There are an infinite number of fractions that can solve this famous math puzzle, but not a single whole number can do the job.
April 13, 2026
Original Paper
Seven squares from three numbers
arXiv · 2604.08729
The Takeaway
In mathematics, we usually expect whole numbers and fractions to behave similarly, but this discovery reveals a 'forbidden' zone for integers. While fractions allow for endless solutions to this specific pattern of squares, whole numbers are fundamentally locked out by the laws of logic.
From the abstract
We study triples {a,b,c} of distinct nonzero rational numbers such that a+1,b+1,c+1,ab+1,ac+1,bc+1 and abc+1 are all perfect squares. We prove that there exist infinitely many such triples. In contrast, we show that no triple of positive integers has this property.