Re: Can you prove 10101 isn't prime in any base?

"mensanator@xxxxxxx" <mensanator@xxxxxxx> writes:
Representation systems such as those required to make sense of
10101 in "base 1" are in fact used quite regularly by millions
of machines every day. E.g. anyone using fingers in their bignums
is using the technique, which probably means that a reasonable
proportion of people who use ssh/https will be using such a
representation. So it may not be standard by name, but it's
really quite far from unusual. And of course, anyone who's doing
an FFT multiply will also be doing the same.

Ok, but how, exactly, does it work?

Just view the representation as a polynomial evaluated at its
base, and relax the restriction about the range of numbers
that may be in each place.

Of course - you lose unique representation as soon as you make
this relaxation, which is a drawback.

So <1,0,1,0,1> represents b^4+b^2+1 in any base b.
Even base 1 when you accept {0,1} to be valid limbs.

It's not uncommon in computer arithmetic to have base 2^28 and
permit intermediate values to temporarily be passed around with
limbs up to +/-2^31 or 2^32.

In FFT arithmetic, you'd use base 2^20, say, and intermediates
could reach +/-2^50 or more.

Of course, you want to be canonicalising quite frequently in
these situations. It's here where "base 1" shows how broken it
is as a concept, as there's no possible canonicalisation.

Phil
--
What is it: is man only a blunder of God, or God only a blunder of man?
-- Friedrich Nietzsche (1844-1900), The Twilight of the Gods
.

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