9 What is a primitive polynomial? I was looking into some random number generation algorithms and 'primitive polynomial' came up a sufficient number of times that I decided to look into it in more detail. I'm unsure of what a primitive polynomial is, and why it is useful for these random number generators.
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 here How common is the use of the term "primitive" to mean "antiderivative"?, some languages such as Dutch only use the term, primitive.
I have read that, but essentially what I want to know is, can a primitive root be defined in a simpler, easier to understand way? For my level of mathematics, some of the more formal definitions can be hard to understand sometimes
A polynomial with integer coefficients is primitive if its content (the GCD of its coefficients) is 1. You can simply enumerate the primitive monic quadratic polynomials (depicted as ordered triples of coefficients in descending order of order):
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 understand even powers can't be primitive roots, also we have shown $2^3$ can't be a primitive root above but what about $2^9$?
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 sequence with known structure) you encounter the following task.
PrimitiveIdentification requires the group to be a primitive group of permutations, not just a group that can be primitive in some action. You will need to convert to a permutation group, most likely by acting on the set of $23^2$ vectors.
how to find all primitive triples when one value of (a,b,c) (a, b, c) is given? For example in this case a = 45 a = 45. What is the procedure to find the primitive triples ? Conditions for primitive triples are: a2 +b2 = c2 a 2 + b 2 = c 2 (a,b) = 1 (a, b) = 1 a a and c c are odd and b b is even.
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 $m\ge0$
2 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 different powers until the result is 1 to have a primitive root.
