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The Stronger Feit-Thompson conjecture states that:

There exist no distinct prime numbers $p$ and $q$ such that: $\dfrac {p^q - 1} {p - 1}$ and $\dfrac {q^p - 1} {q - 1}$ are not coprime.

This was refuted by N.M. Stephens in July 1971 who published On the Feit-Thompson Conjecture in Mathematics of Computation volume 25 no. 115 (p. 625) where he showed that for $p = 17$ and $q = 3313$, the above expressions have the common prime factor $112 \, 643$.

The Wikipedia article Feit–Thompson conjecture suggests his first name might have been Nelson, but I have been unable to corroborate this.

Does anyone have any biographical details about N.M. Stephens?

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I’m Nelson Stephens’ daughter. He was born on 6th May 1941 and was a Professor of Mathematics at the University of London. He passed away on 8th January 2024. Not sure I can be much help on the maths but very cool to see what an impact he had :)

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    $\begingroup$ Condolences for you and your family's recent loss. Thank you for finding this post and letting us know. $\endgroup$ Jan 19 at 1:03
  • $\begingroup$ Thank you for responding. My condolences to your family. I'm remembering now I was always fascinated by a couple of relatively minor articles of his. They are sciencedirect.com/science/article/pii/0022314X77900324 and academic.oup.com/blms/article-abstract/11/1/52/277885 which I have to go back to now. $\endgroup$
    – Mike
    Feb 23 at 1:34
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    $\begingroup$ Sophie - condolences on the passing of your Dad. We attended the same grammar school and we were good friends. The year group last met up in 2016. Can you contact me on 'Holdstock.groups (at) gmail.com'? I have written to his email address, if you are able to access that. John Holdstock $\endgroup$ Mar 16 at 14:15
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A google search for [Stephens + On the Feit-Thompson Conjecture" led me to an online copy of the paper, and the bottom of the first page of this paper gives Stephens' affiliation at that time as "Atlas Computer Laboratory".

A google-books search for "Stephens, Nelson" + "Atlas Computer Laboratory" led to a snippet view of the London Mathematical Society's "List of Members" for 1972 that shows the full entry of Stephen's entry:

STEPHENS, NELSON MALCOLM, B.Sc., Ph.D.; Joint Research Fellow, Atlas Computer Laboratory; Pembroke College, Oxford. [1966

A google-books search for "Stephens, Nelson Malcolm" gives several hits, and in particular shows that Donald E. Knuth gave Stephens' full name in one of Knuth's The Art of Computer Programming volumes AND a snippet view of the London Mathematical Society's "List of Members" for 1976 that shows the full entry of Stephen's entry:

STEPHENS, NELSON MALCOLM, B.Sc., Ph.D.; University College, Cardiff CFI IXL [1966

A google search for "Nelson Malcolm Stephens" shows some Pembroke College Records for 1970, 1971, 1972 that are accessible online.

With this information in hand you can try various other search words/phrases, such as "Nelson M Stephens" + "Manchester", for items missed by the previous searches.

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  • $\begingroup$ Thank you, that's a good start. Probably safe to conjecture that he is British and that his date of birth is probably in the late 1930s or early 1940s. I will continue hunting. $\endgroup$ Jul 27, 2022 at 9:37
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    $\begingroup$ The January 2020 newsletter of the London Mathematical Society lists Mr. Stephens as a member with over 50 years of membership. $\endgroup$
    – njuffa
    Jul 27, 2022 at 10:33
  • $\begingroup$ Stephens' Ph.D. thesis (Manchester 1965; see B. J . Birch and N. M. Stephens, "The parity of the rank of the Mordell-Weil group", Topology, Vol. 5, No. 4, Nov. 1966, pp. 295-299) is referred to in several places as "as-yet-unpublished" and I have not had any luck in tracking it down online. Math Genealogy Project dates the Ph.D. thesis to 1966 instead: "The Birch--Swinnerton-Dyer Conjecture for Selmer curves of positive rank". $\endgroup$
    – njuffa
    Jul 27, 2022 at 10:52
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    $\begingroup$ N. M. Stephens, "Conjectures concerning elliptic curves", Bulletin of the American Mathematical Society. Vol. 73, No. 1, Jan. 1967, pp. 160-163, contains this footnote: "These results are part of the author's doctoral thesis submitted to Manchester University, England, in 1965. The author wishes to thank Dr. B. J. Birch for his guidance." $\endgroup$
    – njuffa
    Jul 27, 2022 at 10:58
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    $\begingroup$ @njuffa: Math Genealogy Project dates the Ph.D. thesis to 1966 instead -- There is sometimes a difference between a year printed on the title page of a dissertation and the year the degree is awarded. For example, see my last comment here. I suspect that is the case here, with the Math Genealogy Project using the year the degree was awarded because the primary focus there is the degree and not the dissertation, whereas the citation in "The parity of the rank of the Mordell-Weil group" is using the year affixed to the dissertation. $\endgroup$ Jul 27, 2022 at 12:53

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