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How to prove following statement :

Conjecture :

Let $p$ be a prime number of the form : ${\color{BlueViolet}{p=16 \cdot n^4+1}}$

If $n$ is an odd prime greater than $3$ then $3$ is a primitive root modulo $p$ .

I wrote small Maple program (see below) in order to find counterexample , but I haven't found anyone . For the first $204231$ primes of this form $3$ is a primitive root modulo $p$ .

There is theorem that states :

$a$ is a primitive root modulo $p$ iff $\operatorname{ord}_p(a) = \phi(p)$

One can show that only possible values of $\operatorname{ord}_p(3)$ are :

$16n , 16 n^2 , 16n^3 ~\text{or}~ 16n^4$

If we roll out first three possibilities then we have a proof , but I don't see any reason why $\operatorname{ord}_p(3)$ couldn't be : $ 16 n , 16n^2 ~\text{or}~ 16n^3$ .

for n from 4 to 44400000 do
if isprime(n) then
if isprime(16*n^4+1) then
if not(primroot(16*n^4+1)=3) then
end if;
end if;
end if;
end do;
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You're producing one after another of these "conjectures". I showed you in some detail for the previous one why the results you found noteworthy were in fact statistically expected. Why do you simply post another question in this "series" without linking to that other question and without first performing an analysis along the lines of my answer to the other one? – joriki Feb 20 '12 at 9:51
@joriki I would like to see opinions of other members...By the way I proved that $p$ cannot be power of an odd prime in previous question... – pedja Feb 20 '12 at 10:06
I was writing in the hope of convincing you that by not citing related questions you waste people's time and you should refer to related questions the next time you post, and also to indicate to others that you've already been shown how to deal with such a question and there's no need for them to do it for you again if you don't invest the effort yourself. – joriki Feb 20 '12 at 10:36
I'm cleaning up some of the comments here. Please keep comments civil and refrain from slinging accusations. – Willie Wong Feb 20 '12 at 12:06
pedja: I have to agree with @joriki that it is in general a good idea to link to related questions when asking a new one. This will (a) establish that the question grew out of a useful line of inquiry and (b) show other readers what have been tried before to tackle the problem. – Willie Wong Feb 20 '12 at 12:08

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