Re: factoring
milo gardner
http://en.wikipedia.org/wiki/Least_common_multiple
The 2/21 example meant that multiple 2 scaled 2/21 to 4/42 such that
2/21 = (3 + 1)/42 = 1/14 + 1/42
found LCM 42.
The method was used by Ahmes, an ancient Egyptian scribe, in 1650 BCE to find the best solution by writing (3 + 1) in red ink. Ahmes added 49 other 2/n conversions by finding the best LCMs and best red numbers by:
http://rmprectotable.blogspot.com/
Taking a long view of LCMs, very little intellectual change has taken place in the last 4,000 years.
Milo Gardner
From: Thys van der Veer <iim...@gmail.com>
To: milog...@yahoo.com
Sent: Sat, May 7, 2011 6:18:22 AM
Subject: factoring
kirby urner
both methods: using Euclid's Algorithm to get the GCD, or prime
factoring.
Once gcd is in hand, then lcm(M,N) is just (M*N)/gcd(M,N) where *
means "to multiply".
In another thread, I was decrying a lack of attention to Euclid's
Algorithm per college, when its expression in a computer language is
as simple as:
def gcd(a, b):
while b:
a, b = b, a % b
return a
You needn't worry whether a < b or like that. Then:
def lcm(a, b):
return (a * b)/gcd(a, b)
We're defining functions, iterative loops, getting a lot of concepts
flowing. But we're not using a calculator, we're using a
full-featured computer language, interactively, like a calculator.
The point I was hammering home is we want to follow Neal Stephenson's
model in telling real world war stories, like the cracking of the
Enigma code by the folks at Bletchley Park.
Fast forward and you get this later story of RSA, originally
discovered in pretty much the same form by GHCQ cryptographers, but
kept mum about.
However, with the rise of the Web and the need for casual secure
transactions among strangers, it became a commercial interest to make
RSA publicly available.
For any of this to make sense mathematically, one needs to focus on
the near impossibility of finding the two needle in haystack numbers,
p, q that were coughed up as probable primes by a Miller-Rabin filter.
And yet we're still able to do gcd operations, such as test for coprimehood.
Oh, so does that mean there's a way to get the gcd (ergo lcm) of two
numbers without cracking them down into primes? You bet you booty
there is.
Once you allow a real (free no cost) computer language to filter into
the math labs (workshops, making and baking areas), then you can start
playing with "math objects" that multiply just fine, but they're not
you're ordinary numbers. They're not even a species of N, Z, Q, R or
C for that matter. I'm speaking of permutations, such as are on
display in permworlds.py at my 4dsolutions.net/ocn/python/OST tank.
Kirby
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