In 1993, Hungarian mathematician Paul Erdős—one of the most prolific mathematicians of the 20th century—posed a question with two components seemingly at odds with one another: Could a Sidon set be an “asymptotic basis of order three?”
Let us explain.
Named after another Hungarian mathematician, Simon Sidon, these sets are basically a collection of numbers where no two numbers in the set add up to the same integer. For example, in the simple Sidon set (1, 3, 5, 11), when any of the two numbers in the set are added together, they equal a unique number. Constructing a Sidon set with only four numbers is extremely easy, but as the set increases in size, it just gets harder and harder. As soon as two sums are the same, the collection of numbers is no longer considered a Sidon set.
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The second element of Erdős’ problem—that scary-sounding “asymptotic basis of order three” part—means that:
- a set must be infinitely large
- any large enough integer can be written as the result of adding together at most 3 numbers in the set.
So, this 30-year-old conundrum centered on whether or not these two elements could exist in the same set of numbers. For decades, the answer seemed to be no.
But in March of this year, Oxford graduate student Cédric Pilatte published a proof confirming the existence of such a Sidon set. Reaching that milestone wasn’t easy. In 2010, mathematicians proved that a Sidon set can be be an asymptotic basis of order 5, and three years later, they proved that is was also possible for a Sidon set to “be an asymptotic basis of order 4.” But “order 3” remained elusive—some considered it theoretically possible but incredibly difficult (and potentially impossible) to prove.
“They’re pulling in opposite directions,” Pilatte told Quanta Magazine. “Sidon sets are constrained to be small, and an asymptotic basis is constrained to be large. It was not obvious that it could work.”
So how did Pilatte get a mathematically square peg to fit a seemingly round hole? He took an unconventional approach and turned to geometry rather than the probabilistic method championed by Erdős and what’s called additive number theory. Pilatte replaced numbers with polynomials and made use of the recent work of Columbia University mathematicians. Combining these ideas, Pilatte successfully created a Sidon set dense enough and random enough to finally solve Erdős’s original problem.
Pilatte’s work relied on the discoveries of many mathematicians across different disciplines, and even combined seemingly unrelated fields of mathematics to answer the question. “It’s cool that these very deep techniques from algebraic geometry can also be used for this simple and concrete question about sets of numbers,” Pilatte told Quanta Magazine.
And with that, yet another “impossible” math question is found to be very much possible.
Darren lives in Portland, has a cat, and writes/edits about sci-fi and how our world works. You can find his previous stuff at Gizmodo and Paste if you look hard enough.
