The Top Twenty--a Prime Page Collection

Proth

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The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

(up) Definitions and Notes

[To be written soon, for now:] In 1878 Francois Proth (a self-taught farmer) published a short note stating four theorems related to primes, including the on now known as Proth's theorem [Proth1878]:
Proth's Theorem : Let n = h.2k+1 with 2k > h.   If there is an integer a such that a(n-1)/2 = -1 (mod n), then n is prime.
The Proth primes are those that meet the criteria of Proth's theorem.

Though Proth did not publish a proof, he did state in a letter that he had one (and Williams believes him [Williams98]).  The earliest proof I have seen is by Robinson in the 1950's; but I find it hard to believe that this is the first published proof since the proof is about two sentences long [Robinson57b]. Robinson [Robinson1958] listed the earlier tables: [Seelhoff1886, Cunningham1927, Kraitchik1924]; other early articles were [MW1977], [Shippee1978], and [Baillie1979].

(up) Record Primes of this Type

rankprime digitswhowhencomment
110223 · 231172165 + 1 9383761 SB12 Nov 2016  
2168451 · 219375200 + 1 5832522 L4676 Sep 2017  
319249 · 213018586 + 1 3918990 SB10 May 2007  
4193997 · 211452891 + 1 3447670 L4398 Apr 2018  
53 · 210829346 + 1 3259959 L3770 Jan 2014 Divides GF(10829343, 3), GF(10829345, 5)
627653 · 29167433 + 1 2759677 SB8 Jun 2005  
790527 · 29162167 + 1 2758093 L1460 Jun 2010  
828433 · 27830457 + 1 2357207 SB7 Dec 2004  
9161041 · 27107964 + 1 2139716 L4034 Jan 2015  
103 · 27033641 + 1 2117338 L2233 Feb 2011 Divides GF(7033639, 3)
1133661 · 27031232 + 1 2116617 SB11 Oct 2007  
126679881 · 26679881 + 1 2010852 L917 Aug 2009 Cullen
131582137 · 26328550 + 1 1905090 L801 Apr 2009 Cullen
147 · 25775996 + 1 1738749 L3325 Nov 2012  
159 · 25642513 + 1 1698567 L3432 Nov 2013  
16258317 · 25450519 + 1 1640776 g414 Jul 2008  
1727 · 25213635 + 1 1569462 L3760 Mar 2015  
183 · 25082306 + 1 1529928 L780 Apr 2009 Divides GF(5082303, 3), GF(5082305, 5)
195359 · 25054502 + 1 1521561 SB6 Dec 2003  
2013 · 24998362 + 1 1504659 L3917 Jul 2014  

(up) Related Pages

(up) References

Baillie1979
R. Baillie, "New primes of the form k · 2n + 1," Math. Comp., 33:148 (October 1979) 1333--1336.  MR 80h:10009 (Abstract available)
BCW81
Baillie, R., Cormack, G. and Williams, H.C., "The problem of Sierpinski concerning k · 2n + 1," Math. Comp., 37:155 (1981) 229--231.  MR 83a:10006a [Corrigenda: [BCW1982]]
Chen2003
Chen, Yong-Gao, "On integers of the forms kr-2n and kr2n+1," J. Number Theory, 98:2 (2003) 310--319.  MR1955419
Cunningham1927 (pp. 56-73)
A. J. C. Cunningham, Quadratic and linear tables, F. Hodgson, 1927.
HB1975
J. C. Hallyburton, Jr. and J. Brillhart, "Two new factors of Fermat numbers," Math. Comp., 29 (1975) 109--112.  Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday.  MR 51:5460
Kraitchik1924 (pp. 12-13)
M. Kraitchik, Recherches sur la th'eorie des nombres, W. W. Norton \& Co., Vol, 1, Gauthier-Vilars, 1924.
MW1977
G. Matthew and H. C. Williams, "Some new primes of the form k· 2n+1," Math. Comp., 31 (1977) 797--798.  MR 55:12605
Proth1878
F. Proth, "Théorèmes sur les nombres premiers," C. R. Acad. Sci. Paris, 85 (1877) 329-331.
Robinson57b
R. M. Robinson, "The converse of Fermat's theorem," Amer. Math. Monthly, 64 (1957) 703--710.  MR 20:4520
Robinson58
R. M. Robinson, "A report on primes of the form k· 2n + 1 and on factors of Fermat numbers," Proc. Amer. Math. Soc., 9 (1958) 673--681.  MR 20:3097
Seelhoff1886
P. Seelhoff, "Die Zahlen von der Form k· 2n+1," Zeitschrift fur Mathematik und Physik, 31 (1886) 380.
Shippee1978
D. E. Shippee, "Four new factors of Fermat numbers," Math. Comp., 32:143 (1978) 941. (Abstract available)
Williams98 (pp. 121-140)
H. C. Williams, Édouard Lucas and primality testing, Canadian Math. Soc. Series of Monographs and Adv. Texts Vol, 22, John Wiley \& Sons, New York, NY, 1998.  pp. x+525, ISBN 0-471-14852-0. MR 2000b:11139 (Annotation available)
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