How the primality of R19 was discovered
The above website contains the answer to a question that has
interested me for several years today. I knew R23 was shown to be
prime in the late 1920s, but I had no clues when R19 was found to be
However, the site above shows the exact date when the proof of the
primality of R19 was published - February 14, 1918, just before the
end of World War I, by an American named Oscar Hoppe.
Whilst there is no clue as to the exact method by which Hoppe proved
R19 to be a prime, there is an interesting story about the discovery.
Before Hoppe's proof, factors were known for all composite repunits up
to R31 (presumably for R41 and R43 as they have very small factors 83
and 173 respectively), but no effort to test R19, R23 or (composite)
R37 for primality had been made. It is odd that the smaller factor of
R37 had not been found given that Edourd Lucas apparently factored R17
as early as the 1880s.
Almost, seemingly, in violation of scientific method, it is recorded
that Hoppe wrote R19 was prime before beginning his proof! Can you
imagine doing that today?!
- --- In email@example.com,
"julienbenney" <jpbenney@...> wrote:
> there is no clue as to the exact method by which Hoppe provedhttp://www.ams.org/journals/bull/1930-36-12/S0002-9904-1930-05077-6/S0002-9904-1930-05077-6.pdf
> R19 to be a prime
Another application of the principle was made by Hoppe in
the investigation of (10^19l)/9. Two proofs of the
primality of this number were submitted by him to the London
Mathematical Society. The first proof consisted in isolating
an unbroken sequence of the 73 smallest prime residues, in
fact all those <=761. He submitted this proof to Cole, who
did not consider it sufficient. This seems to indicate that
Cole had some doubt as to the rigor of his own method. It
was probably Cole's reply that prompted Hoppe to give an
independent proof of the primality of his number, as
described in his second communication to the Society.
Thursday, December 6th, 1917.
Lt.-Col. Cunningham and Dr. Western communicated a paper by
Mr. 0. Hoppe, "Proof of the Primality of N = (1/9)*(10^19-1)."*
* A further paper on this subject by Mr. 0. Hoppe was
communicated on February 14th, 1918; and an Abstract
containing the results of both papers will be found in the
Records of that meeting.
Thursday, February 14th, 1918.
Lt.-Col. Cunningham and Dr. Western gave an account of a
further investigation by Mr. 0. Hoppe* on "The Primality of
* See Records of Proceedings, December 6th, 1917.
"The Primality of (1/9)*(10^19-1)."
Mr. Oscar Hoppe.
*Mr. Oscar Hoppe's papers deal with the number
N=(1/9)*(10^19-1) and contain a summary of the results of
his calculations. He first employed the process for
factorising large numbers contained in Prof. F. N. Cole's
paper (Bulletin American Math. Soc., Ser. 2, Vol. 10, p.
134). He thus proved that p is a quadratic residue of every
factor of N, where p represents any prime, less than 761, of
which N is a quadratic residue, those of the form 4n+3
being taken with a minus sign.
He then searched for possible prime factors of N up to a
limit about 23*10^6, as follows. From the properties of such
factors to moduli 19, 8, 3, 5, 7, 11, 13, every prime factor
must be congruent to one of 360 residues (mod 2282280). All
the possible factors of these forms were excluded by using
the tests of the quadratic character (mods 17, 19, ..., 127).
Finally, he searched for solutions of N=x^2-y^2, excluding
possible values of x in a similar manner. The result was that
no factors were discovered,and that therefore N is a prime.
He also found that 3^(N-1) = 1 (mod N), which does not
furnish a proof that N is prime, but confirms the accuracy
of the other calculations.
*This Abstract (prepared by Dr. Western) gives a summary of
the results obtained in two papers communicated at the
meetings on December 6th, 1917, and February 14th, 1918.