Re: FLTMA: A little group theory

The Dougster wrote:

Chip Eastham wrote:
The converse is not true, though. Consider the residue
in Z/15Z. The order of residue 4 is 2, but clearly half that
order does not give a power of 4 equal to -1 in Z/15Z.
In fact you can run through all possible powers of 4 in
Z/15Z without getting to -1.

That follows from 2 prime, I think, if the above was valid.

More directly, if the order of an element w mod z is 2,
then the only possible solution is the trivial one:

w^1 = w = -1 mod z.

since the sequence of positive powers of w repeats:

w, 1, w, 1,...

Other prime orders are oddly enough, odd, and we
know if some power of w is -1, then the order of w
must be divisible by the order of -1, which is 2 (in
that z > 2).

However w may fail to have a power equal to -1
in spite of the order of w mod z being even and
composite. For example in Z/15Z*, the powers
of 2 are:

2, 4, 8, 1, (repeat)

so the order of 2 is 4, but no power of 2 is -1.

regards, chip

.

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