Jared Duker Lichtman (Balliol 2019, DPhil Mathematics), a third-year postgraduate student, has resolved a longstanding mathematical conjecture.
The conjecture deals with primitive sets: sequences of numbers in which no number divides any other, which were introduced by the mathematician Paul Erdős in the 1930s. An example of a primitive set is the set of all prime numbers, since each prime number can only be divided by 1 and itself.
In 1935 Erdős found that for any primitive set, including infinite ones, there is an ‘Erdős sum’ that is always finite: whatever the primitive set, its ‘Erdős sum’ will always be less than or equal to some number. In 1988 Erdős conjectured that the maximum possible Erdős sum would be the one for the set of all prime numbers, which comes out to about 1.64. Over the decades, mathematicians made partial progress towards proving this conjecture — showing, for instance, that it was true for particular types of primitive sets — but full proof remained elusive.
Fascinated by the question, Jared started working on the Erdős primitive set conjecture while he was an undergraduate at Dartmouth College in the US. He and his adviser, Carl Pomerance, found that a primitive set’s Erdős sum could be no greater than around 1.78.
Since coming to Oxford to read for his DPhil with Professor James Maynard (Balliol 2009 and Professor of Number Theory at Oxford’s Mathematical Institute), Jared has been working mainly on other problems relating to prime numbers. However, he continued to think about the Erdős problem, and in February 2022 he released ‘A proof of the Erdős primitive set conjecture’ – and in so doing he has proved that prime numbers hold a special place among primitive sets.
Jared explains his fascination with the conjecture and introduces it in a short video here.