Problem Statement
Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a<b$ nonnegative integers are distinct?
Categories:
Number Theory Powers
Progress
Erdős and Graham describe this problem as 'very annoying'. Probably $f(x)=x^5$ should work. The Lander, Parkin, and Selfridge conjecture would imply that $f(x)=x^n$ has this property for all $n\geq 5$.Source: erdosproblems.com/324 | Last verified: January 14, 2026