## Re: [PrimeNumbers] Carmichael Number Reduction

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• ... presumably called C below ... Yes! ... No it s not. And even if it were, what precisely is the this that you are refering too? The fact that (P^792)
Message 1 of 2 , Jan 27, 2007
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--- miltbrown@... wrote:
> For Carmichael Numbers

presumably called 'C' below

> (P^(C-1)) mod C = 1 for all primes P except those that divide C, for example

Yes!

> (P^1050984) mod 1050985 = 1 for all primes P except 5,13,19,23, and 37
>
> But, also
>
> (P^792) mod 1050985 = 1 for all primes P except 5,13,19,23, and 37
>
> This is the Carmichael Reduction

No it's not.

And even if it were, what precisely is the 'this' that you are refering too?
The fact that "(P^792) mod 1050985 = 1 for all primes P except 5,13,19,23, and
37"? That single fact is henceforth to be called the "Carmichael Reduction"? I
can assure you there's no point in coining such a term, as it really isn't
going to be used more than once.

> (P^(R-1)) mod C = 1, where C is a Carmichael Number
> and P ranges over the primes
>
> (P^(R-1)) mod C =/= 1 when P is a factor of C.
>
> R is determined by R-1 = (C-1) / q where q is the largest prime factor of C-1

So in the case C=1729, R is 576+1?
But (P^36) mod 1729 = 1.

What's the use of your R, when the functions we already have in our toolkits
give superior results?

> Another example is
>
> C = 63973 = 2^2*3^2*1777+1 and R = 36 = 2^2*3^2

You forgot the '-1'.

What about C=2465, R-1 is 224, but (P^112) mod 2465 = 1
Or C=8911, R-1 is 810, but (P^990) mod 8911 = 1

Wait a tick - 990 !| 810.

Dang - I should just instinctively reject all your posts, as it seems there's
always something mathematically wrong with them.

C = 8911 = 2*3^4*5*11+1 and R-1 = 810 = 2*3^4*5
And hitting Pari:
? p=1;while(p=nextprime(p+1),if(gcd(p,8911)==1&&Mod(p,8911)^810!=1,print("p="p"
shows that Milton is wrong")))
p=2 shows that Milton is wrong
p=3 shows that Milton is wrong
p=5 shows that Milton is wrong
p=11 shows that Milton is wrong
p=13 shows that Milton is wrong
p=17 shows that Milton is wrong
p=23 shows that Milton is wrong
p=31 shows that Milton is wrong
p=41 shows that Milton is wrong
p=43 shows that Milton is wrong
p=47 shows that Milton is wrong
p=53 shows that Milton is wrong
p=59 shows that Milton is wrong
p=61 shows that Milton is wrong
p=71 shows that Milton is wrong
p=73 shows that Milton is wrong
p=79 shows that Milton is wrong
p=83 shows that Milton is wrong
p=89 shows that Milton is wrong
...
p=4950613 shows that Milton is wrong
p=4950623 shows that Milton is wrong
p=4950661 shows that Milton is wrong
p=4950683 shows that Milton is wrong
p=4950703 shows that Milton is wrong
p=4950707 shows that Milton is wrong
p=4950713 shows that Milton is wrong
p=4950733 shows that Milton is wrong
p=4950739 shows that Milton is wrong
p=4950779 shows that Milton is wrong
p=4950797 shows that Milton is wrong
p=4950817 shows that Milton is wrong
p=4950833 shows that Milton is wrong
p=4950853 shows that Milton is wrong
...

Make it stop, make it stop!

Does anyone actually bother reading to the end of these? DO NOT answer that, in
the interest of list SNR.

Phil

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