Proth

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 though is not about an archivable form, but rather about a form which is tolerated on the current list, and the primes with this comment only appear on the list if the prime there for some other reason.

(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  
2202705 · 221320516 + 1 6418121 L5181 Dec 2021  
37 · 220267500 + 1 6101127 L4965 Jul 2022  
4168451 · 219375200 + 1 5832522 L4676 Sep 2017  
57 · 218233956 + 1 5488969 L4965 Oct 2020 Divides Fermat F(18233954)
63 · 216408818 + 1 4939547 L5171 Oct 2020 Divides GF(16408814, 3), GF(16408817, 5)
737 · 215474010 + 1 4658143 L4965 Nov 2022  
8215317227 + 27658614 + 1 4610945 L5123 Jul 2020 Gaussian Mersenne norm 41?, generalized unique
937 · 214166940 + 1 4264676 L4965 Jun 2022  
1099739 · 214019102 + 1 4220176 L5008 Dec 2019  
11404849 · 213764867 + 1 4143644 L4976 Mar 2021 Generalized Cullen
1225 · 213719266 + 1 4129912 L4965 Sep 2022 Generalized Fermat
1381 · 213708272 + 1 4126603 L4965 Oct 2022 Generalized Fermat
1481 · 213470584 + 1 4055052 L4965 Oct 2022 Generalized Fermat
159 · 213334487 + 1 4014082 L4965 Mar 2020 Divides GF(13334485, 3)
1619249 · 213018586 + 1 3918990 SB10 May 2007  
1781 · 212804541 + 1 3854553 L4965 Sep 2022  
189 · 212406887 + 1 3734847 L4965 Mar 2020 Divides GF(12406885, 3)
1927 · 212184319 + 1 3667847 L4965 Feb 2021  
2037 · 211855148 + 1 3568757 L4965 May 2022  

(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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