Selfridge received his Ph.D. in 1958 from the University of California, Los Angeles under the supervision of Theodore Motzkin.
[3] He was a founder of the Number Theory Foundation,[4] which has named its Selfridge prize in his honour.
[6] [7] 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 Paul Erdős, Selfridge solved a 150-year-old problem, proving that the product of consecutive numbers is never a power.
[9] 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 thanked "R. B. Eggleton for reorganizing and writing the paper in its final form".
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.
In support of his conjecture he showed a sufficient (but not necessary) condition for its truth is the existence of another Fermat prime beyond the five known (3, 5, 17, 257, 65537).