Researching John Selfridge

Selfridge received his Ph.D. in 1958 from the University of California, Los Angeles under the supervision of Theodore Motzkin.

In 1962, he proved that 78,557 is a Sierpinski number; he showed that, when k = 78,557, all numbers of the form k2ⁿ + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. Five years later, he and Sierpiński proposed the conjecture that 78,557 is the smallest Sierpinski number, and thus the answer to the Sierpinski problem. A distributed computing project called Seventeen or Bust is currently trying to prove this statement, only five of the original seventeen possibilities remain.

In 1964, Selfridge and Alexander Hurwitz proved that the 14th Fermat number $2^\{2^\{14\}\}\; +\; 1$ was composite.

However, their proof did not provide a factor. It was not until 2010 that the first factor of the 14th Fermat number was found.

In 1975 John Brillhart, Derrick Henry Lehmer and Selfridge developed a method of proving the primality of p given only partial factorizations of p − 1 and p + 1.

Together with Samuel Wagstaff they also all participated in the Cunningham project.

Together with Paul Erdős, Selfridge solved a 150-year-old problem, proving that the product of consecutive numbers is never a power. It took them many years to find the proof and John made extensive use of computers, but the final version of the proof requires only a modest amount of computation, namely evaluating an easily computed function f(n) for 30,000 consecutive values of n. Selfridge suffered from writer's block and paid a former student to write up the result, even though it is only two pages long.

As a mathematician, Selfridge was one of the most effective number theorists with a computer. He also had a way with words. On the occasion that another computational number theorist, Samuel Wagstaff, was lecturing at the semiannual Bloomington Illinois Number Theory Conference on his computer investigations into Fermat's Last Theorem, someone a little too pointedly asked him what methods he was using and kept insisting on an answer. Wagstaff stood there like a deer blinded in headlights, totally at a loss what to say, until Selfridge helped him out. "He used the principle of computer fooling-aroundedness." Wagstaff said later that you probably wouldn't want to use that phrase in a research proposal asking for funding, such as an NSF proposal.

Selfridge also developed the Selfridge–Conway discrete procedure for creating an envy-free cake-cutting among three people. Selfridge developed this in 1960, and John Conway independently discovered it in 1993. Neither of them ever published the result, but Richard Guy told many people Selfridge's solution in the 1960s, and it was eventually attributed to the two of them in a number of books and articles.

Selfridge served on the faculties of the University of Illinois at Urbana-Champaign and Northern Illinois University from 1971 to 1991 (retirement), chairing the Department of Mathematical Sciences 1972–1976 and 1986–1990.He was executive editor of Mathematical Reviews from 1978 to 1986, overseeing the computerization of its operations . He was a founder of the Number Theory Foundation, which has named its Selfridge prize in his honour.