Why Are these Relative Primes?

[NotLKH]

I just made the following statement in something I was documenting:

(3*c_Rows²+1)/4 and c_Rows are relatively prime

Now, c_Rows is always odd, > 1, so 3*c_Rows²+1 is always divisible by 4.

So my question, am I right that they are relatively prime?
Can anyone prove it?

[sql_lall]

Let X = c_Rows (make it cleaner)

using the fact that gcd(a,b) <= gcd(Ya, b), and <= gcd(a^2, b)

gcd((3*X^2+1)/4, X) <= gcd(3X^2+1, X) <= gcd(3X^2+1, 3X^2)

of course, gcd(p, p+1) = 1
so, gcd(3X^2+1, 3X^2) = 1
so, gcd((3*X^2+1)/4, X) <= 1

of course, it can't be smaller, so it must = 1 !!

[NotLKH]

Looks Good!

I was pretty sure it was something like that.
I'll give you proper credit in my little project.

Thanks!
-Lou

[NotLKH]

Hmmm,
I just typed the following:

With this Identity, it is easy to see that c_C and c_Rows are relative primes, that is, they share ABSOLUTELY NO COMMON DIVISOR WHATSOEVER, except for 1.
Let me Quote sql_lall when she helpfully pointed out the following:

[:clearsthroat:] Sorry for not knowing, but am I correct in referencing you as a she?

[sql_lall]

hehe...

yeah, u are actually incorrect, i'm a he.
Its ok though...
i'm guessing u were taking it from my pic, or my lack of aggression or anger on other posts...the former understandable, the latter is a compliment

Thanks for the reference anyway...i feel proud

[bugzpodder]

(3*x^2+1)/4 and x

suppose some prime p divides both numbers, then p|x (p divides x) and p|3x^2+1

but since p|x, then p|3x^2, hence p|1, so p=1

[NotLKH]

Hmmm,....Let me jot this down,...Got It!



Thanks Bugz!

[NotLKH]

Hmmm,

BTW, I corrected the she to a he, sql_lall, and you are still included in my little document.

Which I just finished.

Partially titled: "Why One"