[Algebraic equations and systems of equations (other)]
Formulas for abbreviated multiplication (other)

The number $a$ is a root of the equation $x^{11} + x^{7} + x^{3} = 1$. For which natural values of $n$ does the equality $a^{4} + a^{3} = a^{n} + 1$ hold?
By substitution into the equation, we verify that $a \neq 0$ and $a \neq \pm 1$.

From the condition, it follows that $a^{11}+a^{7}+a^{3}=1$. Multiplying both sides of this equality by $a^{4}-1$, we get $a^{15}-a^{3}=a^{4}-1 \Leftrightarrow a^{4}+a^{3}=a^{15}+1$. Comparing the obtained equality with the equality given in the condition, we get $a^{n}=a^{15}$.

Since $a$ in modulus is different from zero and one, then $n=15$.

## Answer

For $n=15$.