# 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.

### 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].

### 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
• Finding Primes and Proving Primality's n-1 tests

### 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)