## Re: Every Carmichael Number is a Non-centered Polygonal Number

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• Thursday, August 19, 2010, Bill Sindelar wrote: Every Carmichael number is a member of an infinite sequence of increasing odd positive composite integers N,
Message 1 of 3 , Aug 21, 2010
Thursday, August 19, 2010, Bill Sindelar wrote:

"Every Carmichael number is a member of an infinite sequence of increasing odd positive composite integers N, where for each distinct prime factor P of N, there exists an associated positive integer S, such that the expression A=(((S-2)*(P^2))-((S-4)*P))/2 evaluates to N." <snip

Friday, August 20, 2010, Jaroslaw Wroblewski wrote:

<snip> "Carmichael number is such an odd integer N>1 that 1*(N-1) is divisible by P-1 for each prime factor P of N. Your condition: "an odd integer N>1 such that 2*(N-1) is divisible by P-1 for each prime factor P of N, is in fact a weakened Carmichael condition"

Jarek, a belated personal thanks for taking the time to comment. I appreciate it very much. I did not know what you meant by a Carmichael condition, much less a weakened one, so I looked it up. It says that a number n satisfies a Carmichael condition, if and only if for all prime divisors p of n, (p-1) divides (n-1).

You cleverly show that any N in the sequence satisfies a condition very similar to a Carmichael condition. This is of course true. But Carmichael numbers and conditions were far from my mind, when I constructed the sequence of 25 N's. I was not looking for C's. I was just interested in generating N's. I noticed then that some N's looked familiar. They were the first nine existing C's. I also noticed that their S's were even integers in contrast to the other N's.

Then I wondered, if I could extend the sequence to any arbitrary length, would ALL existing C's up to the last N always show up in the sequence?. Would they all have even S's? Were there other algebraic expressions in 2 variables with the same property? I tried the one for centered polygonal numbers but it does not work. Do you or anyone have any opinions on this?

With best regards
Bill Sindelar

Thursday, August 19, 2010, Bill Sindelar wrote:

"Every Carmichael number is a member of an infinite sequence of increasing odd positive composite integers N, where for each distinct prime factor P of N, there exists an associated positive integer S, such that the expression A=(((S-2)*(P^2))-((S-4)*P))/2 evaluates to N." <snip

Friday, August 20, 2010, Jaroslaw Wroblewski wrote:

<snip> "Carmichael number is such an odd integer N>1 that 1*(N-1) is divisible by P-1 for each prime factor P of N. Your condition: "an odd integer N>1 such that 2*(N-1) is divisible by P-1 for each prime factor P of N, is in fact a weakened Carmichael condition"

Jarek, a belated personal thanks for taking the time to comment. I appreciate it very much. I did not know what you meant by a Carmichael condition, much less a weakened one, so I looked it up. It says that a number n satisfies a Carmichael condition, if and only if for all prime divisors p of n, (p-1) divides (n-1).

You cleverly show that any N in the sequence satisfies a condition very similar to a Carmichael condition. This is of course true. But Carmichael numbers and conditions were far from my mind, when I constructed the sequence of 25 N's. I was not looking for C's. I was just interested in generating N's. I noticed then that some N's looked familiar. They were the first nine existing C's. I also noticed that their S's were even integers in contrast to the other N's.

Then I wondered, if I could extend the sequence to any arbitrary length, would ALL existing C's up to the last N always show up in the sequence?. Would they all have even S's? Were there other algebraic expressions in 2 variables with the same property? I tried the one for centered polygonal numbers but it does not work. Do you or anyone have any opinions on this?

With best regards
Bill Sindelar

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