω = 1 2 ( − 1 + i 3 ) = e 2 π i / 3 {\displaystyle \omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, 25.

There are 6 non-equivalent ways in which 100 can be expressed as the sum of two prime numbers: (3 + 97), (11 + 89), (17 + 83), (29 + 71), (41 + 59) and (47 + 53). [14]

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Six is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist; sixty (10 × 6) and ninety (15 × 6) are the next two. [7]