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Bill picture

A Bill picture is a sequence of nested regular polygons in which subsequent polygons are each rotated so that they begin one vertex further. The term was coined by Trott (2004, pp. 88-89) and commemorates Swiss artist Max Bill, who in 1938 created a picture showing a similar arrangement of the equilateral triangle through octagon (Huttingerr 1978, Bill 1987).The figure above shows the Bill picture including regular polygons up through theregular dodecagon.

Plato's numbers

The positive integers 216 and appear in an obscure passage in Plato's The Republic. In this passage, Plato alludes to the fact that 216 is equal to , where 6 is one of the numbers representing marriage since it is the product to the female 2 and the male 3. Plato was also aware of the fact the sum of the cubes of the 3-4-5 Pythagorean triple is equal to (Livio 2002, p. 66).In Laws, Plato suggests that is the optimal number of citizens in a state because 1. It is the product of 12, 20, and 21. 2. The 12th part of it can still be divided by 12. 3. It has 59 proper divisors, including all numbers for 1 to 12 except 11, and 5038--which is very close to 5040--is divisible by 11 (Livio 2002, p. 65).

Partition function p congruences

The fraction of odd values of the partition function P(n) is roughly 50%, independent of , whereas odd values of occur with ever decreasing frequency as becomes large. Kolberg (1959) proved that there are infinitely many even and odd values of .Leibniz noted that is prime for , 3, 4, 5, 6, but not 7. In fact, values of for which is prime are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (OEIS A046063), corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575). Numbers which cannot be written as a product of are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (OEIS A046064), corresponding to numbers of nonisomorphic Abelian groups which are not possible for any group order.Ramanujan conjectured a number of amazing and unexpected congruences involving . In particular, he proved(1)using Ramanujan's identity (Darling 1919; Hardy and Wright 1979; Drost 1997; Hardy 1999, pp. 87-88; Hirschhorn 1999). Ramanujan (1919) also showed that(2)and..

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