How would you find a primitive root of a prime number such as 761? How do you pick the primitive roots to test? Randomly? Thanks
Checking per Gerry's suggestion, a quick spreadsheet for the 40 primitive roots mod 101 shows that twenty-six (26) of them are square-free and fourteen (14) of them are not. We are helped in this by the fact that 2 is the smallest primitive root mod 101, so taking powers of 2 with exponents coprime to 100 gives all forty of the primitive roots (reduced mod 101).
In some contexts, the word primitive is used to mean a polynomial whose coefficients are relatively prime. In other contexts the word primitive is used to mean a polynomial a root of which generates a field under discussion. A polynomial that is primitive in the second sense must be irreducible.
field theory - How can I prove a polynomial to be primitive ...
This is because all odd squares are $1\pmod 8$. Any primitive triple must have two odd squares, whose difference is therefore a multiple of $8$, and so the third number must be a multiple of $4$. (You can't have two odd numbers with even hypotenuse, since the sum of two odd squares is $2\pmod 4$, but every even square is $0\pmod 4$.)
Are all natural numbers (except 1 and 2) part of at least one primitive ...
A character is non-primitive iff it is of the form $1_ {\gcd (n,k)=1} \psi (n)$ with $\psi$ a character $\bmod m$ coprime with $k$. A character $\bmod p^2$ can be primitive with conductor $p$.