How can you tell if an equation has infinitely many solutions?

Dec 19, 2017

A few thoughts...

Explanation:

Here are a few possibilities:

• The equation simplifies to the point that it no longer contains a variable, but expresses a true equation, e.g. $0 = 0$. For example: $2 x + 2 = 2 \left(x + 1\right)$ simplifies in this way.

• The equation has an identifiable solution and is periodic in nature. For example: ${\tan}^{2} x + \tan x - 5 = 0$ has infinitely many solutions since $\tan x$ has period $\pi$.

• The equation has a piecewise behaviour and simplifies within at least one of the intervals to a true equation without variables. For example: $\left\mid x + 1 \right\mid + \left\mid x - 1 \right\mid = 2$, which simplifies suitably for $x \in \left[- 1 , 1\right]$.

• The equation has more than one variable and does not force uniqueness. For example: ${x}^{2} + {y}^{2} = 1$ has infinitely many solutions, but ${x}^{2} + {y}^{2} = 0$ has one solution (assuming $x , y \in \mathbb{R}$).

Note that it may be extremely difficult to determine the number of solutions in the case of Diophantine equations - equations where the values of the variables are limited to integers or to positive integers.

For example, Euler conjectured that the equation:

${x}^{4} + {y}^{4} + {z}^{4} = {w}^{4}$

had no non-trivial solutions, but Noam Elkies found one in 1988, hence there are an infinite number of non-trivial solutions, since any solution can be multipled by a fourth power.