Number of superparticular ratios in a certain limit

Is there a proof that any given prime limit contains only a finite number of superparticular ratios? For example, I proved to myself today that 2/1, 3/2, 4/3, and 9/8 are the only 3-limit superparticular ratios, using some modular arithmetic. I tried to do 5-limit too but that seemed messy (there are 6 cases) and I wasn't motivated enough. I wrote a program to print out all the ones it could find, and they all seemed to stop pretty soon (the last 5-limit one was 81/80), but that doesn't prove anything, really. I want a proof!

Keenan Pepper

--- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:
>
> Is there a proof that any given prime limit contains only a finite
number of
> superparticular ratios? For example, I proved to myself today that
2/1, 3/2,
> 4/3, and 9/8 are the only 3-limit superparticular ratios, using some
modular
> arithmetic.

Indeed there is; this is Stormer's theorem. He proved it in a manner
which allows a constructive listing of the p-limit superparticulars,
but if you have Baker's theorem a nonconstructive proof is easy.

I tried to do 5-limit too but that seemed messy (there are 6 cases)
> and I wasn't motivated enough. I wrote a program to print out all
the ones it
> could find, and they all seemed to stop pretty soon (the last
5-limit one was
> 81/80), but that doesn't prove anything, really. I want a proof!

Stormer proved it, and the paper to read is by Dick Lehmer: "On a
problem of Stormer", Illinois Journal of Mathematics 8(1964), 57-79.
Well worth reading, and I've discussed it on this list.

Gene Ward Smith wrote:
> Indeed there is; this is Stormer's theorem. He proved it in a manner
> which allows a constructive listing of the p-limit superparticulars,
> but if you have Baker's theorem a nonconstructive proof is easy.

Is this Carl Fredrik M�lertz St�rmer, the same guy who did the aurora stuff?

> Stormer proved it, and the paper to read is by Dick Lehmer: "On a
> problem of Stormer", Illinois Journal of Mathematics 8(1964), 57-79.
> Well worth reading, and I've discussed it on this list. I just read it but I think it needs some time to sink in. =P It's amazing how those superparticular ratios just pop out of Pell's equation.

Keenan

--- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:

> Is this Carl Fredrik Mülertz Størmer, the same guy who did the
aurora stuff?

I presume so, but you can check the references in Lehmer's paper and
find out. I used to have a copy, but sadly I think the cleaning lady
threw it out.

> I just read it but I think it needs some time to sink in. =P It's
amazing how
> those superparticular ratios just pop out of Pell's equation.

Yeah, that's neat, and they don't need to be superparticular to apply
the method.

I have been reading with interest the various messages on this topic,
particularly those of Gene Ward Smith and Paul Erlich.

I'm a research assistant sponsored by Prof R.P. Brent at the
Australian National University. The general area I'm working is the
study of algorithms for computing the elementary functions (at very
high precisions).

Recent work has led me to Derrick Lehmer's 1964 paper, "On a Problem
of Størmer", which I call [L64], and it is currently a subject of
intense interest to me, because I have found a very practical use for
these "Superparticular Ratios" (aka Smooth Triangular Numbers,
or "smooth pairs").

When I considered the question of how I might find them, I found
precious little help was available in the body of published works.
This is a shame (although perhaps to my advantage, as I think that I
could make a case for a PhD thesis on this topic alone!)

I did find [L64], of course and have been immersed in it (and the
implementation of its method) ever since. This has kept me off the
streets for some time now, and I have also been looking into the
historical aspects - the problem has only been seriously considered
twice in over a century, but in both cases, Lehmer and Størmer, we
have particularly interesting (not to mention bright) characters.

Carl Størmer was indeed the "Aurora" man, as somebody asked. There is
a good summary of his life and work that appeared as an obituary
piece by Viggo Brun in "Acta Mathematica" (v100, Sep 1958), and a
nice picture!

Størmer solved the STN enumeration problem when he was very young -
writing in 1896/1897 he would have been just 22, so would probably
have barely completed his Bachelor's degree. What turned his
mind temporarily to this particular problem is unclear, it might
simply have been a puzzle that caught his fancy while on summer
holiday.

In any case, having dealt with the problem, he put together a paper,
sent it off, and moved on to other things - the problem then remained
undisturbed in the archives of oblivion until dusted off 67 years
later by DHL. What exactly prompted this interest is another
interesting, but unanswered, question!

Størmer's early papers are notoriously difficult to locate. His
original 1897 paper I have not been able to obtain, nor have I been
able to find the 1908 note he published in which he applied his
result specifically to the problem in "Lehmer terms" (which is the
way we generally view it).

His logic can be "reverse-engineered" by inference from Lehmer's
paper, however, which gives a more accurate appreciation of the
precise nature of Lehmer's contribution. That contribution, by the
way, came from Lehmer's intimate knowledge of the properties of the
sets of multiple solutions of Pell equations. Lehmer published a
paper on this topic back in 1938(9?), a paper written when he too
was just 22 years old!

Some specific observations, results

a) Members here would call this a "157-limit comma", I believe?

6318268857746831540296874 =
2, 11, 17, 29^2, 31^7, 37^2, 43, 79, 157

6318268857746831540296875 =
3, 5^6, 7^4, 19, 47^2, 89^2, 97^2, 131, 137

b) Is there a bigger 157-limit comma?

Even with Lehmer's refinement, and today's mega-fast computers,
the requirement to solve 2^N Pell equations with increasingly large
parameter values is daunting, to say the least.

For p <= 157, we have N = 2^37, so are looking at over
100,000,000,000 Pell equations to test.

This would explain why the known results have not been pushed much
further than p <= 97, which is a limit that seems implicit in the
relevant OEIS entries (Sloane's Online Encyclopedia of Integer
Sequences)

I have a "refinement" of Lehmer's refinement, one that promises to
dramatically reduce the computational cost of this method ... I am
working on a proof-of-correctness right now (for obvious reasons)
before I can state with unimpeachable certitude that any enumerations
I produce are complete

Meanwhile, though, I think it safe to assume that this really is
the maximal case for 157.

c) The Missing entries in Lehmer's 1964 Tables

Ref: [L64]

* the tables involving the superparticulars, ie Table I (S, S-1),
are correct, only Tables II (S, S-2) and III (S, S-4) are
affected

* pg 68, the count given for Table II is 101, it should be 109

* same page, the count for Table III is 99, it should be 103

* the 8 missing entries in table II are:

(t = 7)
2025
(t = 10)
10935, 12901781, 26578125

(t = 11)
4807, 12495, 16337, 89375

* the 4 missing entries in Table III are:

(t = 11)
961, 8649, 277729, 9503329

Cheers to all ...

Jim White
Math'l Sciences Institute
ANU, Canberra

--- In tuning-math@yahoogroups.com, "mathimagics" <mathimagics@...>
wrote:
>
> I have been reading with interest the various messages on this topic,
> particularly those of Gene Ward Smith and Paul Erlich.
>
> I'm a research assistant sponsored by Prof R.P. Brent at the
> Australian National University. The general area I'm working is the
> study of algorithms for computing the elementary functions (at very
> high precisions).

Very interesting. Here is a problem of musical interest, whcih I think
could also be relevant to applications in computing. Given a prime p,
there can be found (I presume, I haven't considered a proof) phi(p)
superparticular ratios such that the exponent vectors define a
unimodular matrix, and hence which form a set of generators for the
group of p-limit positive rationals under multiplication.

Examples:

3: [4/3, 9/8]
5: [16/15, 25/24, 81/80]
7: [126/125, 225/224, 2401/2400, 4375/4374]

Question:

For each p, there is going to be a minimax superparticular ratio;
that is, the smallest of the largest (least height) ratios. For
instance, that is 4/3 for 3, 16/15 for 5, 126/125 for 7. How can we
find this minimax comma expeditiously? How may we enumerate all the
sets of superparticulars with this minimax comma which form a set of
generators?

If you want to try your hand at calculating some answers for small p,
I'd be interested to see them here.

--- In tuning-math@yahoogroups.com, "mathimagics" <mathimagics@...>
wrote:

> c) The Missing entries in Lehmer's 1964 Tables

Do you think you could make a corrected, and perhaps extended, listing
available? I would find a list of numerators in ascii format separated
by commas convenient (it could easily be edited to feed into Maple.)

"Gene Ward Smith" <genewardsmith@...> wrote:
>
> Do you think you could make a corrected, and perhaps extended,
> listing available? I would find a list of numerators in ascii
> format separated by commas convenient (it could easily be
> edited to feed into Maple.

I will be providing a set of tables for primes up to 127 and beyond
via Richard Brent's website sometime soon, with a choice of comma-
delimited lists and EVF (exponent-vector format, as you see in
Lehmer's Table IB).

It will be a few weeks before that's up and running, though.

Meanwhile I could easily post here the sets of numerators for the
smaller primes, if you like. I can begin with the existing Lehmer
Table I data - the 869 41-limit numerators, grouped by highest-prime-
dividing as per Lehmer, and then extend that by increments.

The incremental counts for the primes from 47 to 61, are
284 p=43
349 p=47
428 p=53
524 p=59
652 p=61

Let me know if any of this is what you actually want.

Also let me know if you want any of the auxiliary sets, as per
Lehmer's Tables II and II - ie the pairs of the form (S, S-2) and (S,
S-4).

Cheers

"Gene Ward Smith" <genewardsmith@...> wrote:

> Given a prime p, there can be found (I presume, I haven't
> considered a proof) phi(p) superparticular ratios such that
> the exponent vectors define a unimodular matrix, and hence which
> form a set of generators for the group of p-limit positive
> rationals under multiplication.

Yes, and as you add more primes, increasing the dimension, there are
ever-increasing numbers of combinations that will form a basis, and
mnany of these form unimodular matrices.

> For each p, there is going to be a minimax superparticular ratio;
> that is, the smallest of the largest (least height) ratios. For
> instance, that is 4/3 for 3, 16/15 for 5, 126/125 for 7. How can we
> find this minimax comma expeditiously?

Have I interpreted this correctly? =>

"Given N primes, what is the largest sup-ratio S which can form a
unimodular matrix when combined with (N-1) larger ratios?"

Such a value clearly exists, if my reading of your definition is
correct

> How may we enumerate all the sets of superparticulars with this
> minimax comma which form a set of generators?

From the little experience I've had, fairly slowly .... :-)

I was in fact doing just this sort of thing a few weeks ago
(improving a result by Xavier Gourdon and Pascal Sebah related to
computing tables of logarithms of the small primes)

LA is not one of my strengths (!) so my method was fairly crude. I
was only after one good result, and had to write multiple-precision
rational matrix routines from scratch, to get something usable.

I don't really know what the best general method is, though, for
finding a basis among a large batch of vectors, so I just used a
combination generator to select and test combinations of N vectors,
one at a time until I found a hit.

It's fast enough, however, to allow me to provide such answers for p
up to 41 or so, I would think.

Cheers

--- In tuning-math@yahoogroups.com, "mathimagics" <mathimagics@...>
wrote:
> "Given N primes, what is the largest sup-ratio S which can form a
> unimodular matrix when combined with (N-1) larger ratios?"

Modulo some confusion about "larger" and "smaller" yes--of course, the
largest ratios have the smallest height--ie, the smallest numerators.

> I was in fact doing just this sort of thing a few weeks ago
> (improving a result by Xavier Gourdon and Pascal Sebah related to
> computing tables of logarithms of the small primes)

> It's fast enough, however, to allow me to provide such answers for p
> up to 41 or so, I would think.

Results for the smaller prime limits might be of interest to people
here.

--- In tuning-math@yahoogroups.com, "mathimagics" <mathimagics@...>
wrote:

> Meanwhile I could easily post here the sets of numerators for the
> smaller primes, if you like.

The smaller primes are the most interesting for us.

Here are the complete lists for primes up to 31.

Each group corresponds to a specific prime, P. It is a list of all
the integers S having the property that the greatest prime divisor
of S x (S-1) is P (or, equivalently, that S/(S-1) is a P-limit
superparticular ratio).

The groups are self-identifying, each begins with the value P.

Similar data (necessarily involving groups of increasing size) can be
provided on request for any prime up to P = 127.

Cheers
Jim White
ANU, Canberra

=================================

3, 4, 9

5, 6, 10, 16, 25, 81

7, 8, 15, 21, 28, 36,
49, 50, 64, 126, 225, 2401,
4375

11, 12, 22, 33, 45, 55,
56, 99, 100, 121, 176, 243,
385, 441, 540, 3025, 9801

13, 14, 26, 27, 40, 65,
66, 78, 91, 105, 144, 169,
196, 325, 351, 352, 364, 625,
676, 729, 1001, 1716, 2080, 4096,
4225, 6656, 10648, 123201

17, 18, 34, 35, 51, 52,
85, 120, 136, 154, 170, 221,
256, 273, 289, 375, 442, 561,
595, 715, 833, 936, 1089, 1156,
1225, 1275, 1701, 2058, 2431, 2500,
2601, 4914, 5832, 12376, 14400, 28561,
31213, 37180, 194481, 336141

19, 20, 39, 57, 76, 77,
96, 133, 153, 171, 190, 209,
210, 286, 324, 343, 361, 400,
456, 476, 495, 513, 969, 1216,
1331, 1445, 1521, 1540, 1729, 2376,
2432, 2926, 3136, 3250, 4200, 5776,
5929, 5985, 6175, 6860, 10241, 10830,
12636, 13377, 14080, 14365, 23409, 27456,
28900, 43681, 89376, 104976, 165376, 228096,
601426, 633556, 709632, 5909761, 11859211

23, 24, 46, 69, 70, 92,
115, 161, 162, 208, 231, 253,
276, 300, 323, 391, 392, 460,
484, 507, 529, 576, 736, 760,
875, 897, 1105, 1197, 1288, 1496,
1863, 2024, 2025, 2185, 2300, 2646,
2737, 3060, 3381, 3520, 3888, 4693,
4761, 5083, 7866, 8281, 8625, 10626,
11271, 11662, 12168, 16929, 19551, 21505,
21736, 23276, 25025, 25921, 43264, 52326,
71875, 75141, 76545, 104329, 122452, 126225,
152881, 202125, 264385, 282625, 328510, 2023425,
4096576, 5142501

29, 30, 58, 88, 116, 117,
145, 175, 204, 232, 261, 290,
320, 378, 406, 494, 551, 552,
609, 638, 726, 783, 784, 841,
1015, 1045, 1276, 1450, 1596, 1625,
1683, 2001, 2002, 2176, 2205, 2262,
2465, 2640, 2755, 2784, 3249, 3451,
3510, 4641, 4785, 4901, 5104, 5888,
5916, 6670, 7106, 7425, 7889, 8671,
9802, 10557, 11340, 12006, 12673, 13225,
13311, 13312, 13456, 19228, 20736, 23751,
24795, 25840, 27000, 30625, 30856, 35322,
47125, 53361, 72501, 83521, 87465, 136851,
158950, 166635, 168751, 176001, 176176, 184093,
240787, 244036, 303601, 410670, 418761, 613089,
949026, 1163800, 1235169, 1243840, 1625625, 1852201,
2697696, 4004001, 4090625, 8268800, 10556001, 18085705,
96059601, 177182721

31, 32, 63, 93, 125, 155,
156, 187, 217, 248, 280, 341,
342, 435, 465, 496, 528, 589,
621, 651, 714, 806, 837, 868,
900, 931, 961, 1024, 1054, 1210,
1365, 1426, 1519, 1520, 1768, 1860,
2016, 2233, 2945, 2976, 3565, 3751,
3876, 3969, 4186, 4960, 4992, 5643,
5797, 6076, 6138, 6293, 6325, 6480,
6728, 7657, 7905, 7936, 8092, 8464,
8526, 8960, 9425, 10881, 11781, 11935,
12122, 13300, 13455, 15625, 17577, 19251,
19344, 19965, 21142, 22816, 23375, 23716,
24025, 27405, 29792, 31465, 31900, 32799,
41262, 42688, 49011, 58311, 78337, 96876,
98736, 102487, 108376, 111321, 111476, 116281,
116964, 122265, 174097, 175770, 178126, 190464,
207576, 212382, 227448, 240065, 245025, 260338,
268801, 278784, 288145, 314433, 453376, 459173,
509796, 773605, 863940, 912951, 1147125, 1154440,
1255501, 1594176, 2307361, 2310400, 2345057, 3206269,
3301376, 3346110, 3897166, 14753025, 16093000, 76271625
80061345, 133920000, 181037025, 370256250, 1611308700

> 3: [4/3, 9/8]
> 5: [16/15, 25/24, 81/80]
> 7: [126/125, 225/224, 2401/2400, 4375/4374]

I think the answer for p = 11 is 176/175.

I looked at the 12 highest (whoops, I mean shortest!) values:

1. [-3 -1 -1 0 2] = 121 / 120
2. [ 1 2 -3 1 0] = 126 / 125
3. [ 4 0 -2 -1 1] = 176 / 175
4. [-5 2 2 -1 0] = 225 / 224
5. [-1 5 0 0 -2] = 243 / 242
6. [-7 -1 1 1 1] = 385 / 384
7. [-3 2 -1 2 -1] = 441 / 440
8. [ 2 3 1 -2 -1] = 540 / 539
9. [-5 -1 -2 4 0] = 2401 / 2400
10. [-4 -3 2 -1 2] = 3025 / 3024
11. [-1 -7 4 1 0] = 4375 / 4374
12. [-3 4 -2 -2 2] = 9801 / 9800

There are 664 unimodular combinations in total. From the set 3...12
there are 69, but you can't find any at all if you try the set 4...12.

This log extract shows the "Top 20":

07:23:25 645. Rows = 3, 6, 8, 10, 11 < ok > det = -1
07:23:25 646. Rows = 3, 6, 8, 10, 12 < ok > det = 1
07:23:25 647. Rows = 3, 6, 8, 11, 12 < ok > det = 1
07:23:25 648. Rows = 3, 6, 9, 10, 11 < ok > det = 1
07:23:25 649. Rows = 3, 6, 9, 10, 12 < ok > det = -1
07:23:25 650. Rows = 3, 6, 9, 11, 12 < ok > det = -1
07:23:25 655. Rows = 3, 7, 8, 10, 11 < ok > det = -1
07:23:25 656. Rows = 3, 7, 8, 10, 12 < ok > det = 1
07:23:25 657. Rows = 3, 7, 8, 11, 12 < ok > det = 1
07:23:25 658. Rows = 3, 7, 9, 10, 11 < ok > det = 1
07:23:25 659. Rows = 3, 7, 9, 10, 12 < ok > det = -1
07:23:25 660. Rows = 3, 7, 9, 11, 12 < ok > det = -1
07:23:25 662. Rows = 3, 8, 9, 10, 11 < ok > det = 1
07:23:25 663. Rows = 3, 8, 9, 10, 12 < ok > det = -1
07:23:25 664. Rows = 3, 8, 9, 11, 12 < ok > det = -1

> How can we find this minimax comma expeditiously?

My research includes both the search for the ratios and the
identification of an optimal basis. Optimal for me means that it
must
contain your "minimax" value.

I have only developed a very rough hack so far, the identification of
the ratios in the first place has been the main order of the day. So
your request for specific answers is a useful prod.

The current code is quite rudimentary, it does not yet take advantage
of failures (singularity) to identify their root causes (dependent
subsets) and thus trim the search tree.

As you can see above, we can't combine all of rows 10, 11 and 12, but
we can use any two of them. On inspection this is bleedin' obvious
because:

4375 9801 3025
---- X ---- = ----
4374 9800 3024

So we should just park one of those off to the side and move on (I
think), which means we can add a new one without increasing the size
of the tree.

Anyhow, I will provide additional minimax values for you, and
hopefully advance the method along somewhat as a result.

These will form a sequence that I would think worth an entry over at
Sloane's Integer Sequence Database - in fact it's a lot more
interesting than a large swag of those there already!

> How may we enumerate all the sets of superparticulars with this
> minimax comma which form a set of generators?

Ah well, that's a nice idea, and one that has crossed my mind, but I
inevitably
get derailed by the fact that any basis at all, regardless of the
nice
attributes (eg height) of its components, is still just another
generator for the entire set of smooth ratios.

And we still need to find the minimax value in the first place - or
are
you thinking about using one result, for N primes say, in order that
we might more quickly complete the enumeration problem for N+1 primes?

Cheers

Jim White
ANU, Canberra

PS: Regarding the "incremental enumeration" problem, if you are
familiar with Lehmer's Method, then you might also know that it is
not
guaranteed to work as an incremental enumerator - eg, I have all
results for p=127, now I want to fill in the next "section", p=131.

If I restrict the the number of equations to be solved (ie the number
of D values) by only using multiples of 131, I will miss some values,
because Lehmer's scheme produces them only among the multiple
solutions
for D values with greatest prime divisor <= 127.

So we effectively have to do a complete run every time.

One of the results I have in this area is a refinement of the
Lehmer/Stoermer method that targets just the subset with the
specified
prime divisor. It's most useful, and time saving ...

This, along with two other key results, are now being targeted for
proof-of-correctness ....

Another result, I've given the fattest (hopefully unambiguous!) vector
only ...

P = 13, N = 6
1. [ 5 -3 0 0 1 -1] = 352 / 351
2. [ 2 -1 0 1 -2 1] = 364 / 363
3. [-4 -1 4 0 0 -1] = 625 / 624
4. [ 2 -3 -2 0 0 2] = 676 / 675
5. [-3 6 0 -1 0 -1] = 729 / 728
6. [-3 0 -3 1 1 1] = 1001 / 1000
7. [ 2 1 -1 -3 1 1] = 1716 / 1715
8. [ 5 -3 1 -1 -1 1] = 2080 / 2079
9. [12 -2 -1 -1 0 -1] = 4096 / 4095
10. [-7 -1 2 0 -1 2] = 4225 / 4224
11. [ 9 0 -1 0 -3 1] = 6656 / 6655
12. [ 3 -2 0 -1 3 -2] = 10648 / 10647

12:27:49 1667. Rows = 5, 6, 10, 11, 12, 13 < ok > det = 1

If I remember rightly

Sorry, it fell off the end of the table ...

13. [-6 6 -2 -1 -1 2] = 123201 / 123200

The minimax(p) sequence below was thought to be accurate right up
until I pasted it here - but now I see that this is no longer certain.

When I select my test batch from the shortest ratios available, I
have been inadvertently selecting them only from the pool of ratios
using the p-limit in question, rather than select from the whole pool.

But I post it anyway, because it begs some interesting questions

Might this affect the minimax value itself? Why?

I do know already that I am unlikely to find the optimal basis - I'd
previously done this for p=41, and the optimal result has 4 out of 13
members that are from the 31- or 37- pools.

That basis, by the way suggests this result for minimax(41):

[2 -2 0 -3 0 0 -2 3 1 -1 0 0 1] = 25872148 / 25872147

And here is today's version of the minimax sequence.

My gut feeling is that they won't change ... but that could just be
indigestion .. we shall see tomorrow.

Cheers

================================================================
Minimax(p) Sequence

3. [ 2 -1] = 4 / 3
5. [ 4 -1 -1] = 16 / 15
7. [ 1 2 -3 1] = 126 / 125
11. [ 4 0 -2 -1 1] = 176 / 175
13. [ -3 6 0 -1 0 -1] = 729 / 728
17. [ -3 2 -2 0 0 -1 2] = 2601 / 2600
19. [ 6 1 -1 0 1 1 -2 -1] = 27456 / 27455
23. [ 8 -2 0 0 -1 2 0 -1 -1] = 43264 / 43263
29. [ -3 2 -1 1 0 0 2 -2 1 -1] = 418761 / 418760
31. [ -2 0 0 -1 1 3 0 1 -2 0 -1] = 459173 / 459172

A sample basis for the case p = 31 is given below.

1. [ -2 0 0 -1 1 3 0 1 -2 0 -1] = 459173 / 459172
10. [ -5 -1 -1 4 -1 0 0 -1 -1 0 2] = 2307361 / 2307360
14. [ 13 0 -3 -4 -1 1 0 0 0 0 1] = 3301376 / 3301375
15. [ 1 9 1 0 0 -1 1 -2 -1 0 -1] = 3346110 / 3346109
16. [ 1 -1 -1 3 0 1 -2 1 1 -1 -1] = 3897166 / 3897165
18. [ 3 -4 3 1 2 -1 -1 1 0 -1 -1] = 16093000 / 16092999
20. [ -5 3 1 4 -2 1 0 1 -1 -1 -1] = 80061345 / 80061344
21. [ 8 3 4 0 0 0 -2 -1 0 -3 1] = 133920000 / 133919999
22. [ -5 4 2 -1 0 2 0 0 2 -2 -2] = 181037025 / 181037024
23. [ 1 1 5 2 -7 1 0 -1 0 0 1] = 370256250 / 370256249
24. [ 2 6 2 -4 -1 -2 0 -2 1 0 2] = 1611308700 / 1611308699
================================================================

--- In tuning-math@yahoogroups.com, "mathimagics" <mathimagics@...>
wrote:
> A sample basis for the case p = 31 is given below.
>
> 1. [ -2 0 0 -1 1 3 0 1 -2 0 -1] = 459173 /
459172
> 10. [ -5 -1 -1 4 -1 0 0 -1 -1 0 2] = 2307361 /
2307360
> 14. [ 13 0 -3 -4 -1 1 0 0 0 0 1] = 3301376 /
3301375
> 15. [ 1 9 1 0 0 -1 1 -2 -1 0 -1] = 3346110 /
3346109
> 16. [ 1 -1 -1 3 0 1 -2 1 1 -1 -1] = 3897166 /
3897165
> 18. [ 3 -4 3 1 2 -1 -1 1 0 -1 -1] = 16093000 /
16092999
> 20. [ -5 3 1 4 -2 1 0 1 -1 -1 -1] = 80061345 /
80061344
> 21. [ 8 3 4 0 0 0 -2 -1 0 -3 1] = 133920000 /
133919999
> 22. [ -5 4 2 -1 0 2 0 0 2 -2 -2] = 181037025 /
181037024
> 23. [ 1 1 5 2 -7 1 0 -1 0 0 1] = 370256250 /
370256249
> 24. [ 2 6 2 -4 -1 -2 0 -2 1 0 2] = 1611308700 /
1611308699
> ================================================================
>

If you take the matrix this defines and invert it, the colums are
now "vals" or mappings to primes for various equal temperaments:

[<148418, 235237, 344616, 416662, 513442, 549212, 606653, 630469,
671378, 721012, 735292|, <396564, 628539, 920793, 1113296, 1371886,
1467461,
1620941, 1684575, 1793882, 1926500, 1964656|,
<301040, 477137, 698993, 845126, 1041427, 1113980, 1230490, 1278796,
1361773,
1462446, 1491411|,
<109590, 173696, 254460, 307658, 379119, 405531, 447945, 465530,
495737, 532386, 542930|,
<155945, 247167, 362093, 437793, 539481, 577065, 637419, 662443,
705427,
757578, 772582|,
<562132, 890958, 1305230, 1578104, 1944657, 2080135, 2297694,
2387896, 2542839, 2730826, 2784912|,
<59737, 94681, 138705, 167703, 206656,
221053, 244173, 253759, 270224, 290201, 295949|,
<-382216, -605798, -887478, -1073016, -1322250, -1414367, -1562294, -
1623626,
-1728978, -1856798, -1893573|,
<184417, 292294, 428203, 517724, 637978, 682424, 753798, 783390,
834222,
895894, 913638|,
<16808, 26640, 39027, 47186, 58146, 62197, 68702, 71399,
76032, 81653, 83270|, <337948, 535635, 784691, 948740, 1169108,
1250556,
1381350, 1435579, 1528729, 1641745, 1674261|]

These give the exponents to use on the commas to express the 31-
limit. The ets 16808 and 148418 are particularly notable and have
been discussed before.

One question I have is how you are defining "optimal". These vals, as
well as the orginal commas, could be used for such definitions.

> And here is today's version of the minimax sequence.
>
> My gut feeling is that they won't change

Ah, how wrong was my conjecture!

Minimax(p) Sequence

3. [ 2 -1] = 4 / 3
5. [ 4 -1 -1] = 16 / 15
7. [ 1 2 -3 1] = 126 / 125
11. [ 4 0 -2 -1 1] = 176 / 175
13. [ -3 6 0 -1 0 -1] = 729 / 728
17. [ -3 2 -2 0 0 -1 2] = 2601 / 2600
19. [ 6 1 -1 0 1 1 -2 -1] = 27456 / 27455
23. [ 8 -2 0 0 -1 2 0 -1 -1] = 43264 / 43263
29. [ -3 2 -1 1 0 0 2 -2 1 -1] = 418761 / 418760
31. [ -2 0 0 -1 1 3 0 1 -2 0 -1] = 459173 / 459172

Revised:
17. [ -7 -1 2 0 -1 2 0] = 4225 / 4224
19. [ 2 -2 2 0 0 -2 2 -1] = 28900 / 28899
23. [ -1 -3 5 0 -3 0 0 0 1] = 71875 / 71874
29. [ -5 6 0 -2 0 0 -1 0 -1 2] = 613089 / 613088
31. [ 2 2 -1 2 -1 -1 2 0 -1 0 -1] = 509796 / 509795

Case 29 is interesting - the sequence is not strictly increasing,
although with these numbers that is par for the course (the sequence
of maximum superparticular numerators for p behaves similarly, as we
know).

Also notice that case p=17 is one where the minimax value is itself
is in fact 13-smooth.

Cheers
Jim White
ANU

--- In tuning-math@yahoogroups.com, "mathimagics" <mathimagics@...>
wrote:
>
> > And here is today's version of the minimax sequence.
> >
> > My gut feeling is that they won't change
>
> Ah, how wrong was my conjecture!

Are you going to send it off to OEIS?

--- "Gene Ward Smith" <genewardsmith@...> wrote:

> Are you going to send it off to OEIS?

Gene

I sent you an email using the gmail id given at OEIS - do you still
monitor that mailbox?

Cheers
Jim