Hello Guys I am wondering if this mathematical statement is correct (not the result but the grammar).
I want to find out if $35$ is a primitive root of $97$ which it is not as I calculated.
Is this correct?:
$$ 35^{48} \pmod{97} \equiv 1 \implies 35 \ \text{not a primitive root in } \mathbb Z_{97}^\ast $$
Example of fast exponentiation $35^{48} \bmod 97$
Furthermore:
Is this correct?:
$$ 48_{(10)} = 110000_{(2)} $$
$$ 1: \ 35 \ \equiv \ 35 \pmod{97} $$ $$ 1: \ 35^2*35 \ \equiv \ 1\pmod{97}$$ $$ 0: \ 1^2 \ \equiv \ 1 \pmod{97}$$ $$ 0: \ 1^2 \ \equiv \ 1 \pmod{97}$$ $$ 0: \ 1^2 \ \equiv \ 1 \pmod{97}$$ $$ 0: \ 1^2 \ \equiv \ 1 \pmod{97}$$ $$35^{48} \bmod 97 \equiv 1$$
I have to know if this is correct if I write this as an answer in my exam or if it is grammatically wrong.
The best way to write down the result of this exercise is :
A base $b$ is called a primitive root of a positive integer $n$ , if the smallest positive integer $k$ with $$b^k\equiv 1\mod n$$ is equal to $\varphi(n)$
In the example , $n$ is prime, hence $\varphi(n)=n-1=96$ But because of $$35^{48}\equiv 1\mod 97$$ $96$ is not the smallest $k$, hence $35$ is not a primitive root of $97$.
The calculation of the power modulo the prime is correct , but I agree lulu that you should write it down clearer (If you actually are supposed to calculate the result by hand).