> 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

No it's not.

>

> But, also

>

> (P^792) mod 1050985 = 1 for all primes P except 5,13,19,23, and 37

>

> This is the Carmichael Reduction

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

So in the case C=1729, R is 576+1?

> 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

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

You forgot the '-1'.

>

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

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.

Retracing my/your steps:

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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