Counterexample/Examples

Examples of Counterexamples

Sum of Cubes of Digits

Let $P$ be the statement:

There exists no integer which is the sum of the cubes of its digits.

A counterexample to $P$ is the number $153$, as can be seen in Pluperfect Digital Invariants: $3$ Digits:

\(\ds 153\)

\(=\)

\(\ds 1 + 125 + 27\)

\(\ds \)

\(=\)

\(\ds 1^3 + 5^3 + 3^3\)

Sum of Sine and Cosine equals $1$

Let $P$ be the statement:

$\forall x \in \R: \cos x + \sin x = 1$

A counterexample to $P$ is the real number $\dfrac \pi 4 \in \R$:

\(\ds \cos \dfrac \pi 4\)

\(=\)

\(\ds \dfrac {\sqrt 2} 2\)

Cosine of $\dfrac \pi 4$

\(\ds \sin \dfrac \pi 4\)

\(=\)

\(\ds \dfrac {\sqrt 2} 2\)

Sine of $\dfrac \pi 4$

\(\ds \leadsto \ \ \)

\(\ds \cos \dfrac \pi 4 + \sin \dfrac \pi 4\)

\(=\)

\(\ds \sqrt 2\)

\(\ds \ne 1\)