Wilfrid Keller

Current address:
Badestrasse 5
20148 Hamburg
Germany

e-mail:  wilfrid.keller@gmx.net

Curriculum Vitae

Member of Mathematische Gesellschaft in Hamburg, Deutsche Mathematiker-Vereinigung (DMV), American Mathematical Society (AMS).

Publications

Thesis

Asymptotische Aussagen und Fehlerabschätzungen für eine Klasse linearer Integro-Differential-Differenzengleichungen als Folge von Monotonieeigenschaften gewöhnlicher Anfangswertaufgaben (Supervisor: Lothar Collatz). Universität Hamburg, 1970, 75 pages.

Technical report

Ingo Büchel und____, Ein Programmsystem für Rationale Arithmetik: Einführung und Beispielsammlung, Bericht Nr. 8004. Rechenzentrum der Universitä Hamburg, April 1980, 123 pages + 12 hard copy tables.

Abstracts

1. Large twin prime pairs related to Mersenne numbers, 4 (1983), 490.
2. New factors of Fermat numbers, 5 (1984), 391−392.
3. The 17th prime of the form 5 · 2ⁿ + 1, 6 (1985), 121.
4. The Carmichael numbers up to 10¹³, 9 (1988), 328−329.
5. The least prime of the form k · 2ⁿ + 1 for certain values of k, 9 (1988), 417−418.
6. New prime solutions p of a ^p-1 = 1 (mod p²), 9 (1988), 503.
7. Richard G. E. Pinch and ____, The Carmichael numbers up to 10¹⁵, 13 (1992), 505.
8. ____ and François Morain, The complete factorization of some large Mersenne composites, 13 (1992), 506.
9. Harvey Dubner and ____, New Fibonacci and Lucas primes, 16 (1995), 267.
10. Prime solutions p of a ^p-1 = 1 (mod p²) for prime bases a, 19 (1998), 394.
11. ____ and Jörg Richstein, Prime solutions p of a ^p-1 = 1 (mod p²) for prime bases a, II, 20 (1999), 228.
12. Harvey Dubner and ____, Some large prime primitive parts of Fibonacci and Lucas numbers, 21 (2000), 248.
13. Günter Löh and ____, Generalized Cullen primes, 25 (2004), 440−441.

Papers

1. Factors of Fermat numbers and large primes of the form k · 2ⁿ + 1, Mathematics of Computation 41 (1983), 661−673. MR 85b:11117.
2. ____ and Günter Löh, The criteria of Kummer and Mirimanoff extended to include 22 consecutive irregular pairs, Tokyo Journal of Mathematics 6 (1983), 397−402; Supplement, ibid, 487. MR 85h:11014.
3. Woher kommen die größten derzeit bekannten Primzahlen?, Mitteilungen der Mathematischen Gesellschaft in Hamburg XII (1991), 211−229. MR 92j:11006.
4. Harvey Dubner and ____, Factors of generalized Fermat numbers, Mathematics of Computation 64 (1995), 397−405. MR 95c:11010.
5. New Cullen primes, Mathematics of Computation 64 (1995), 1733−1741; Supplement (with Wolfgang Niebuhr), ibid, S39−S46. MR 95m:11015.
6. ____ et Leopoldo Kulesz, Courbes algébriques de genre 2 et 3 possédant de nombreux points rationnels, Comptes Rendus de l'Académie des Sciences (I) 321 (1995), 1469−1472. MR 96i:11072.
7. Harvey Dubner and ____, New Fibonacci and Lucas primes, Mathematics of Computation 68 (1999), 417−427; Supplement, ibid, S1−S12. MR 99c:11008.
8. ____ and Jörg Richstein, Solutions of the congruence a ^p-1 = 1 (mod p^r), Mathematics of Computation 74 (2005), 927−936. MR 2005i:11004.
9. Chris K. Caldwell, Angela Reddick, Yeng Xiong, and ____, The history of the primality of One: A selection of sources, Journal of Integer Sequences 15 (2012), Article 12.9.8, 40 pages. MR 3005523.

Preprints

1. Factors of Fermat numbers and large primes of the form k · 2ⁿ + 1. II, 40 pages + 20 tables, 27 September 1992.
2. Factors of generalized Fermat and Mersenne numbers, 13 pages + 5 tables, 7 September 1995.

Tables

1. Factors of Mersenne numbers M_p for p < 100000, Hamburg, 1977; cf. abstract #8.
2. Primes of the form k · 2ⁿ + 1, k odd, for k < 1200 and n ≤ 1000, Hamburg, 1980; cf. paper #4 and preprint #1. 
3. Factors of Cullen numbers C_n and Woodall numbers W_n for 300 < n ≤ 1000; cf. paper #5.
   For an up-to-date account, see the tables maintained by Paul Leyland, https://www.brnikat.com/nums/cullen_woodall/cw.html .
4. Factors of Fibonacci numbers F_n (n odd) and Lucas numbers L_n for 1000 < n < 9750; cf. paper #7.
   For an extended and up-to-date account, see the site maintained by Blair Kelly, http://mersennus.net/fibonacci/index.html.

Web pages

1. Ray Ballinger and ____, List of primes k · 2ⁿ + 1 for k < 300,
   http://www.prothsearch.com/riesel1.html, since April 1998; extended to k < 600 in September 2000; extended to k < 1200 in June 2005. 
2. Prime factors k · 2ⁿ + 1 of Fermat numbers F_m and complete factoring status,
   http://www.prothsearch.com/fermat.html, since April 1998. 
3. ____ and Jörg Richstein, Fermat quotients q_p(a) that are divisible by p (currently not available).
   
4. The Sierpiński problem: Definition and status,
   http://www.prothsearch.com/sierp.html, since November 1998, unchanged since November 2002, reworked February 2010. 
5. The Riesel problem: Definition and status,
   http://www.prothsearch.com/rieselprob.html, since November 1998, unchanged since July 2008, reworked December 2010. 
6. List of primes k · 2ⁿ − 1 for k < 300,
   http://www.prothsearch.com/riesel2.html, since December 1998. 
7. Prime factors k · 2ⁿ + 1 of generalized Fermat numbers, since December 2001:  
   • numbers F_m(6) with complete factoring status, see base 6,
   • numbers F_m(10) with complete factoring status, see base 10,
   • numbers F_m(12) with complete factoring status, see base 12.
8. Factors of generalized Fermat numbers found after Björn & Riesel,
   http://www.prothsearch.com/GFNfacs.html, since December 2003. 
9. Factorization of generalized Fermat numbers for small indices m,
   http://www.prothsearch.com/GFsmall.html, since January 2004. 
10. Prime factors k · 2ⁿ + 1 of generalized Fermat numbers, since December 2008:  
    • numbers F'_m(3) = F_m(3) / 2 with complete factoring status, see base 3,
    • numbers F'_m(5) = F_m(5) / 2 with complete factoring status, see base 5.
11. Prime factors k · 2ⁿ + 1 of generalized Fermat numbers, since November 2016:  
    • numbers F'_m(7) = F_m(7) / 2 with complete factoring status, see base 7,
    • numbers F'_m(11) = F_m(11) / 2 with complete factoring status, see base 11.
12. History of the search for primes of the form k · 2ⁿ + 1,
    http://www.prothsearch.com/history.html, since February 2022.

Participation in a book project

Updates to "Paulo Ribenboim, Die Welt der Primzahlen: Geheimnisse und Rekorde", translated from the English by Jörg Richstein. Springer-Verlag, Berlin and Heidelberg, September 2006, xxiii + 356 pages; second, completely revised edition, March 2011, xxv + 366 pages.