[P] **Fallout of amateur research** by akdevaraj Aug 4We can use pari to find the smallest divisor of phi(n) for which
a congruence holds good. Example: consider the pseudoprime 341.
This is pseudo to the base 2. To find d, the smallest divisor
we run the program {p(n) = (2^n -1)/341}. It was found that
when n = 10, the function is exactly divisible by 341. Application:
When the program {p(n) = (2^n+97)/341} was run for n = 1,10, no divisibility
was found. Hence 2^n + 97 is not divisible for any value of n.
However (2^n + 1007) is divisible by 341 before the program reaches
10. Similarly 40 is the smallest divisor of phi(561) ( 561 is
a Carmichael number). Hence any relevant program pertaining to
an exponential expression has to be run only till n reaches 40.

[P] **message system** by pahio Jul 31Thanks Joe,
Now the messages are visible again!
The search not...

[P] **messages...** by jac Jul 28...seem to work for me (I see your message, do you see this?)

[P] **Message system** by pahio Jul 27Message system does not work =o(

[P] **which isomorphism?** by pahio Jul 27Ok, $\mathbb{Z}_{31}$ is an additive group of order 31 (and in fact a Galois field since 31 is prime).
But I'm interested in the isomorphism you are speaking of. The expression
$\frac{2^{5n}-1}{31}$ says me nothing $-$ excuse me!

[P] **Which isomorphism?** by pahio Jul 27Ok, $\mathbb{Z}_{31}$ is an additive group of order 31 (and forms in fact a field since 31 is prime).
But I am interested which isomorphism you are speaking of. The expression $\frac{2^{5n}-1}{31}$ says
me nothing $-$ excuse me!

[P] **Which isomorphism?** by pahio Jul 27Ok, $\mathbb{Z}_{31}$ is an additive group of order 31 (and forms in fact a field since 31 is prime).
But I am interested which isomorphism you are speaking of. The expression $\frac{2^{5n}-1}{31}$ says
me nothing $-$ excuse me!

[P] **A property of exponential functions.** by akdevaraj Jul 27Let f(n) = a^n + c (a,n and c belong to N, n
is not fixed ). Let M_p be a Mersenne prime.
If M-p does not exactly divide f(n) for n = 1 to p
then M_p does not exactly divide f(n) for any
value of n, however large n may be.

[P] **A property of exponential functions.** by akdevaraj Jul 27Let f(n) = a^n + c (a,n and c belong to N, n
is not fixed ). Let M_p be a Mersenne prime.
If M-p does not exactly divide f(n) for n = 1 to p
then M_p does not exactly divide f(n) for any
value of n, however large n may be.

[P] **A property of exponential functions.** by akdevaraj Jul 27Let f(n) = a^n + c (a,n and c belong to N, n
is not fixed ). Let M_p be a Mersenne prime.
If M-p does not exactly divide f(n) for n = 1 to p
then M_p does not exactly divide f(n) for any
value of n, however large n may be.

[P] **Carmichael numbers - pari code** by akdevaraj Jul 27Since we know that 561 is not a Carmichael number in Z(i) we need to
know the code for searching those bases for which 561
is a pseudoprime. Code in pari: {p(n) = ((n + i)^40 - 1)/561}.
Incidentally 153 + i is also a valid base for pseudoprimality
of 561; needless to say its associates 153 - i etc. are also
valid bases.

[P] **Isomorphism - an example** by akdevaraj Jul 27Pahio, I meant the finite group of remainders (mod 31)-sorry I
typed " ïnfinite ".

[P] **No comment** by Ascold1 Jul 25Hi, BCI1!
It's me, Ascold1.
Bye!

[P] **Isomorphism?** by pahio Jul 25Deva, can you please explain in detail which is the (infinite) group you mean? I see only the expression
$\frac{2^{5n}-1}{31}$.
The group $\mathbb{Z}_{31}$ is finite.