Re: [tuning-math] Digest Number 1011

Erv Wilson should be apprised of Stormer's theorem and Lehmer's result
as it was he who asked me to compute the 23-limit up to 10 million.

--John

--- In tuning-math@yahoogroups.com, John Chalmers <JHCHALMERS@U...> wrote:

> Erv Wilson should be apprised of Stormer's theorem and Lehmer's result
> as it was he who asked me to compute the 23-limit up to 10 million.

Does Erv have an email address?

I have the Lehmer paper, which computes all 41-limit superparticular
ratios. It also gives an improvement on Stoermer's result, which
allows faster computation. Aside from the fact that the paper is by
someone I knew, another thing that gets me is that I had heard that
Stoermer had solved the 5-limit superparticular ratio problem, showing
that the smallest such comma is 81/80. Apparently it was a mere minor
detail his own computations went up to the 7-limit, and that his
result was much more general!

Lehmer proves the following:

(Stoermer-Lehmer)

Let 2=q1 < q2 < ... < qt be prime, let Q be {q1^a1 ... qt^at}, and let
Q' be the set of all 2^t-1 square-free members of Q, excepting 2.
Suppose (S+1)/S is a superparticular ratio belonging to Q. Then
S = (xn-1)/2, where (xn, yn) is a solution of the Pell's equation

x^2 - 2D y^2 = 1

where D is in Q', 1 <= n <= max(3, (1+qt)/2)), and yn is in Q.

Conversely, if (xn, yn) is a solution to a Pell's equation with the
above conditions, then (S+1)/S belongs to Q.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

By the way, I misread the statement of Stoermer's theorem in Haluska;
Stoermer only proves his theorem for (S+2)/S and (S+1)/S. Haluska,
however, is aware that the more general result follows easily from
Baker's theorem. I showed the stonger result that for any epimericity
less than 1, we have only a finite list, in fact; also using Baker's
theorem.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

4375/4374 is named "ragisma" on Manuel's list, but Stoermer was the
first to know it was the largest 7-limit superparticular, and possibly
the first to know about it at all, so naming it after him would seem
logical. Dick Lehmer was the first to know about the whole lot of them
up to the 41-limit, and I suppose a lehmerisma of 9801/9800 might be
appropriate.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Dick Lehmer was the first to know about the whole lot of them
> up to the 41-limit, and I suppose a lehmerisma of 9801/9800 might be
> appropriate.

Or perhaps 3025/3024 is a better lehmerisma, as Lehmer mentions it on
the first page of his paper. In any case, 9801/9800 is called a
"kalisma" for reasons unknown to me, but Manuel does not list
3025/3024, so attaching Dick's name to it seems like a good idea. He
was a wonderful gentlemen and a hell of a mathematician, and deserves
a comma as much as anyone.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Or perhaps 3025/3024 is a better lehmerisma, as Lehmer mentions it on
> the first page of his paper. In any case, 9801/9800 is called a
> "kalisma" for reasons unknown to me, but Manuel does not list
> 3025/3024, so attaching Dick's name to it seems like a good idea. He
> was a wonderful gentlemen and a hell of a mathematician, and deserves
> a comma as much as anyone.

Lehmer unfortunately does not give a cite, but he claims Gauss
mentioned 9801/9800, so I think I'll propose gaussisma for that, and
lehmerisma for 3025/3024.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, John Chalmers <JHCHALMERS@U...>
wrote:
>
> > Erv Wilson should be apprised of Stormer's theorem and Lehmer's
result
> > as it was he who asked me to compute the 23-limit up to 10
million.
>
> Does Erv have an email address?
>
> I have the Lehmer paper, which computes all 41-limit superparticular
> ratios.

Great. Xenharmonikon readers might like to know: are there any 23-
limit superparticulars above 10,000,000?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Great. Xenharmonikon readers might like to know: are there any 23-
> limit superparticulars above 10,000,000?

Nope.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Great. Xenharmonikon readers might like to know: are there any 23-
> > limit superparticulars above 10,000,000?
>
> Nope.

I spoke too soon, after only looking at the strict 23 limit table. For
some reason probably connected to the fact that 19 is the larger of a
twin prime pair, there are two 19-limit commas smaller than any
strictly 23-limit comma. They are:

5909761/5909760
|-8 -5 -1 0 2 2 2 -1>

11859211/11859210
|-1 -4 -1 1 -4 1 0 4>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > > limit superparticulars above 10,000,000?
> > >
> > > Nope.
> >
> > I spoke too soon, after only looking at the strict 23 limit
table.
> For
> > some reason probably connected to the fact that 19 is the larger
of
> a
> > twin prime pair, there are two 19-limit commas smaller
>
> You mean larger?

Never mind; larger numbers, smaller in cents.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > Great. Xenharmonikon readers might like to know: are there any
23-
> > > limit superparticulars above 10,000,000?
> >
> > Nope.
>
> I spoke too soon, after only looking at the strict 23 limit table.
For
> some reason probably connected to the fact that 19 is the larger of
a
> twin prime pair, there are two 19-limit commas smaller than any
> strictly 23-limit comma. They are:
>
> 5909761/5909760
> |-8 -5 -1 0 2 2 2 -1>
>
> 11859211/11859210
> |-1 -4 -1 1 -4 1 0 4>

As expected, the former appears in XH17; the latter doesn't. So John
Chalmers could inform the readership that if the 19-limit list is
supplemented with 11859211/11859210, the "less than 10^7"
qualification can be dropped, and the lists are complete. The total
number of superparticulars in the 23-prime-limit would then be 241.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> As expected, the former appears in XH17; the latter doesn't. So John
> Chalmers could inform the readership that if the 19-limit list is
> supplemented with 11859211/11859210, the "less than 10^7"
> qualification can be dropped, and the lists are complete. The total
> number of superparticulars in the 23-prime-limit would then be 241.

That and a cite of Dick Lehmer's article would require only a short
paragraph in the next edition of XH.

I calculated the 29- and 31-limit superparticulars.
Pretty sure they are complete too, tested until 7*10^7.
Only the denominators are shown.

29-limit: 28, 29, 57, 87, 115, 116, 144, 174, 203, 231,
260, 289, 319, 377, 405, 493, 550, 551, 608, 637, 725, 782, 783,
840,
1014, 1044, 1275, 1449, 1595, 1624, 1682, 2000, 2001, 2175, 2204,
2261,
2464, 2639, 2754, 2783, 3248, 3450, 3509, 4640, 4784, 4900, 5103,
5887,
5915, 6669, 7105, 7424, 7888, 8670, 9801, 10556, 11339, 12005,
12672,
13224, 13310, 13311, 13455, 19227, 20735, 23750, 24794, 25839,
26999,
30624, 30855, 35321, 47124, 53360, 72500, 83520, 87464, 136850,
158949,
166634, 168750, 176000, 176175, 184092, 240786, 244035, 303600,
410669,
418760, 613088, 949025, 1163799, 1235168, 1243839, 1625624,
1852200,
2697695, 4004000, 4090624, 8268799, 10556000, 18085704

31-limit: 30, 62, 92, 124, 154, 155, 186, 247, 279, 340,
341, 434, 464, 495, 527, 588, 620, 650, 713, 805, 836, 867, 899,
930,
960, 1053, 1209, 1364, 1425, 1518, 1519, 1767, 1859, 2015, 2232,
2944,
2975, 3564, 3750, 3875, 3968, 4185, 4959, 4991, 5642, 5796, 6075,
6137,
6292, 6324, 6479, 6727, 7656, 7904, 7935, 8091, 8463, 8525, 8959,
9424,
10880, 11780, 11934, 12121, 13299, 13454, 15624, 17576, 19250,
19343,
19964, 21141, 22815, 23374, 23715, 24024, 27404, 29791, 31464,
31899,
32798, 41261, 42687, 49010, 58310, 78336, 96875, 98735, 102486,
108375,
111320, 111475, 116280, 116963, 122264, 174096, 175769, 178125,
190463,
207575, 212381, 227447, 240064, 245024, 260337, 268800, 278783,
288144,
314432, 453375, 459172, 509795, 773604, 863939, 912950, 1147124,
1154439,
1255500, 1594175, 2307360, 2310399, 2345056, 3206268, 3301375,
3346109,
3897165, 14753024, 16092999

Manuel