The important fact is that the only numbers $n$ that have primitive roots modulo $n$ are of the form $2^\varepsilon p^m$, where $\varepsilon$ is either $0$ or $1$, $p$ is an odd prime, and $m\ge0$

I thought $\varphi (12)$ counts the number of coprimes to $12$.. Why does this now suddenly tell us the number of primitive roots modulo $13$? How have these powers been plucked out of thin air? I …

May 16, 2023 · 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

Jun 18, 2015 · Moreover the other primitive elements are obtained as powers of $2$ that are not the ones relevant to checking that $2$ is primitive, so you'd have to separately compute those powers …

Jan 6, 2019 · While antiderivative, primitive, and indefinite integral are synonymous in the United States, other languages seem not to have any equivalent terms for antiderivative. As others have pointed out …

Jun 6, 2016 · Primes have not just one primitive root, but many. So you find the first primitive root by taking any number, calculating its powers until the result is 1, and if p = 13 you must have 12 …

Jul 6, 2019 · I have been reading about Pythagorean triples from the wiki page link here. It says that a pythagorean triple consists of 3 positive integer's $ a, b, c $ such that $ a^2 + b^2 = c^2 $. Also if a...

After you have one primitive polynomial, you often want to find other closely related ones. For example, when calculating generating polynomials of a BCH-code or an LFSR of a Gold sequence (or other …

Jul 27, 2020 · This follows from the fact that $2$ is primitive $\pmod3$, since $2^ {3-1}\equiv 4\not\equiv 1\pmod9$. See Henri Cohen, A Course in Computational Algebraic Number Theory, 1993, p.26.

A nonnegative, irreducible matrix is primitive if and only if it is aperiodic (ibid. Theorem 8.5.3). According to this source, we wouldn't ever call a reducible matrix primitive. Though a reducible matrix can still …