# John Selfridge

John Selfridge
Born February 17, 1927
Died October 31, 2010 (aged 83) [1]
Nationality American
Alma mater University of California, Los Angeles
Scientific career
Fields Analytic number theory
Institutions University of Illinois at Urbana-Champaign
Northern Illinois University

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 April 2017 only five of the original seventeen possibilities remain.

In 1964, Selfridge and Alexander Hurwitz proved that the 14th Fermat number ${\displaystyle 2^{2^{14}}+1}$ was composite. [3] However, their proof did not provide a factor. It was not until 2010 that the first factor of the 14th Fermat number was found. [4] [5]

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. [6] 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 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 [1]. He was a founder of the Number Theory Foundation [2], which has named its Selfridge prize in his honour.

## Selfridge's Conjecture about Fermat Numbers

Selfridge made the following conjecture about the Fermat numbers Fn = 22n + 1 . Let g(n) be the number of distinct prime factors of Fn (sequence A046052 in the OEIS). As to 2016, g(n) is known only up to n = 11, and it is monotonic. Selfridge conjectured that contrary to appearances, g(n) is NOT monotonic. 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). [7]

## Selfridge's Conjecture about Primality Testing

This conjecture is also called the PSW conjecture, after Selfridge, Carl Pomerance, and Samuel Wagstaff.

Let p be an odd number, with p ≡ ± 2 (mod 5). Selfridge conjectured that if

• 2p-1 ≡ 1 (mod p) and at the same time
• fp+1 ≡ 0 (mod p),

where fk is the kth Fibonacci number, then p is a prime number, and he offered \$500 for an example disproving this. He also offered \$20 for a proof that the conjecture was true. The Number Theory Foundation will now cover this prize. An example will actually yield you \$620 because Samuel Wagstaff offers \$100 for an example or a proof, and Carl Pomerance offers \$20 for an example and \$500 for a proof. Selfridge requires that a factorization be supplied, but Pomerance does not. The conjecture was still open August 23, 2015. The related test that fp-1 ≡ 0 (mod p) for p ≡ ±1 (mod 5) is false and has e.g. a 6-digit counterexample.[8][9] The smallest counter example for +1 (mod 5) is 6601 = 7 × 23 × 41 and the smallest for -1 (mod 5) is 30889 = 17 × 23 × 79.

## References

1. ^ a b "John Selfridge (1927–2010)". DeKalb Daily Chronicle. November 11, 2010. Retrieved November 13, 2010.
2. ^
3. ^ J. L. Selfridge; A. Hurwitz (January 1964). "Fermat numbers and Mersenne numbers". Mathematics of Computation. 18 (85): 146–148. JSTOR 2003419. doi:10.2307/2003419.
4. ^ Rajala, Tapio (3 February 2010). "GIMPS' second Fermat factor!". Retrieved 9 April 2017.
5. ^ Keller, Wilfrid. "Fermat factoring status". Retrieved 11 April 2017.
6. ^ John Brillhart; D. H. Lehmer; J. L. Selfridge (April 1975). "New Primality Criteria and Factorizations of 2m ± 1". Mathematics of Computation. 29 (130): 620–647. JSTOR 2005583. doi:10.1090/S0025-5718-1975-0384673-1.
7. ^ Prime Numbers: A Computational Perspective, Richard Crandall and Carl Pomerance, Second edition, Springer, 2011 Look up Selfridge's Conjecture in the Index.
8. ^ According to an email from Pomerance.
9. ^ Carl Pomerance, Richard Crandall, Prime Numbers: A Computational Perspective, Second Edition, p. 168, Springer Verlag, 2005.

## 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. MR 0823257. doi:10.1216/rmj-1985-15-2-459.
• Lacampagne, C. B.; Selfridge, J. L. (1986). "Pairs of squares with consecutive digits". Math. Mag. 59 (5): 270–275. MR 0868804. doi:10.2307/2689401.
• 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. MR 1540993. doi:10.2307/2322936.
• Guy, R. K.; Lacampagne, C. B.; Selfridge, J. L. (1987). "Primes at a glance". Math. Comp. 48 (177): 183–202. MR 0866108. doi:10.1090/s0025-5718-1987-0866108-3.
• 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. MR 1541238. doi:10.2307/2323449.
• 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. MR 0992073. doi:10.2307/2323195.
• Lacampagne, C. B.; Nicol, C. A.; Selfridge, J. L. (1990). "Sets with nonsquarefree sums". de Gruyter: 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. MR 1085395. doi:10.1017/s0305004100069747.
• 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. MR 1199990. doi:10.1090/s0025-5718-1993-1199990-6.
• 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. MR 1626189. doi:10.2307/2589404.
• Blecksmith, Richard; Erdos, Paul; Selfridge, J. L. (1999). "cluster primes". Amer. Math. Monthly. 106 (1): 43–48. MR 1674129. doi:10.2307/2589585.
• 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. MR 1692286. doi:10.1016/s0012-365x(98)00332-x.
• Granville, Andrew; Selfridge, J. L. (2001). "Product of integers in an interval, modulo squares". El. J. Combin. 8 (1): #R5. MR 1814512.