Which Diophantine equations have solutions?

One equation The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The solutions are described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b.

What are integer coefficients?

In mathematics, an integer-valued polynomial (also known as a numerical polynomial) is a polynomial whose value is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial. takes on integer values whenever t is an integer.

What is integer solution for a linear equation?

The notion of “general integer solution” of a linear equation with two unknowns is extended to linear equations with n unknowns and then, to linear systems. The properties of the general integer solution are determined (both of a linear equation and of a linear system).

What are the examples of integer?

An integer includes whole numbers and negative whole numbers. Integers can be positive, negative, or zero. For example: 1, -1, 0, 101 and -101. There are an infinite number of integers.

Are fractions integers?

The integers are the set of whole numbers and their opposites. Fractions and decimals are not included in the set of integers. For example, 2,5,0,−12,244,−15 and 8 are all integers.

What does solve over the integers mean?

Or you may want to solve something over the integers. That means the solutions to the equation must be integers. For example, for integer values a, b, it’s straightforward to solve ax + by = c over reals (you get a line). But solving over the integers may be impossible (e.g., 2x + 2y = 1)

How many integer solutions does x3 = y2 + 2 have?

The Diophantine equation X 3 = Y 2 + 2 has only one integer solution, namely ( x, y) = ( 3, ± 5). Proof. Evidently y and 2 are coprime.

Is x3 = y2 + 2 a coprime equation?

The Diophantine equation X 3 = Y 2 + 2 has only one integer solution, namely ( x, y) = ( 3, ± 5). Proof. Evidently y and 2 are coprime. By the corollary, we must have b = 1 = s ( 3 r 2 − 2 s 2) for integers r and s.

What are the integral solutions to the first problem?

The only integral solutions to your first problem are ( 3, ± 5). The general class of equations are known as Mordell’s equation. A fairly elaborate discussion and case by case analysis is provided here.