Modulo one of the primes 151 and 157 the polynomial x^2+ 2 x-1 has two roots, and modulo the oth

Modulo one of the primes 151 and 157 the polynomial x^2+ 2...

Key Concepts

Completing the Square

Completing the square is a classical algebraic technique that rewrites a quadratic expression in a form that isolates a perfect square. In the context of modular arithmetic, this process can simplify quadratic congruences to an equivalent form, often reducing the problem to checking whether a particular number is a quadratic residue. This method is particularly useful when the quadratic equation is monic and can be recast into a more tractable form.

Legendre and Jacobi Symbols

Legendre symbols (and their generalization, Jacobi symbols) are notations used to determine the quadratic character of a number modulo a prime or an odd composite number. They simplify the process of testing whether an integer is a quadratic residue modulo p, and tables of these symbols, like Jacobi's tables, provide a convenient reference to quickly assess solvability conditions in quadratic congruences.

Quadratic Congruences

A quadratic congruence is an equation of the form ax² + bx + c ? 0 (mod p) where p is a prime, and its solutions are sought within the arithmetic of integers modulo p. The existence and number of solutions depend on the discriminant of the equation and the concept of quadratic residues, which indicate whether a given number has a square root modulo p.

Discriminant and Its Role

For a quadratic equation, the discriminant b² - 4ac is central in deciding the nature of its roots. In modular arithmetic, if the discriminant is a quadratic residue modulo p, then the quadratic congruence has two distinct solutions; if not, there are no solutions. The discriminant’s quadratic residuosity is thus the key to determining the solvability of the equation.

Quadratic Residues

A number a is called a quadratic residue modulo p if there exists some integer x such that x² ? a (mod p). Whether a given number is a quadratic residue is crucial in solving quadratic congruences. In practice, methods such as using Legendre symbols (or Jacobi symbols in more general settings) facilitate the test of quadratic residuosity.

Transcript

00:01 A question here is x in us to a proof that 2x to the 3rd plus 5x square plus 6x plus 1 is equivalent to 0, mod 7.

00:17 That equation of that expression has three solutions.

00:24 Would you use the theorem 2 .29 from textbook? and if you want to look at the textbook, it was specifically the fifth edition of the introduction to the number, a theory of numbers by ivan navan at all, sockerman montgomery.

00:45 Okay, so let's look at that theory first.

00:52 That theory goes along the lines of, by the way, i have already worked in the problem.

01:01 And had it typed out.

01:03 So we're going to go through it.

01:06 Theorem, the theorem goes, let ffx be a polynomial of degree n with integer coefficients and p a prime.

01:18 Then the congruence, ffx equivalent into zero mod p has at most n incongruent solutions, modular p, right, has at most at most and in congruent solutions mod p.

01:37 So basically, look at this very important.

01:41 Look at the degree, look at the module of p.

01:43 So anyway, that's what the theorem says.

01:45 So let's look through how we solve this.

01:49 All right, so i got two steps here for you.

01:52 The first step is, again, i want to make it as very simple as possible.

01:58 First step would be to compute the modular 7, right, for the x values.

02:07 0, 1, 2, 3, all the weight is 6.

02:11 Okay? so that's the first thing we're going to do...