Factorial square products are products of distinct factorials that are perfect squares. For an integer 
,
let 
be the least 
for which there are integers 
such that
 |
(1)
|
for an integer 
. Set 
if there is no representation. For example, 
since 
,
and 
since 
.
No prime number has a representation, while every
composite number has 
(Erdős and Graham 1976).
Define the counting function
 |
(2)
|
Problem 374 in the Erdős problems collection asks for the orders of growth of these functions for 
(Bloom 2026). Yudin (2026) proved, for every 
, that
 |
(3)
|
where 
denotes big-O notation and
 |
(4)
|
with 
and 
the squarefree part of 
. Each distinct member of 
is counted once. Yudin (2026) also proved
 |
(5)
|
Here 
means bounded above and below by positive constant multiples
for large 
.
Together with 
and 
for 
,
this determines all orders of growth. The notation 
denotes the floor function.
Whether 
has a limit for 
, 5, and 6 remains open (Yudin 2026).
The integers requiring six factors begin 527, 611, 713, 731, 779, 893, 923, 1003, 1037, ... (OEIS A389148). Yudin (2026) reports using ChatGPT and Claude to develop and check arguments, locate references, and edit the manuscript, and takes responsibility for the results.
