Source: Quanta, Feb 2019
They proposed that no matter what set of numbers you use to generate your grid — whether the numbers are consecutive or nonconsecutive, powers of 2 or county-by-county vote totals — either the number of distinct sums, the number of distinct products, or both has to be close to N2.
the sum-product conjecture, which says that no set of numbers can produce a grid with simultaneously few distinct sums and few distinct products, is really making a statement about the relationship between arithmetic and geometric progressions.
“What the conjecture is really about is asking, ‘Can a set look both like an arithmetic progression and a geometric progression at the same time?’ And the answer is, ‘No, it can’t, because these are very different objects,’” Ford said.
“You’re thinking about sums and products and all of a sudden you’re drawing points and lines,” Shakan said. “It’s not obvious there should be any connection, but there is.”
“Addition and multiplication are two of the most basic operations in math,” Shakan said. “It shows how ignorant we are, because we can’t say everything about this problem.”