Find all functions satisfying $ f \left( m ^ 2 + f ( n ) \right) = f ( m ) ^ 2 + n $, for all $ m , n \in \mathbb N $
31698
$f(m^2 + f(n)) = f(m)^2 + n$ for all natural numbers
55485
Recursive Function - $f(n)=f(an)+f(bn)+n$
132093
Solve $f(f(x))=f(x)$
12910
For the Fibonacci numbers, show for all $n$: $F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$
29647
Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?
23485
Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$
88683
Functional equation $f\left(\frac{x+y}{2}\right)+f\left(\frac{2xy}{x+y}\right)=f(x)+f(y)$ implies $2f(\sqrt{xy})=f(x)+f(y)$
32432
Strategies to find the set of functions $f:\mathbb R\to\mathbb R$ satisfing the functional equation: $f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$
66772
$f(f(x)f(y))+f(x+y)=f(xy)$
151947
Finding a pair of functions with properties
98255
Find $f(x)$ such that $f(f(x)) = x^2 - 2$
73519
Determine all functions satisfying $f(x + f(x + y)) + f(xy) = x + f(x + y) +yf(x)$
62254
Show $F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n$ for all $n \in \mathbb{N}$
141938
Find all continuous functions from positive reals to positive reals such that $f(x)^2=f(x^2)$
83080
Question regarding $f(n)=\cot^2\left(\frac\pi n\right)+\cot^2\left(\frac{2\pi}n\right)+\cdots+\cot^2\left(\frac{(n-1)\pi}n\right)$
14975
Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$
31732
If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$