Jacobi indices

This page is an appendix to "Irish Logarithms Part 2"

The Jacobi index Ind(z) of an integer z can be given by

z = g Ind(z) modulo p                            (1)

The trick is to find values of g and p for which this equation applies and for which Ind(z) is unique for each value of z in a selected set of numbers. If p is prime and the set of z's consists of all integers between 1 and p − 1, then there is a g for which each z has an unique index between 0 and p − 2. If so, then: 

Ind(z1×z2 ) = Ind(z1) + Ind(z2 )             (2)

So using Ind(), we can multiply by adding.[1]

For a calculator like Verea's, we only need the indices of the 36 integers in the simple multiplication table.

So for a calculator the set of z's differs from the set of all integer between 1 and p − 1: there are "gaps" in the collection and the collection does not end with a prime ( p − 1 = 81).

Because in the calculator only numbers less than 10 are multiplied, equation (2) does not have to apply for all z1 and z2, but only for z1, z2 <10. The resulting index Ind(z1×z2 ) must be unique for all unique simple products.

So we do not have to apply the Jacobi indices literally for the multiplication table of a calculating machine. We can still try using equation (1) to generate indices that meet our goal. There is no guarantee that the indices that we find are the smallest numbers causing Ind(z1 × z2) = Ind(z1) + Ind(z2). Our final goal is to minimize  the largest index.

Using equation (1) with p = 11 and g = 2, I found two sets of indices with the largest index less than 100. The following table shows the indices for the integers <10: 

 z 1 2 3 4 5 6 7 8 9
 Ind(z) 0 1 18 2 44 19 7 4 36
 Ind(z) 0 1 8 2 44 9 27 4 16

Other choices of p and g may provide better (i.e. lower) indices.

The indices of Schumacher's slide rule are generated with p = 101 and g = 2.

Notes

  1. Strictly speaking: the sum of the indices modulo p − 1.

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