Another game one can play with lattice basis reduction finds p-limit

elements which are represented by a small number of steps in a given

et. Ratios of these can then produce kernel elements.

For instance, h72 in the 11-limit is represented by

[72,114,167,202,249]. From this we obtain the five lattice vectors

[72,1,0,0,0,0,0], [114,0,1,0,0,0], [167,0,0,1,0,0], [202,0,0,0,1,0],

and [249,0,0,0,0,1]. The first entry is the number of steps in the 72-

et for the given element, and the rest are that element in standard

prime-power notation--the ones I give above being the primes

themselves. We now reduce the above using LLL, and obtain

[1,1,2,-3,1,0], meaning 126/125 represented by 1 step of the 72 et,

[2,1,0,2,-2,0], meaning 50/49 represented by 2 steps of the 72 et,

and so forth for 1 and 99/98; -1 and 242/245; and 2 and 55/54. We

then may find (99/98)/(126/125) = 1375/1372, (50/49)/(55/54) =

540/539, (245/242)/(126/125) = 4375/4356, etc as commas for the 72 et.

I forgot to carry out the second part of my example; I've just now

done it, and find the results a little surprising, but interesting.

Since my original lattice was unimodular, the steps I get from the

lattice basis reduction are unimodular also, forming a basis for a

notation. We have

[ 1 2 -3 1 0] [14 11 4 -8 12]

[ 1 0 2 -2 0]^(-1) [22 17 7 -13 19]

[-1 2 0 -2 1] = [32 25 10 -19 28]

[ 1 0 -1 -2 2] [39 30 12 -23 34]

[-1 -3 1 0 1] [48 37 15 -28 42]

This is a notation, but I was suprised at the somewhat exotic 14, 11

and 8 entries; which, however seem fruitful scale possibilities. I

expect it is due in good part to the fact that I went all the way to

the 11-limit, and it makes me wonder what would happen if I went

totally overboard and looked at 311.