# John Selfridge

John Selfridge
Born February 17, 1927
Died October 31, 2010 [1]
Nationality American
Fields Analytic number theory
Institutions University of Illinois at Urbana-Champaign
Northern Illinois University
Alma mater University of California, Los Angeles

John Lewis Selfridge (February 17, 1927 in Ketchikan, Alaska – October 31, 2010 in DeKalb, Illinois[1]), was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics. He co-authored 14 papers with Paul Erdős (giving him an Erdős number of 1).

Selfridge received his Ph.D. in 1958 from the University of California, Los Angeles under the supervision of Theodore Motzkin.[2]

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, as of January 2011 only six of the original seventeen possibilities remain.

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 250 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 a function f(n) for 30,000 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 division of a resource 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 60s, and it was eventually attributed to them in a number of books.

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 [1]. He was a founder of the Number Theory Foundation [2], which has named its Selfridge prize in his honour.

## Selfridge's Conjecture

John L. Selfridge made an intriguing conjecture about the Fermat numbers. Let g(n) be the number of distinct prime factors of 22n + 1. Then g(n) is not monotonic (nondecreasing). If another Fermat prime exists, that would imply the conjecture.[3]

## References

1. ^ a b "John Selfridge (1927–2010)". DeKalb Daily Chronicle. November 11, 2010. Retrieved November 13, 2010.
2. ^
3. ^ Prime Numbers: A Computational Perspective, Richard Crandall and Carl Pomerance, Second edition, Springer, 2011 Look up Selfridge's Conjecture in the Index.

## Publications

• Pirani, F. A. E.; Moser,, Leo; Selfridge, John (1950). "Elementary Problems and Solutions: Solutions: E903". Amer. Math. Monthly 57 (8): 561–562. MR 1527674.
• Eggan, L. C.; Eggan, Peter C.; Selfridge, J. L. (1982). "Polygonal products of polygonal numbers and the Pell equation". Fibonacci Quarterly 20 (1): 24–28. MR 0660755.
• Erdos, P; Selfridge, J. L. (1982). "Another property of 239 and some related questions". Congr. Numer: 243–257. MR 0681710.
• Lacampagne, C. B.; Selfridge, J. L. (1985). "Large highly powerful numbers are cubeful". Rocky Mountain Journal of Mathematics 15 (2): 459. doi:10.1216/rmj-1985-15-2-459. MR 0823257.
• Lacampagne, C. B.; Selfridge, J. L. (1986). "Pairs of squares with consecutive digits". Math. Mag. 59 (5): 270–275. doi:10.2307/2689401. MR 0868804.
• Blair, W. D.; Lacampagne, C. B.; Selfridge, J. L. (1986). "Notes: Factoring Large Numbers on a Pocket Calculator". Amer. Math. Monthly 93 (10): 802–808. doi:10.2307/2322936. MR 1540993.
• Guy, R. K.; Lacampagne, C. B.; Selfridge, J. L. (1987). "Primes at a glance". Math. Comp. 48 (177): 183–202. doi:10.1090/s0025-5718-1987-0866108-3. MR 0866108.
• Trench, William F.; Rodriguez, R. S.; Sherwood, H.; Reznick, Bruce A.; Rubel, Lee A.; Golomb, Solomon W.; Kinnon, Nick M.; Erdos, Paul; Selfridge, John (1988). "Problems and Solutions: Elementary Problems: E3243–E3248". Amer. Math. Monthly 95: 50–51. doi:10.2307/2323449. MR 1541238.
• Erdos, P.; Lacampagne, C. B.; Selfridge, J. L. (1988). "Prime factors of binomial coefficients and related problems". Acta Arith. 49 (5): 507–523. MR 0967334.
• Bateman, P. T.; Selfridge, J. L.; Wagstaff, S. S. (1989). "The New Mersenne conjecture". Amer. Math. Monthly 96 (2): 125–128. doi:10.2307/2323195. MR 0992073.
• Lacampagne, C. B.; Nicol, C. A.; Selfridge, J. L. (1990). "Sets with nonsquarefree sums". de Gruyter. pp. 299–311{{inconsistent citations}}
• Howie, John M.; Selfridge, J. L. (1991). "A semigroup embedding problem and an arithmetical function". Math. Proc. Cambr. Phil. Soc. 109 (2): 277–286. doi:10.1017/s0305004100069747. MR 1085395.
• Eggleton, R. B.; Lacampagne, C. B.; Selfridge, J. L. (1992). "Eulidean quadratic fields". Amer. Math. Monthly 99 (9): 829–837. MR 1191702.
• Erdos, P.; Lacampagne, C. B.; Selfridge, J. L. (1993). "Estimates of the least prime factor of a binomial coefficient". Math. Comp. 61 (203): 215–224. doi:10.1090/s0025-5718-1993-1199990-6. MR 1199990.
• Lin, Cantian; Selfridge, J. L.; Shiue, Peter Jau-shyong (1995). "A note on periodic complementary binary sequences". J. Combin. Math. Combin. Comput. 19: 225–29. MR 1358509.
• Blecksmith, Richard; McCallum, Michael; Selfridge, J. L. (1998). "3-smooth representations of integers". Amer. Math. Monthly 105 (6): 529–543. doi:10.2307/2589404. MR 1626189.
• Blecksmith, Richard; Erdos, Paul; Selfridge, J. L. (1999). "cluster primes". Amer. Math. Monthly 106 (1): 43–48. doi:10.2307/2589585. MR 1674129.
• Erdos, Paul; Malouf,, Janice L.; Selfridge, J. L.; Szekeres, Esther (1999). "Subsets of an interval whose product is a power". Discr. Math. 200 (1–3): 137–147. doi:10.1016/s0012-365x(98)00332-x. MR 1692286.
• Granville, Andrew; Selfridge, J. L. (2001). "Product of integers in an interval, modulo squares". El. J. Combin. 8 (1): #R5. MR 1814512.