## I find it very interesting...

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• Dear Group, ... I made an observation of a conjecture that was posted on Carlos Rivera s website. It was named The Goldbach Temptation. Can someone prove
Message 1 of 6 , May 31 7:31 PM
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Dear Group,
...
I made an observation of a conjecture that was posted
on Carlos Rivera's website. It was named The Goldbach
Temptation. Can someone prove whether phi(n^2 -k^2) =
(n-1)^2 -k^2, working for ONLY some n's and some k's,
is equivalent to Goldbach's conjecture? I proposed it
just recently as ... phi(n^2-k^2)=2*phi(n+k)*phi(n-k)
such that 'n' is prime, gcd(n,k) = 1 w/the hopes that
someone might find a way to prove the latter equation.
Are both representations equivalent??? I believe that
they are... but people were too busy to question it.
...
Rewards, Bill
...
P.S. I remember an old quote... "If you want something
done, give it to a busy person."
• ... Let n and k be positive integers with n k+1. Then eulerphi(n^2-k^2) = (n-1)^2 - k^2 if and only if n+k and n-k are both prime. David
Message 2 of 6 , Jun 1, 2013
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--- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...> wrote:

> phi(n^2-k^2) = (n-1)^2 - k^2, working for ONLY some n's and some k's

Let n and k be positive integers with n > k+1. Then
eulerphi(n^2-k^2) = (n-1)^2 - k^2
if and only if n+k and n-k are both prime.

David
• See conjecture 33 at primepuzzles Enviado desde Yahoo! Mail con Android [Non-text portions of this message have been removed]
Message 3 of 6 , Jun 1, 2013
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See conjecture 33 at primepuzzles

Enviado desde Yahoo! Mail con Android

[Non-text portions of this message have been removed]
• Also, I wondered about the algebra of Euler s phi function. Is it true that phi(n^3 -k^3) = y*phi(n -k)*phi(n^2 +nk +k^2)? and... Is it true that phi(n^3
Message 4 of 6 , Jun 1, 2013
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Also, I wondered about the "algebra" of Euler's phi function.
Is it true that phi(n^3 -k^3) = y*phi(n -k)*phi(n^2 +nk +k^2)?
and...
Is it true that phi(n^3 +k^3) = z*phi(n +k)*phi(n^2 -nk +k^2)?
...
of course, when 'n' is prime, and gcd(n, k) = 1; it appears
that y, z = 1 in the smaller cases of 'n' and 'k', but when
I tried n= 5 and k= 4, I noticed that z= 3/2; at what point
do things tend to break down?? or Can 'y' and 'z' be computed
in terms of n's and k's.
Bill

>
> --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@> wrote:
>
> > phi(n^2-k^2) = (n-1)^2 - k^2, working for ONLY some n's and some k's
>
> Let n and k be positive integers with n > k+1. Then
> eulerphi(n^2-k^2) = (n-1)^2 - k^2
> if and only if n+k and n-k are both prime.
>
> David
>
• ... You may find this useful: phi(a*b)=phi(a)*phi(b)*g/phi(g), where g=gcd(a,b). David
Message 5 of 6 , Jun 1, 2013
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"leavemsg1" <leavemsg1@...> wrote:

> I wondered about the "algebra" of Euler's phi function

You may find this useful:
phi(a*b)=phi(a)*phi(b)*g/phi(g), where g=gcd(a,b).

David
• That surely derives from the fact that if both n-k and n+k are primes then phi(n^2-k^2) = phi((n-k)(n+k)) = phi(n-k) * phi (n+k) = (n-k-1) * (n+k-1) = n^2 - 2n
Message 6 of 6 , Jun 1, 2013
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That surely derives from the fact that if both n-k and n+k are primes then

phi(n^2-k^2) = phi((n-k)(n+k)) = phi(n-k) * phi (n+k) = (n-k-1) * (n+k-1) =
n^2 - 2n + 1 - k^2 = (n-1)^2 - k^2
Example
n = 105, k = 4

phi(105^2-4^2) = (105-1)^2 - 4^2 = 10800 = (101-1)*(109-1)

But my question is how we can approach to an additive feature of EulerPhi?

If there was one such a feature then the primality tests would become so
easy. As an example consider n!+1, we know phi(n!) without any computation
and using that dreamed feature we could easily check if phi(n!+1) equals n!
or not :-)

http://math.stackexchange.com/questions/249982/how-to-calculate-euler-totient-from-nearby-values

> **
>
>
> --- In primenumbers@yahoogroups.com, "leavemsg1" <leavemsg1@...> wrote:
>
> > phi(n^2-k^2) = (n-1)^2 - k^2, working for ONLY some n's and some k's
>
> Let n and k be positive integers with n > k+1. Then
> eulerphi(n^2-k^2) = (n-1)^2 - k^2
> if and only if n+k and n-k are both prime.
>
> David
>
>
>

--
"Mathematics is the queen of the sciences and number theory is the queen of
mathematics."
--Gauss

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