By Idun Reiten, Sverre O. Smalø, Øyvind Solberg
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Extra info for Algebras and Modules One
D|n 40 2. Products and Divisibility So, for example, N ∗ u(n) = N (d)u(n/d) = d|n d · 1 = (n), d|n or, omitting mention of the variables, N ∗ u = . 1. What well-loved function is u ∗ u? The divisors of an integer n come in pairs: For every d dividing n, the integer c = n/d is another divisor. For that reason, we can write the deﬁnition of convolution in a more symmetric way, f ∗ g(n) = f (d)g(c), c,d c·d=n and from this it is clear that convolution is commutative, that is, that f ∗ g = g ∗ f .
Determine whether n is deﬁcient, perfect, or abundant. Look for patterns when n is odd or even. Is 32 2. Products and Divisibility s(n) odd or even? ) You can compute s(n) easily in Mathematica by ﬁrst deﬁning it as a function s[n ] := DivisorSigma[1, n] - n The input s returns the answer 7. In Maple, after loading the numtheory package (see p. xiii), you can deﬁne the function using s:=n->sigma(n)-n; The input s(12); returns the answer 16. 15. The number 284 = 22 · 71 isn’t perfect; s(284) = 220 = 22 · 5 · 11.
To him, individual numbers were symbols of individual gods of the Greek pantheon. For example, seven is the only number among the ﬁrst ten that is neither divisible by nor a divisor of any of the others. For this reason, it represents Athena, the virgin goddess. One and two were not even considered numbers. One is the “monad,” representing unity or the absence of multitude. Two is the “dyad,” representing duality. Strange as this may seem, consider that ordinary language conventions still support this.