Erdős's function f(n) explodes toward infinity instead of staying small, defying a limit the legendary mathematician set decades ago.
April 29, 2026
Original Paper
Unbounded logarithmic limsup in Erdős problem 684
arXiv · 2604.23784
The Takeaway
Paul Erdős was a legendary mathematician who left behind a series of unsolved problems regarding the density of number sequences. Experts in the field widely expected that a specific function within problem 684 would never exceed a logarithmic bound. This new proof shows that the function actually grows toward infinity at a much faster rate. The result overturns decades of assumptions about how these sequences behave in the long run. It proves that our intuition about mathematical limits can be dangerously wrong even for simple functions. This breakthrough resets the search for the true boundaries of number theory.
From the abstract
For $0\le k\le n$, write $\binom nk=uv$ where the primes dividing $u$ are at most $k$ and the primes dividing $v$ exceed $k$, and let $f(n)$ be the least $k$ with $u>n^2$; Erdős problem 684 asks for bounds on $f(n)$. We resolve the problem at the order level. By a short-multiplier construction $n_M=tL_M-1$, where $L_M=\operatorname{lcm}(1,\ldots,M)$ and $t$ is a multiplier of size $\exp(o(M))$ extracted from a Fourier sieve, we prove that for every fixed $C>1$ there exist integers $n$ with $$f(n