Arabic Six Children's Stories

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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] 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] is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, 25. 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]

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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} .

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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. 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