5
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For a fixed positive integer $n$, the Diophantine equation

$$x^2 + y^2 + z^2 = n$$

was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the quaternion algebra $B_{-1,-1}$, characterized by the relations

$$ i^2 = -1, j^2 = -1, k^2 = (ij)^2 = -1. $$

More precisely, it is equivalent to the norm equation

$$ \operatorname{nrd}(xi + yj + zk) = n, $$

where $\operatorname{nrd}$ denotes the reduced norm of the (integral) pure quaternion $xi + yj + zk$. For me, the most interesting fact here is that for $n \not \equiv 0 \pmod 4$ the number of solutions to this equation is some multiple the Hurwitz class number $H(n)$. In the special case $n \equiv 7 \pmod 8$, it is a multiple of zero.

Now, I wonder if the same phenomenon was observed for other Diophantine equations of this kind. For example, in the quaternion algebra $B_{-1, -3}$ defined by the relations

$$ i^2 = -1, j^2 = -3, k^2 = (ij)^2 = -3, $$

we can consider the norm equation

$$ \operatorname{nrd}\left(ix + \frac{1+j}{2}y + \frac{i + k}{2}z\right) = n, $$

which is equivalent to the Diophantine equation

$$ x^2 + y^2 + z^2 + xz = n. $$

Is there a formula for the number of its solutions? (see the OEIS sequence A014453) Is it true that, for some $n$, it is equal to the multiple of the (Hurwitz?) class number of some number field, say $\mathbb Q(\sqrt{-n})$? Perhaps, a Diophantine equation

$$ x^2 + 3y^2 + 3z^2 = n $$

was studied in detail? I'd be thankful for any references.

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7
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One of the beautiful (and sometimes flummoxing) aspects of the theory of ternary quadratic forms is that you can find answers and questions coming from many different points of view! Using modular forms will give you an answer along the lines that Henri Cohen suggests. I'd like to put in a pitch for a quaternionic approach, which has other features.

A reference is my book http://quatalg.org. The theorem of Gauss is Theorem 30.1.3. You can mimic the proof of this theorem for your ternary quadratic forms, to express the number of primitive representations in terms of (Hurwitz) class numbers, because the associated quaternion orders have type number one.

Here are a few more details--if you'd like a more complete write up, just ask. The main input is Theorem 30.4.7, which sums the number of optimal embeddings = number of primitive representations over the right class set of an order, expressing this in terms of the class number and a local correction factor. The two quadratic forms you list arise as the reduced norm on the trace zero submodule of orders $\mathcal{O}$ in the quaternion algebra $B=(-1,-3\,|\mathbb{Q})$ of discriminant $3$. The first form $x^2+y^2+z^2+xz$ arises from $\mathcal{O}$ a maximal order one: this order is in fact Euclidean under the reduced norm, and so has class number $1$, which means the average is just over the one term. The second form $x^2+3y^2+3z^2$ arises from an order of reduced discriminant $36$ and index $12$; I compute using Magma that this order has class number $2$ but type number $1$, which means that the sum is over two terms that are equal, so we just need to divide the right-hand side by $2$.

So to finish, we need to compute the local embedding numbers, which provides a correction term depending on $n$ modulo a power of $2$ and $3$. This gets a bit technical, so I would just stare at the numerical answer provided by Will Jagy. For the first form, the order is maximal so the local embedding numbers are in 30.5 (just a correction at $3$); unfortunately for the second form requires a separate calculation (not covered by 30.5 or 30.6).

Happy to say more if you'd like!

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  • $\begingroup$ Thanks for the response, Prof. Voight! I learned the basics of quaternion algebras from your book, as well as from the lecture notes of M.F. Vignéras. Will take a look at the theorems you referenced. P.S. Thank you very much for your fundamental monograph! $\endgroup$ – Anton Jul 4 '18 at 1:29
  • $\begingroup$ You're most welcome, and please call me John! Let me know if you have any comments or questions (on anything in the book!). Of course, I have a slight quaternionic bias, but you can also do all of this (equivalently) in the language of ternary quadratic forms: Siegel has given an expression for the average number of representations over the genus, and the fact that there is only one class in the genus for each of your ternary quadratic forms means you have a formula. There is again a "local mass" (equivalent to the local embedding numbers). $\endgroup$ – John Voight Jul 5 '18 at 17:50
  • $\begingroup$ For the modular forms proof, the fact of class number one translates into no cusp forms, and then the local calculation is the determination the precise (linear combination of) Eisenstein series. $\endgroup$ – John Voight Jul 5 '18 at 17:52
10
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The theta function associated to your quadratic form (here $\theta(\tau)\theta(3\tau)^2$) belongs to the modular form space $M_{3/2}(\Gamma_0(12))$, and you are in luck because there are no cusp forms, so the space is entirely generated by three Eisenstein series whose coefficients are similar to Hurwitz class numbers and easily computable, so there is an explicit formula analogous to Gauss in your case (which I did not take time to work out). This works as long as there are no cusp forms. However, for instance $S_{3/2}(\Gamma_0(28))$ is nonzero, so you could not do the same with $x^2+7y^2+7z^2$.

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2
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Here are some computations about primitive representations by two forms, $g_0$ is the underlying $x^2 + y^2 + 3 z^2 + xy,$ while $g_1$ is your $x^2 + 3 y^2 + 3 z^2.$ I was a little surprised to find a distinction requiring mod 72, mod 36 was not enough. Compare $7 \pmod {72}$ to $43 \pmod {72}.$ Alright, it amounts to $\pmod {24}.$ When I have time I will compute the cases divisible by 2 or 3 or both.

As is the case (analogous) for the sum of three squares, for the number $1$ we do not get the same ratios as for other numbers $1 \pmod {72},$ those being together in lines with the output N 1 in the column for %72

The class numbers are for positive binary quadratic forms. See Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares? and Gerry's reference in comment, as well as request for clarification from John Voight (which eventually worked out).

Made a little summary: the ratios $(r_0/h(-4M), r_1/h(-4M))$ are split three ways when $\gcd(M,6) = 1.$ We get ratios $(12,4)$ when $M \equiv 1, 13,25,37,49,61 \pmod {72},$ which is all $M \equiv 1 \pmod {12}.$ We get ratios $(24,16)$ when $M \equiv 7,31,55 \pmod {72},$ which is all $M \equiv 7 \pmod {24}.$We get ratios $(16,8)$ when $M \equiv 19,43,67 \pmod {72},$ which is all $M \equiv 19 \pmod {24}.$

---------------------------------------
 g0  :          9 :     1    1    3    0    0    1 auto 24
 g1  :         36 :     1    3    3    0    0    0 auto 16
---------------------------------------
---------------------------------------

      M     r0   r1  h(4M)  %72  r0/h r1/h
      1      6    2      1  N 1    6    2           1 =  1 
      3      8    4      1  N 3    8    4           3 = 3
      4     12    8      1  N 4   12    8           4 = 2^2
      6     12    4      2  N 6    6    2           6 = 2 * 3
      7     24   16      1  N 7   24   16           7 = 7
     10     24    8      2  N10   12    4          10 = 2 * 5
     12     12    8      2  N12    6    4          12 = 2^2 * 3
     13     24    8      2  N13   12    4          13 = 13
     15     24   16      2  N15   12    8          15 = 3 * 5
     16     24   16      2  N16   12    8          16 = 2^4
     19     48   24      3  N19   16    8          19 = 19
     21     24    8      4  N21    6    2          21 = 3 * 7
     22     24    8      2  N22   12    4          22 = 2 * 11
-------------------------------------------------------------------------
      1      6    2      1  N 1    6    2           1 =  1 
     73     48   16      4  N 1   12    4          73 = 73
    145     96   32      8  N 1   12    4         145 = 5 * 29
    217     96   32      8  N 1   12    4         217 = 7 * 31
    289     96   32      8  N 1   12    4         289 = 17^2
    361    120   40     10  N 1   12    4         361 = 19^2
    433    144   48     12  N 1   12    4         433 = 433
    505     96   32      8  N 1   12    4         505 = 5 * 101
    577     96   32      8  N 1   12    4         577 = 577
    649    240   80     20  N 1   12    4         649 = 11 * 59
    721    192   64     16  N 1   12    4         721 = 7 * 103
    793     96   32      8  N 1   12    4         793 = 13 * 61
    865    192   64     16  N 1   12    4         865 = 5 * 173
    937    240   80     20  N 1   12    4         937 = 937
   1009    240   80     20  N 1   12    4        1009 = 1009
   1081    192   64     16  N 1   12    4        1081 = 23 * 47
   1153    192   64     16  N 1   12    4        1153 = 1153
   1225    192   64     16  N 1   12    4        1225 = 5^2 * 7^2
   1297    144   48     12  N 1   12    4        1297 = 1297
   1369    216   72     18  N 1   12    4        1369 = 37^2
------------------------------------------------------------------------
      7     24   16      1  N 7   24   16           7 = 7
     79    120   80      5  N 7   24   16          79 = 79
    151    168  112      7  N 7   24   16         151 = 151
    223    168  112      7  N 7   24   16         223 = 223
    295    192  128      8  N 7   24   16         295 = 5 * 59
    367    216  144      9  N 7   24   16         367 = 367
    439    360  240     15  N 7   24   16         439 = 439
    511    336  224     14  N 7   24   16         511 = 7 * 73
    583    192  128      8  N 7   24   16         583 = 11 * 53
    655    288  192     12  N 7   24   16         655 = 5 * 131
    727    312  208     13  N 7   24   16         727 = 727
    799    384  256     16  N 7   24   16         799 = 17 * 47
    871    528  352     22  N 7   24   16         871 = 13 * 67
    943    384  256     16  N 7   24   16         943 = 23 * 41
   1015    384  256     16  N 7   24   16        1015 = 5 * 7 * 29
   1087    216  144      9  N 7   24   16        1087 = 1087
   1159    384  256     16  N 7   24   16        1159 = 19 * 61
   1231    648  432     27  N 7   24   16        1231 = 1231
   1303    264  176     11  N 7   24   16        1303 = 1303
   1375    480  320     20  N 7   24   16        1375 = 5^3 * 11
------------------------------------------------------------------
     13     24    8      2  N13   12    4          13 = 13
     85     48   16      4  N13   12    4          85 = 5 * 17
    157     72   24      6  N13   12    4         157 = 157
    229    120   40     10  N13   12    4         229 = 229
    301     96   32      8  N13   12    4         301 = 7 * 43
    373    120   40     10  N13   12    4         373 = 373
    445     96   32      8  N13   12    4         445 = 5 * 89
    517    144   48     12  N13   12    4         517 = 11 * 47
    589    192   64     16  N13   12    4         589 = 19 * 31
    661    216   72     18  N13   12    4         661 = 661
    733    168   56     14  N13   12    4         733 = 733
    805    192   64     16  N13   12    4         805 = 5 * 7 * 23
    877    120   40     10  N13   12    4         877 = 877
    949    144   48     12  N13   12    4         949 = 13 * 73
   1021    264   88     22  N13   12    4        1021 = 1021
   1093    120   40     10  N13   12    4        1093 = 1093
   1165    240   80     20  N13   12    4        1165 = 5 * 233
   1237    168   56     14  N13   12    4        1237 = 1237
   1309    288   96     24  N13   12    4        1309 = 7 * 11 * 17
   1381    312  104     26  N13   12    4        1381 = 1381
------------------------------------------------------------------
     19     48   24      3  N19   16    8          19 = 19
     91     96   48      6  N19   16    8          91 = 7 * 13
    163     48   24      3  N19   16    8         163 = 163
    235     96   48      6  N19   16    8         235 = 5 * 47
    307    144   72      9  N19   16    8         307 = 307
    379    144   72      9  N19   16    8         379 = 379
    451    288  144     18  N19   16    8         451 = 11 * 41
    523    240  120     15  N19   16    8         523 = 523
    595    192   96     12  N19   16    8         595 = 5 * 7 * 17
    667    192   96     12  N19   16    8         667 = 23 * 29
    739    240  120     15  N19   16    8         739 = 739
    811    336  168     21  N19   16    8         811 = 811
    883    144   72      9  N19   16    8         883 = 883
    955    192   96     12  N19   16    8         955 = 5 * 191
   1027    192   96     12  N19   16    8        1027 = 13 * 79
   1099    288  144     18  N19   16    8        1099 = 7 * 157
   1171    336  168     21  N19   16    8        1171 = 1171
   1243    192   96     12  N19   16    8        1243 = 11 * 113
   1315    288  144     18  N19   16    8        1315 = 5 * 263
   1387    192   96     12  N19   16    8        1387 = 19 * 73
------------------------------------------------------------------
     25     24    8      2  N25   12    4          25 = 5^2
     97     48   16      4  N25   12    4          97 = 97
    169     72   24      6  N25   12    4         169 = 13^2
    241    144   48     12  N25   12    4         241 = 241
    313     96   32      8  N25   12    4         313 = 313
    385     96   32      8  N25   12    4         385 = 5 * 7 * 11
    457     96   32      8  N25   12    4         457 = 457
    529    144   48     12  N25   12    4         529 = 23^2
    601    240   80     20  N25   12    4         601 = 601
    673    144   48     12  N25   12    4         673 = 673
    745    192   64     16  N25   12    4         745 = 5 * 149
    817    144   48     12  N25   12    4         817 = 19 * 43
    889    192   64     16  N25   12    4         889 = 7 * 127
    961    192   64     16  N25   12    4         961 = 31^2
   1033    144   48     12  N25   12    4        1033 = 1033
   1105    192   64     16  N25   12    4        1105 = 5 * 13 * 17
   1177    144   48     12  N25   12    4        1177 = 11 * 107
   1249    384  128     32  N25   12    4        1249 = 1249
   1321    288   96     24  N25   12    4        1321 = 1321
   1393    192   64     16  N25   12    4        1393 = 7 * 199
-------------------------------------------------------------------
     31     72   48      3  N31   24   16          31 = 31
    103    120   80      5  N31   24   16         103 = 103
    175    144   96      6  N31   24   16         175 = 5^2 * 7
    247    144   96      6  N31   24   16         247 = 13 * 19
    319    240  160     10  N31   24   16         319 = 11 * 29
    391    336  224     14  N31   24   16         391 = 17 * 23
    463    168  112      7  N31   24   16         463 = 463
    535    336  224     14  N31   24   16         535 = 5 * 107
    607    312  208     13  N31   24   16         607 = 607
    679    432  288     18  N31   24   16         679 = 7 * 97
    751    360  240     15  N31   24   16         751 = 751
    823    216  144      9  N31   24   16         823 = 823
    895    384  256     16  N31   24   16         895 = 5 * 179
    967    264  176     11  N31   24   16         967 = 967
   1039    552  368     23  N31   24   16        1039 = 1039
   1111    528  352     22  N31   24   16        1111 = 11 * 101
   1183    336  224     14  N31   24   16        1183 = 7 * 13^2
   1255    288  192     12  N31   24   16        1255 = 5 * 251
   1327    360  240     15  N31   24   16        1327 = 1327
   1399    648  432     27  N31   24   16        1399 = 1399
---------------------------------------------------------------
     37     24    8      2  N37   12    4          37 = 37
    109     72   24      6  N37   12    4         109 = 109
    181    120   40     10  N37   12    4         181 = 181
    253     48   16      4  N37   12    4         253 = 11 * 23
    325    144   48     12  N37   12    4         325 = 5^2 * 13
    397     72   24      6  N37   12    4         397 = 397
    469    192   64     16  N37   12    4         469 = 7 * 67
    541    120   40     10  N37   12    4         541 = 541
    613    120   40     10  N37   12    4         613 = 613
    685    144   48     12  N37   12    4         685 = 5 * 137
    757    120   40     10  N37   12    4         757 = 757
    829    264   88     22  N37   12    4         829 = 829
    901    288   96     24  N37   12    4         901 = 17 * 53
    973    144   48     12  N37   12    4         973 = 7 * 139
   1045    192   64     16  N37   12    4        1045 = 5 * 11 * 19
   1117    168   56     14  N37   12    4        1117 = 1117
   1189    240   80     20  N37   12    4        1189 = 29 * 41
   1261    240   80     20  N37   12    4        1261 = 13 * 97
   1333    240   80     20  N37   12    4        1333 = 31 * 43
   1405    288   96     24  N37   12    4        1405 = 5 * 281
------------------------------------------------------------------
     43     48   24      3  N43   16    8          43 = 43
    115     96   48      6  N43   16    8         115 = 5 * 23
    187     96   48      6  N43   16    8         187 = 11 * 17
    259    192   96     12  N43   16    8         259 = 7 * 37
    331    144   72      9  N43   16    8         331 = 331
    403     96   48      6  N43   16    8         403 = 13 * 31
    475    192   96     12  N43   16    8         475 = 5^2 * 19
    547    144   72      9  N43   16    8         547 = 547
    619    240  120     15  N43   16    8         619 = 619
    691    240  120     15  N43   16    8         691 = 691
    763    192   96     12  N43   16    8         763 = 7 * 109
    835    288  144     18  N43   16    8         835 = 5 * 167
    907    144   72      9  N43   16    8         907 = 907
    979    384  192     24  N43   16    8         979 = 11 * 89
   1051    240  120     15  N43   16    8        1051 = 1051
   1123    240  120     15  N43   16    8        1123 = 1123
   1195    384  192     24  N43   16    8        1195 = 5 * 239
   1267    288  144     18  N43   16    8        1267 = 7 * 181
   1339    384  192     24  N43   16    8        1339 = 13 * 103
   1411    192   96     12  N43   16    8        1411 = 17 * 83
-------------------------------------------------------------------
     49     48   16      4  N49   12    4          49 = 7^2
    121     72   24      6  N49   12    4         121 = 11^2
    193     48   16      4  N49   12    4         193 = 193
    265     96   32      8  N49   12    4         265 = 5 * 53
    337     96   32      8  N49   12    4         337 = 337
    409    192   64     16  N49   12    4         409 = 409
    481    192   64     16  N49   12    4         481 = 13 * 37
    553     96   32      8  N49   12    4         553 = 7 * 79
    625    120   40     10  N49   12    4         625 = 5^4
    697     96   32      8  N49   12    4         697 = 17 * 41
    769    240   80     20  N49   12    4         769 = 769
    841    168   56     14  N49   12    4         841 = 29^2
    913    144   48     12  N49   12    4         913 = 11 * 83
    985    288   96     24  N49   12    4         985 = 5 * 197
   1057    192   64     16  N49   12    4        1057 = 7 * 151
   1129    192   64     16  N49   12    4        1129 = 1129
   1201    192   64     16  N49   12    4        1201 = 1201
   1273    240   80     20  N49   12    4        1273 = 19 * 67
   1345    192   64     16  N49   12    4        1345 = 5 * 269
   1417    192   64     16  N49   12    4        1417 = 13 * 109
------------------------------------------------------------------
     55     96   64      4  N55   24   16          55 = 5 * 11
    127    120   80      5  N55   24   16         127 = 127
    199    216  144      9  N55   24   16         199 = 199
    271    264  176     11  N55   24   16         271 = 271
    343    168  112      7  N55   24   16         343 = 7^3
    415    240  160     10  N55   24   16         415 = 5 * 83
    487    168  112      7  N55   24   16         487 = 487
    559    384  256     16  N55   24   16         559 = 13 * 43
    631    312  208     13  N55   24   16         631 = 631
    703    336  224     14  N55   24   16         703 = 19 * 37
    775    288  192     12  N55   24   16         775 = 5^2 * 31
    847    240  160     10  N55   24   16         847 = 7 * 11^2
    919    456  304     19  N55   24   16         919 = 919
    991    408  272     17  N55   24   16         991 = 991
   1063    456  304     19  N55   24   16        1063 = 1063
   1135    432  288     18  N55   24   16        1135 = 5 * 227
   1207    432  288     18  N55   24   16        1207 = 17 * 71
   1279    552  368     23  N55   24   16        1279 = 1279
   1351    576  384     24  N55   24   16        1351 = 7 * 193
   1423    216  144      9  N55   24   16        1423 = 1423
--------------------------------------------------------------
     61     72   24      6  N61   12    4          61 = 61
    133     48   16      4  N61   12    4         133 = 7 * 19
    205     96   32      8  N61   12    4         205 = 5 * 41
    277     72   24      6  N61   12    4         277 = 277
    349    168   56     14  N61   12    4         349 = 349
    421    120   40     10  N61   12    4         421 = 421
    493    144   48     12  N61   12    4         493 = 17 * 29
    565    144   48     12  N61   12    4         565 = 5 * 113
    637    144   48     12  N61   12    4         637 = 7^2 * 13
    709    120   40     10  N61   12    4         709 = 709
    781    240   80     20  N61   12    4         781 = 11 * 71
    853    120   40     10  N61   12    4         853 = 853
    925    144   48     12  N61   12    4         925 = 5^2 * 37
    997    168   56     14  N61   12    4         997 = 997
   1069    360  120     30  N61   12    4        1069 = 1069
   1141    288   96     24  N61   12    4        1141 = 7 * 163
   1213    120   40     10  N61   12    4        1213 = 1213
   1285    144   48     12  N61   12    4        1285 = 5 * 257
   1357    192   64     16  N61   12    4        1357 = 23 * 59
   1429    264   88     22  N61   12    4        1429 = 1429
----------------------------------------------------------------
     67     48   24      3  N67   16    8          67 = 67
    139    144   72      9  N67   16    8         139 = 139
    211    144   72      9  N67   16    8         211 = 211
    283    144   72      9  N67   16    8         283 = 283
    355    192   96     12  N67   16    8         355 = 5 * 71
    427     96   48      6  N67   16    8         427 = 7 * 61
    499    144   72      9  N67   16    8         499 = 499
    571    240  120     15  N67   16    8         571 = 571
    643    144   72      9  N67   16    8         643 = 643
    715    192   96     12  N67   16    8         715 = 5 * 11 * 13
    787    240  120     15  N67   16    8         787 = 787
    859    336  168     21  N67   16    8         859 = 859
    931    288  144     18  N67   16    8         931 = 7^2 * 19
   1003    192   96     12  N67   16    8        1003 = 17 * 59
   1075    288  144     18  N67   16    8        1075 = 5^2 * 43
   1147    288  144     18  N67   16    8        1147 = 31 * 37
   1219    288  144     18  N67   16    8        1219 = 23 * 53
   1291    432  216     27  N67   16    8        1291 = 1291
   1363    288  144     18  N67   16    8        1363 = 29 * 47
   1435    192   96     12  N67   16    8        1435 = 5 * 7 * 41
-----------------------------------------------------------------------
      M     r0   r1  h(4M)  %72  r0/h r1/h
---------------------------------------
 g0  :          9 :     1    1    3    0    0    1 auto 24
 g1  :         36 :     1    3    3    0    0    0 auto 16
---------------------------------------
---------------------------------------

  M     r0   r1  h(4M)  %72  r0/h r1/h
$\endgroup$
1
$\begingroup$

well, here is the same information when the target number has a factor of 2 or 3. The ratio pair (12,4) is evidently repeated, the two cases are $1,10 \pmod {12}.$

Oh, note that multiples of $9$ are never represented primitively. That is fine, the number of representations is exactly the same as the number of representations of $M/9.$

---------------------------------------
 g0  :          9 :     1    1    3    0    0    1 auto 24
 g1  :         36 :     1    3    3    0    0    0 auto 16
---------------------------------------
---------------------------------------

      M     r0   r1  h(4M)  %72  r0/h r1/h
      3      8    4      1  N 3    8    4           3 = 3
     75     48   24      6  N 3    8    4          75 = 3 * 5^2
    147     48   24      6  N 3    8    4         147 = 3 * 7^2
    219     96   48     12  N 3    8    4         219 = 3 * 73
    291     96   48     12  N 3    8    4         291 = 3 * 97
    363     96   48     12  N 3    8    4         363 = 3 * 11^2
    435     96   48     12  N 3    8    4         435 = 3 * 5 * 29
    507     96   48     12  N 3    8    4         507 = 3 * 13^2
    579    192   96     24  N 3    8    4         579 = 3 * 193
    651    192   96     24  N 3    8    4         651 = 3 * 7 * 31
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
      4     12    8      1  N 4   12    8           4 = 2^2
     76     72   48      6  N 4   12    8          76 = 2^2 * 19
    148     48   32      4  N 4   12    8         148 = 2^2 * 37
    220     96   64      8  N 4   12    8         220 = 2^2 * 5 * 11
    292     96   64      8  N 4   12    8         292 = 2^2 * 73
    364    144   96     12  N 4   12    8         364 = 2^2 * 7 * 13
    436    144   96     12  N 4   12    8         436 = 2^2 * 109
    508    120   80     10  N 4   12    8         508 = 2^2 * 127
    580    192  128     16  N 4   12    8         580 = 2^2 * 5 * 29
    652     72   48      6  N 4   12    8         652 = 2^2 * 163
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
      6     12    4      2  N 6    6    2           6 = 2 * 3
     78     24    8      4  N 6    6    2          78 = 2 * 3 * 13
    150     48   16      8  N 6    6    2         150 = 2 * 3 * 5^2
    222     72   24     12  N 6    6    2         222 = 2 * 3 * 37
    294     72   24     12  N 6    6    2         294 = 2 * 3 * 7^2
    366     72   24     12  N 6    6    2         366 = 2 * 3 * 61
    438     48   16      8  N 6    6    2         438 = 2 * 3 * 73
    510     96   32     16  N 6    6    2         510 = 2 * 3 * 5 * 17
    582     96   32     16  N 6    6    2         582 = 2 * 3 * 97
    654    168   56     28  N 6    6    2         654 = 2 * 3 * 109
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     10     24    8      2  N10   12    4          10 = 2 * 5
     82     48   16      4  N10   12    4          82 = 2 * 41
    154     96   32      8  N10   12    4         154 = 2 * 7 * 11
    226     96   32      8  N10   12    4         226 = 2 * 113
    298     72   24      6  N10   12    4         298 = 2 * 149
    370    144   48     12  N10   12    4         370 = 2 * 5 * 37
    442     96   32      8  N10   12    4         442 = 2 * 13 * 17
    514    192   64     16  N10   12    4         514 = 2 * 257
    586    216   72     18  N10   12    4         586 = 2 * 293
    658     96   32      8  N10   12    4         658 = 2 * 7 * 47
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     12     12    8      2  N12    6    4          12 = 2^2 * 3
     84     48   32      8  N12    6    4          84 = 2^2 * 3 * 7
    156     48   32      8  N12    6    4         156 = 2^2 * 3 * 13
    228     48   32      8  N12    6    4         228 = 2^2 * 3 * 19
    300     72   48     12  N12    6    4         300 = 2^2 * 3 * 5^2
    372     48   32      8  N12    6    4         372 = 2^2 * 3 * 31
    444     96   64     16  N12    6    4         444 = 2^2 * 3 * 37
    516    144   96     24  N12    6    4         516 = 2^2 * 3 * 43
    588     72   48     12  N12    6    4         588 = 2^2 * 3 * 7^2
    660     96   64     16  N12    6    4         660 = 2^2 * 3 * 5 * 11
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     15     24   16      2  N15   12    8          15 = 3 * 5
     87     72   48      6  N15   12    8          87 = 3 * 29
    159    120   80     10  N15   12    8         159 = 3 * 53
    231    144   96     12  N15   12    8         231 = 3 * 7 * 11
    303    120   80     10  N15   12    8         303 = 3 * 101
    375    120   80     10  N15   12    8         375 = 3 * 5^3
    447    168  112     14  N15   12    8         447 = 3 * 149
    519    216  144     18  N15   12    8         519 = 3 * 173
    591    264  176     22  N15   12    8         591 = 3 * 197
    663    192  128     16  N15   12    8         663 = 3 * 13 * 17
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     16     24   16      2  N16   12    8          16 = 2^4
     88     48   32      4  N16   12    8          88 = 2^3 * 11
    160     96   64      8  N16   12    8         160 = 2^5 * 5
    232     48   32      4  N16   12    8         232 = 2^3 * 29
    304    144   96     12  N16   12    8         304 = 2^4 * 19
    376    192  128     16  N16   12    8         376 = 2^3 * 47
    448     96   64      8  N16   12    8         448 = 2^6 * 7
    520     96   64      8  N16   12    8         520 = 2^3 * 5 * 13
    592     96   64      8  N16   12    8         592 = 2^4 * 37
    664    240  160     20  N16   12    8         664 = 2^3 * 83
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     21     24    8      4  N21    6    2          21 = 3 * 7
     93     24    8      4  N21    6    2          93 = 3 * 31
    165     48   16      8  N21    6    2         165 = 3 * 5 * 11
    237     72   24     12  N21    6    2         237 = 3 * 79
    309     72   24     12  N21    6    2         309 = 3 * 103
    381    120   40     20  N21    6    2         381 = 3 * 127
    453     72   24     12  N21    6    2         453 = 3 * 151
    525     96   32     16  N21    6    2         525 = 3 * 5^2 * 7
    597     72   24     12  N21    6    2         597 = 3 * 199
    669     72   24     12  N21    6    2         669 = 3 * 223
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     22     24    8      2  N22   12    4          22 = 2 * 11
     94     96   32      8  N22   12    4          94 = 2 * 47
    166    120   40     10  N22   12    4         166 = 2 * 83
    238     96   32      8  N22   12    4         238 = 2 * 7 * 17
    310     96   32      8  N22   12    4         310 = 2 * 5 * 31
    382     96   32      8  N22   12    4         382 = 2 * 191
    454    168   56     14  N22   12    4         454 = 2 * 227
    526    144   48     12  N22   12    4         526 = 2 * 263
    598     96   32      8  N22   12    4         598 = 2 * 13 * 23
    670    144   48     12  N22   12    4         670 = 2 * 5 * 67
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     24     24   16      4  N24    6    4          24 = 2^3 * 3
     96     48   32      8  N24    6    4          96 = 2^5 * 3
    168     48   32      8  N24    6    4         168 = 2^3 * 3 * 7
    240     48   32      8  N24    6    4         240 = 2^4 * 3 * 5
    312     48   32      8  N24    6    4         312 = 2^3 * 3 * 13
    384     96   64     16  N24    6    4         384 = 2^7 * 3
    456     96   64     16  N24    6    4         456 = 2^3 * 3 * 19
    528     96   64     16  N24    6    4         528 = 2^4 * 3 * 11
    600     96   64     16  N24    6    4         600 = 2^3 * 3 * 5^2
    672     96   64     16  N24    6    4         672 = 2^5 * 3 * 7
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     28     24   16      2  N28   12    8          28 = 2^2 * 7
    100     48   32      4  N28   12    8         100 = 2^2 * 5^2
    172     72   48      6  N28   12    8         172 = 2^2 * 43
    244    144   96     12  N28   12    8         244 = 2^2 * 61
    316    120   80     10  N28   12    8         316 = 2^2 * 79
    388     96   64      8  N28   12    8         388 = 2^2 * 97
    460    144   96     12  N28   12    8         460 = 2^2 * 5 * 23
    532     96   64      8  N28   12    8         532 = 2^2 * 7 * 19
    604    168  112     14  N28   12    8         604 = 2^2 * 151
    676    144   96     12  N28   12    8         676 = 2^2 * 13^2
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     30     24    8      4  N30    6    2          30 = 2 * 3 * 5
    102     24    8      4  N30    6    2         102 = 2 * 3 * 17
    174     72   24     12  N30    6    2         174 = 2 * 3 * 29
    246     72   24     12  N30    6    2         246 = 2 * 3 * 41
    318     72   24     12  N30    6    2         318 = 2 * 3 * 53
    390     96   32     16  N30    6    2         390 = 2 * 3 * 5 * 13
    462     48   16      8  N30    6    2         462 = 2 * 3 * 7 * 11
    534    120   40     20  N30    6    2         534 = 2 * 3 * 89
    606     72   24     12  N30    6    2         606 = 2 * 3 * 101
    678    120   40     20  N30    6    2         678 = 2 * 3 * 113
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     33     24    8      4  N33    6    2          33 = 3 * 11
    105     48   16      8  N33    6    2         105 = 3 * 5 * 7
    177     24    8      4  N33    6    2         177 = 3 * 59
    249     72   24     12  N33    6    2         249 = 3 * 83
    321    120   40     20  N33    6    2         321 = 3 * 107
    393     72   24     12  N33    6    2         393 = 3 * 131
    465     96   32     16  N33    6    2         465 = 3 * 5 * 31
    537     72   24     12  N33    6    2         537 = 3 * 179
    609     96   32     16  N33    6    2         609 = 3 * 7 * 29
    681    120   40     20  N33    6    2         681 = 3 * 227
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     34     48   16      4  N34   12    4          34 = 2 * 17
    106     72   24      6  N34   12    4         106 = 2 * 53
    178     96   32      8  N34   12    4         178 = 2 * 89
    250    120   40     10  N34   12    4         250 = 2 * 5^3
    322     96   32      8  N34   12    4         322 = 2 * 7 * 23
    394    120   40     10  N34   12    4         394 = 2 * 197
    466     96   32      8  N34   12    4         466 = 2 * 233
    538    120   40     10  N34   12    4         538 = 2 * 269
    610    144   48     12  N34   12    4         610 = 2 * 5 * 61
    682    144   48     12  N34   12    4         682 = 2 * 11 * 31
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     39     48   32      4  N39   12    8          39 = 3 * 13
    111     96   64      8  N39   12    8         111 = 3 * 37
    183     96   64      8  N39   12    8         183 = 3 * 61
    255    144   96     12  N39   12    8         255 = 3 * 5 * 17
    327    144   96     12  N39   12    8         327 = 3 * 109
    399    192  128     16  N39   12    8         399 = 3 * 7 * 19
    471    192  128     16  N39   12    8         471 = 3 * 157
    543    144   96     12  N39   12    8         543 = 3 * 181
    615    240  160     20  N39   12    8         615 = 3 * 5 * 41
    687    144   96     12  N39   12    8         687 = 3 * 229
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     40     48   32      4  N40   12    8          40 = 2^3 * 5
    112     48   32      4  N40   12    8         112 = 2^4 * 7
    184     96   64      8  N40   12    8         184 = 2^3 * 23
    256     96   64      8  N40   12    8         256 = 2^8
    328     96   64      8  N40   12    8         328 = 2^3 * 41
    400     96   64      8  N40   12    8         400 = 2^4 * 5^2
    472    144   96     12  N40   12    8         472 = 2^3 * 59
    544    192  128     16  N40   12    8         544 = 2^5 * 17
    616    192  128     16  N40   12    8         616 = 2^3 * 7 * 11
    688    144   96     12  N40   12    8         688 = 2^4 * 43
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     42     24    8      4  N42    6    2          42 = 2 * 3 * 7
    114     48   16      8  N42    6    2         114 = 2 * 3 * 19
    186     72   24     12  N42    6    2         186 = 2 * 3 * 31
    258     48   16      8  N42    6    2         258 = 2 * 3 * 43
    330     48   16      8  N42    6    2         330 = 2 * 3 * 5 * 11
    402     96   32     16  N42    6    2         402 = 2 * 3 * 67
    474    120   40     20  N42    6    2         474 = 2 * 3 * 79
    546    144   48     24  N42    6    2         546 = 2 * 3 * 7 * 13
    618     72   24     12  N42    6    2         618 = 2 * 3 * 103
    690     96   32     16  N42    6    2         690 = 2 * 3 * 5 * 23
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     46     48   16      4  N46   12    4          46 = 2 * 23
    118     72   24      6  N46   12    4         118 = 2 * 59
    190     48   16      4  N46   12    4         190 = 2 * 5 * 19
    262     72   24      6  N46   12    4         262 = 2 * 131
    334    144   48     12  N46   12    4         334 = 2 * 167
    406    192   64     16  N46   12    4         406 = 2 * 7 * 29
    478     96   32      8  N46   12    4         478 = 2 * 239
    550    144   48     12  N46   12    4         550 = 2 * 5^2 * 11
    622    144   48     12  N46   12    4         622 = 2 * 311
    694    120   40     10  N46   12    4         694 = 2 * 347
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     48     24   16      4  N48    6    4          48 = 2^4 * 3
    120     48   32      8  N48    6    4         120 = 2^3 * 3 * 5
    192     48   32      8  N48    6    4         192 = 2^6 * 3
    264     96   64     16  N48    6    4         264 = 2^3 * 3 * 11
    336     96   64     16  N48    6    4         336 = 2^4 * 3 * 7
    408     48   32      8  N48    6    4         408 = 2^3 * 3 * 17
    480     96   64     16  N48    6    4         480 = 2^5 * 3 * 5
    552     96   64     16  N48    6    4         552 = 2^3 * 3 * 23
    624     96   64     16  N48    6    4         624 = 2^4 * 3 * 13
    696    144   96     24  N48    6    4         696 = 2^3 * 3 * 29
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     51     48   24      6  N51    8    4          51 = 3 * 17
    123     48   24      6  N51    8    4         123 = 3 * 41
    195     96   48     12  N51    8    4         195 = 3 * 5 * 13
    267     48   24      6  N51    8    4         267 = 3 * 89
    339    144   72     18  N51    8    4         339 = 3 * 113
    411    144   72     18  N51    8    4         411 = 3 * 137
    483     96   48     12  N51    8    4         483 = 3 * 7 * 23
    555     96   48     12  N51    8    4         555 = 3 * 5 * 37
    627     96   48     12  N51    8    4         627 = 3 * 11 * 19
    699    240  120     30  N51    8    4         699 = 3 * 233
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     52     48   32      4  N52   12    8          52 = 2^2 * 13
    124     72   48      6  N52   12    8         124 = 2^2 * 31
    196     96   64      8  N52   12    8         196 = 2^2 * 7^2
    268     72   48      6  N52   12    8         268 = 2^2 * 67
    340     96   64      8  N52   12    8         340 = 2^2 * 5 * 17
    412    120   80     10  N52   12    8         412 = 2^2 * 103
    484    144   96     12  N52   12    8         484 = 2^2 * 11^2
    556    216  144     18  N52   12    8         556 = 2^2 * 139
    628    144   96     12  N52   12    8         628 = 2^2 * 157
    700    144   96     12  N52   12    8         700 = 2^2 * 5^2 * 7
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     57     24    8      4  N57    6    2          57 = 3 * 19
    129     72   24     12  N57    6    2         129 = 3 * 43
    201     72   24     12  N57    6    2         201 = 3 * 67
    273     48   16      8  N57    6    2         273 = 3 * 7 * 13
    345     48   16      8  N57    6    2         345 = 3 * 5 * 23
    417     72   24     12  N57    6    2         417 = 3 * 139
    489    120   40     20  N57    6    2         489 = 3 * 163
    561     96   32     16  N57    6    2         561 = 3 * 11 * 17
    633    120   40     20  N57    6    2         633 = 3 * 211
    705    144   48     24  N57    6    2         705 = 3 * 5 * 47
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     58     24    8      2  N58   12    4          58 = 2 * 29
    130     48   16      4  N58   12    4         130 = 2 * 5 * 13
    202     72   24      6  N58   12    4         202 = 2 * 101
    274    144   48     12  N58   12    4         274 = 2 * 137
    346    120   40     10  N58   12    4         346 = 2 * 173
    418     96   32      8  N58   12    4         418 = 2 * 11 * 19
    490    144   48     12  N58   12    4         490 = 2 * 5 * 7^2
    562     96   32      8  N58   12    4         562 = 2 * 281
    634    168   56     14  N58   12    4         634 = 2 * 317
    706    288   96     24  N58   12    4         706 = 2 * 353
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     60     24   16      4  N60    6    4          60 = 2^2 * 3 * 5
    132     48   32      8  N60    6    4         132 = 2^2 * 3 * 11
    204     72   48     12  N60    6    4         204 = 2^2 * 3 * 17
    276     96   64     16  N60    6    4         276 = 2^2 * 3 * 23
    348     72   48     12  N60    6    4         348 = 2^2 * 3 * 29
    420     96   64     16  N60    6    4         420 = 2^2 * 3 * 5 * 7
    492     72   48     12  N60    6    4         492 = 2^2 * 3 * 41
    564     96   64     16  N60    6    4         564 = 2^2 * 3 * 47
    636    120   80     20  N60    6    4         636 = 2^2 * 3 * 53
    708     48   32      8  N60    6    4         708 = 2^2 * 3 * 59
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     64     48   32      4  N64   12    8          64 = 2^6
    136     96   64      8  N64   12    8         136 = 2^3 * 17
    208     96   64      8  N64   12    8         208 = 2^4 * 13
    280     96   64      8  N64   12    8         280 = 2^3 * 5 * 7
    352     96   64      8  N64   12    8         352 = 2^5 * 11
    424    144   96     12  N64   12    8         424 = 2^3 * 53
    496    144   96     12  N64   12    8         496 = 2^4 * 31
    568     96   64      8  N64   12    8         568 = 2^3 * 71
    640    192  128     16  N64   12    8         640 = 2^7 * 5
    712    192  128     16  N64   12    8         712 = 2^3 * 89
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     66     48   16      8  N66    6    2          66 = 2 * 3 * 11
    138     48   16      8  N66    6    2         138 = 2 * 3 * 23
    210     48   16      8  N66    6    2         210 = 2 * 3 * 5 * 7
    282     48   16      8  N66    6    2         282 = 2 * 3 * 47
    354     96   32     16  N66    6    2         354 = 2 * 3 * 59
    426    144   48     24  N66    6    2         426 = 2 * 3 * 71
    498     48   16      8  N66    6    2         498 = 2 * 3 * 83
    570     96   32     16  N66    6    2         570 = 2 * 3 * 5 * 19
    642     96   32     16  N66    6    2         642 = 2 * 3 * 107
    714    144   48     24  N66    6    2         714 = 2 * 3 * 7 * 17
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     69     48   16      8  N69    6    2          69 = 3 * 23
    141     48   16      8  N69    6    2         141 = 3 * 47
    213     48   16      8  N69    6    2         213 = 3 * 71
    285     96   32     16  N69    6    2         285 = 3 * 5 * 19
    357     48   16      8  N69    6    2         357 = 3 * 7 * 17
    429     96   32     16  N69    6    2         429 = 3 * 11 * 13
    501     96   32     16  N69    6    2         501 = 3 * 167
    573     96   32     16  N69    6    2         573 = 3 * 191
    645     96   32     16  N69    6    2         645 = 3 * 5 * 43
    717     96   32     16  N69    6    2         717 = 3 * 239
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
     70     48   16      4  N70   12    4          70 = 2 * 5 * 7
    142     48   16      4  N70   12    4         142 = 2 * 71
    214     72   24      6  N70   12    4         214 = 2 * 107
    286    144   48     12  N70   12    4         286 = 2 * 11 * 13
    358     72   24      6  N70   12    4         358 = 2 * 179
    430    144   48     12  N70   12    4         430 = 2 * 5 * 43
    502    168   56     14  N70   12    4         502 = 2 * 251
    574    192   64     16  N70   12    4         574 = 2 * 7 * 41
    646    192   64     16  N70   12    4         646 = 2 * 17 * 19
    718    144   48     12  N70   12    4         718 = 2 * 359
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
      M     r0   r1  h(4M)  %72  r0/h r1/h

---------------------------------------
 g0  :          9 :     1    1    3    0    0    1 auto 24
 g1  :         36 :     1    3    3    0    0    0 auto 16
---------------------------------------
$\endgroup$

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