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Re: example of T3 Sequence

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• ... yes!, if n is even... ... don t know about that complicated mess, but if m IS odd, say 15... then T15 = T8 * T7 - 3 mod N it s that simple!; the author
Message 1 of 3 , Feb 15, 2008
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--- In primenumbers@yahoogroups.com, "Christ van Willegen"
<cvwillegen@...> wrote:
>
> Bill,
>
> just to clarify something for myself..
>
> For even n:
> T(n) = T(n/2)^2 - 2 (MOD 47)
yes!, if n is even...

> T(n + 1) = T(n/2) * T((n+2)/2) - 3 (MOD 47)
don't know about that complicated mess,

but if m IS odd, say 15... then T15 = T8 * T7 - 3 mod N
it's that simple!; the author described it easily...

here's my newest modification... but it's a bit more profound than
the original article; it combines the T3 sequence w/ sqrt N

it'll work with any number; take N = 207

find sqrt N = 14.3 {roughly};

then compute terms T14, T12, T7, T6, T4, T3, T2, and T0 = 2, T1 = 3;

so

T2 = T1^2-2 mod 207 = 7,
T3 = T1*T2-3 mod 207 = 18,
T4 = T2^2-2 mod 207 = 47,
T6 = T3^2-2 mod 207 = 115,
T7 = T4*T3-3 mod 207 = 15,
T12 = T6^2-2 mod 207 = 182,
& T14 = T7^2-2 mod 207 = 16;

Since 182 + 16 < 207, then 207 is composite;
if the sum were > 207, then it would've been prime;
this technique combines SQRT and the T3 sequence.
Try it! or wait 'til you understand the T3 sequence alone.

>
> I was figuring out where the -2 and -3 came from. Is this it?
>
> On Wed, Feb 13, 2008 at 3:57 PM, Bill Bouris <leavemsg1@...> wrote:
> > > take the number 47, for example...
> > > the program calculates the terms, ...
> > >
> > > T47
> > > T24,T23
> > > T12,T11
> > > T6,T5
> > > T3,T2
> > > T1,and T0
> > >
> > > so...
> > > every term is taken MOD 47,
> > > to keep them small...
> > >
> > > T0 = 2 mod 47
> > > T1 = 3 mod 47
> > > T2 is T1^2 -2 mod 47
> > > T3 is T1*T2 -3 ...
> > > T5 is T2*T3 -3 ... etc
> > > T6 is T3^2 -2
> > > T11 is T5*T6 -3
> > > T12 is T6^2 -2
> > > T23 is T11*T12 -3
> > > T24 is T12^2 -2
> > > and T47 is T23*T24 -3
> > >
> > > if the result is T47 = 3, then 47 is prime!
> > > neat, huh?
>
> I can't comment on this, though...
>
> Christ van Willegen
> --
> 09 F9 11 02 9D 74 E3 5B D8 41 56 C5 63 56 88 C0
>
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