Wilson norm: Difference between revisions

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The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at.

If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.

If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.

Note that we have <math>\text{sopfr}(pq) = \text{sopfr}(p) + \text{sopfr}(q)</math>, similar to the logarithm -- as a result, this function is sometimes even referred to as the "integer logarithm." So, equivalently, we can define the Wilson height of a rational number p/q as the Wilson height of p, plus the Wilson height of q.

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 The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at.
-If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.
+If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>.
 Note that we have <math>\text{sopfr}(pq) = \text{sopfr}(p) + \text{sopfr}(q)</math>, similar to the logarithm -- as a result, this function is sometimes even referred to as the "integer logarithm." So, equivalently, we can define the Wilson height of a rational number p/q as the Wilson height of p, plus the Wilson height of q.