9
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This question is inspired by this one. It comes in two parts.

Question 1. Determine all positive integers $k$ such that there are positive integers $a$, $b$, and $c$ such that $$\frac{a^2+b^2+c^2}{bc+ca+ab}=k\,.\tag{*}$$

Question 2. For each positive integer $k$ discovered in Question 1, what are all triples $(a,b,c)$ of positive integers such that the condition (*) is satisfied?

Here are three values of $k$ that have the required property.

  • Case I: $k=1$. All solutions $(a,b,c)$ are of the form $$(a,b,c)=(n,n,n)$$ where $n$ is a positive integer.

  • Case II: $k=2$. It can be proven by Vieta jumping that each solution $(a,b,c)$ is a permutation of $$\big(tm^2,tn^2,t(m+n)^2\big)\tag{#}$$ for some positive integers $t$, $m$, and $n$ (we can assume that $m$ and $n$ are relatively prime). A proof of this claim can be seen in the hidden portion below.

  • Case III: $k=5$. All solutions can be found in this link.

Are there other values of $k$ with the required property? If so, are there infinitely many of them?

Here is a proof sketch for my claim when $k=2$ if you would like to read. Let $S$ denote the set of solutions $(a,b,c)\in\mathbb{Z}_{>0}^3$ to (*). Define a similarity relation $\sim$ on $S$ which is an equivalence relation on $S$ generated by requiring that each triple $(a,b,c)\in S$ is similar to any permutation of $(a,b,c)$, and that $(a,b,c)$ is similar to $(a,b,2a+2b-c)$, provided that $(a,b,2a+2b-c)$ is also in $S$. Pick an equivalence class $C$ of $S$ induced by $\sim$, and suppose that $(a,b,c)$ is its minimal triple in the sense that $a+b+c$ is the smallest among all triples in $C$ that is not of the form (#). We may assume without loss of generality that $a\leq b\leq c$. Note that either $2a+2b-c\leq 0$ or $(a,b,2a+2b-c)$ is a "smaller" triple than $(a,b,c)$ in $C$ that is not of the form (#). Show that $c=2a+2b$ must holds, and this implies $b=c$. It then follows that $(a,b,c)=(t,t,4t)=\big(1^2t,1^2t,(1+1)^2t\big)$ for some positive integer $t$, and this is a contradiction.

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5
  • $\begingroup$ By brute force I can find $k=5,10,14$ for small integers less than $100$. $(2,3,71),(2,5,71),(3,5,41)$ that seems to repeat. If I extend the search some other $k$ appear $17,26,62$. $\endgroup$
    – zwim
    Commented Jul 6, 2020 at 20:46
  • $\begingroup$ @zwim Interesting! Would you please give me examples of $(a,b,c)$ for $k=17,26,62$ (or for any further values of $k$ you have discovered)? This could be put as an answer, if you wish. $\endgroup$ Commented Jul 6, 2020 at 20:59
  • 1
    $\begingroup$ tio.run/##XZDLasQwDEX3/… $\endgroup$
    – zwim
    Commented Jul 6, 2020 at 21:55
  • 1
    $\begingroup$ See OEIS sequence A331605. $\endgroup$ Commented Jul 7, 2020 at 3:43
  • 2
    $\begingroup$ mathoverflow.net/questions/225781/… $\endgroup$
    – individ
    Commented Jul 7, 2020 at 4:25

4 Answers 4

6
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There is such a solution if and only if both $k-1$ and $k+2$ have (well, different) integer expressions as some $u^2 + 3 v^2.$

The justification for that is in several answers I posted at

Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$

$$ $$ $$ $$

Given $$ p^2 + 3 q^2 = 2 + k, $$ $$ r^2 + 3 s^2 = 4(k-1), $$ we can solve $$ (x^2 + y^2 + z^2) = k (yz + zx + xy) $$ with $$ x = 2 p^2 + 6 q^2 - p r - 3 p s + 3 q r - 3 q s, $$ $$ y = 2 p^2 + 6 q^2 - p r + 3 p s - 3 q r - 3 q s, $$ $$ z = 2 p^2 + 6 q^2 + 2 p r + 6 q s. $$

I did not immediately realize, the process of Vieta Jumping lets us take a mixed solution and create one with all the same $\pm$ sign. Suppose $x < 0,$ $y > 0,$ $z>0.$ We do a single jump: $$ x \mapsto k(y+z) - x, $$ where the new $x$ value is then positive!

The permissible values of your $k$ from 2 to 1000 are

  2      5     10     14     17     26     29     37     50     62
 65     74     77     82     98    101    109    110    122    125
145    149    170    173    190    194    197    209    226    242
245    257    269    290    302    305    314    325    334    362
365    398    401    410    434    437    442    469    482    485
497    509    514    530    554    557    577    590    602    605
626    629    674    677    685    689    701    722    725    730
770    773    785    794    830    842    845    869    874    890
901    917    962    965    973    974    989

These all lead to solutions $(a,b,c) $ where it may be that some variables are negative, some positive.

Let me work up some of the smallest such $k,$ see whether positive solutions appear.

$$ k = 17; \; \; \; (377,17,5) $$

$$ k = 26; \; \; \; (418,13,3) $$

$$ k = 29; \; \; \; (1109,11,27) $$

BY RECIPE .........................................

Mon Jul  6 19:11:55 PDT 2020

      2  ( 1, 1 , 4 )  p 1 q 1 r 1 s 1
      5  ( -1, 5 , 17 )   ( 111, 5 , 17 )  p 2 q 1 r 2 s 2
     10  ( 2, -1 , 5 )   ( 2, 71 , 5 )  p 0 q 2 r 3 s 3
     14  ( -1, 2 , 11 )   ( 183, 2 , 11 )  p 2 q 2 r 2 s 4
     17  ( -13, 23 , 47 )   ( 1203, 23 , 47 )  p 4 q 1 r 4 s 4
     26  ( 3, -2 , 13 )   ( 3, 418 , 13 )  p 1 q 3 r 5 s 5
     29  ( -7, 11 , 89 )   ( 2907, 11 , 89 )  p 2 q 3 r 2 s 6
     37  ( -11, 19 , 31 )   ( 1861, 19 , 31 )  p 6 q 1 r 6 s 6
     50  ( -5, 7 , 76 )   ( 4155, 7 , 76 )  p 2 q 4 r 2 s 8
     62  ( -5, 7 , 22 )   ( 1803, 7 , 22 )  p 4 q 4 r 1 s 9
     65  ( -61, 107 , 155 )   ( 17091, 107 , 155 )  p 8 q 1 r 8 s 8
     74  ( 22, -17 , 109 )   ( 22, 9711 , 109 )  p 1 q 5 r 7 s 9
     77  ( -13, 17 , 233 )   ( 19263, 17 , 233 )  p 2 q 5 r 2 s 10
     82  ( 5, -4 , 41 )   ( 5, 3776 , 41 )  p 3 q 5 r 9 s 9
     98  ( -4, 5 , 29 )   ( 3336, 5 , 29 )  p 5 q 5 r 5 s 11
    101  ( -97, 173 , 233 )   ( 41103, 173 , 233 )  p 10 q 1 r 10 s 10
    109  ( -29, 43 , 97 )   ( 15289, 43 , 97 )  p 6 q 5 r 0 s 12
    110  ( -4, 5 , 83 )   ( 9684, 5 , 83 )  p 2 q 6 r 2 s 12
    122  ( 6, -5 , 61 )   ( 6, 8179 , 61 )  p 4 q 6 r 11 s 11
    125  ( -37, 59 , 105 )   ( 20537, 59 , 105 )  p 10 q 3 r 8 s 12
    145  ( 7, -5 , 19 )   ( 7, 3775 , 19 )  p 0 q 7 r 12 s 12
    149  ( -19, 23 , 449 )   ( 70347, 23 , 449 )  p 2 q 7 r 2 s 14
    170  ( -15, 19 , 82 )   ( 17185, 19 , 82 )  p 5 q 7 r 1 s 15
    173  ( -23, 31 , 97 )   ( 22167, 31 , 97 )  p 10 q 5 r 10 s 14
    190  ( 5, -4 , 23 )   ( 5, 5324 , 23 )  p 0 q 8 r 9 s 15
    194  ( -11, 13 , 292 )   ( 59181, 13 , 292 )  p 2 q 8 r 2 s 16
    197  ( -61, 159 , 101 )   ( 51281, 159 , 101 )  p 14 q 1 r 4 s 16
    209  ( -97, 119 , 611 )   ( 152667, 119 , 611 )  p 8 q 7 r 8 s 16
    226  ( 8, -7 , 113 )   ( 8, 27353 , 113 )  p 6 q 8 r 15 s 15
    242  ( 31, -24 , 115 )   ( 31, 35356 , 115 )  p 1 q 9 r 14 s 16
    245  ( -25, 29 , 737 )   ( 187695, 29 , 737 )  p 2 q 9 r 2 s 18
    257  ( 131, -109 , 755 )   ( 131, 227811 , 755 )  p 4 q 9 r 16 s 16
    269  ( -79, 123 , 227 )   ( 94229, 123 , 227 )  p 14 q 5 r 10 s 18
    290  ( 9, -8 , 145 )   ( 9, 44668 , 145 )  p 7 q 9 r 17 s 17
    302  ( -7, 8 , 227 )   ( 70977, 8 , 227 )  p 2 q 10 r 2 s 20
    305  ( -55, 69 , 293 )   ( 110465, 69 , 293 )  p 8 q 9 r 4 s 20
    314  ( 43, -38 , 469 )   ( 43, 160806 , 469 )  p 4 q 10 r 13 s 19
    325  ( -107, 199 , 235 )   ( 141157, 199 , 235 )  p 18 q 1 r 18 s 18
    334  ( -11, 13 , 82 )   ( 31741, 13 , 82 )  p 6 q 10 r 3 s 21
    362  ( 27, -23 , 178 )   ( 27, 74233 , 178 )  p 1 q 11 r 11 s 21
    365  ( -31, 35 , 1097 )   ( 413211, 35 , 1097 )  p 2 q 11 r 2 s 22
    398  ( -14, 19 , 55 )   ( 29466, 19 , 55 )  p 10 q 10 r 1 s 23
    401  ( -79, 101 , 381 )   ( 193361, 101 , 381 )  p 16 q 7 r 20 s 20
    410  ( -59, 67 , 610 )   ( 277629, 67 , 610 )  p 7 q 11 r 7 s 23
    434  ( -17, 19 , 652 )   ( 291231, 19 , 652 )  p 2 q 12 r 2 s 24
    437  ( -121, 179 , 381 )   ( 244841, 179 , 381 )  p 14 q 9 r 4 s 24
    442  ( -34, 41 , 215 )   ( 113186, 41 , 215 )  p 9 q 11 r 6 s 24
    469  ( -137, 211 , 397 )   ( 285289, 211 , 397 )  p 18 q 7 r 12 s 24
    482  ( -4, 5 , 21 )   ( 12536, 5 , 21 )  p 11 q 11 r 7 s 25
    485  ( -481, 905 , 1037 )   ( 942351, 905 , 1037 )  p 22 q 1 r 22 s 22
    497  ( -313, 407 , 1403 )   ( 899883, 407 , 1403 )  p 16 q 9 r 16 s 24
    509  ( -37, 41 , 1529 )   ( 799167, 41 , 1529 )  p 2 q 13 r 2 s 26
    514  ( 44, -37 , 251 )   ( 44, 151667 , 251 )  p 3 q 13 r 18 s 24
    530  ( 151, -125 , 772 )   ( 151, 489315 , 772 )  p 5 q 13 r 23 s 23
    554  ( -29, 33 , 274 )   ( 170107, 33 , 274 )  p 7 q 13 r 5 s 27
    557  ( -283, 347 , 1613 )   ( 1092003, 347 , 1613 )  p 14 q 11 r 14 s 26
    577  ( -191, 361 , 409 )   ( 444481, 361 , 409 )  p 24 q 1 r 24 s 24
    590  ( -10, 11 , 443 )   ( 267870, 11 , 443 )  p 2 q 14 r 2 s 28
    602  ( 61, -50 , 291 )   ( 61, 211954 , 291 )  p 4 q 14 r 23 s 25
    605  ( -81, 95 , 593 )   ( 416321, 95 , 593 )  p 10 q 13 r 8 s 28
    626  ( 13, -12 , 313 )   ( 13, 204088 , 313 )  p 11 q 13 r 25 s 25
    629  ( -511, 743 , 1661 )   ( 1512627, 743 , 1661 )  p 22 q 7 r 22 s 26
    674  ( 133, -116 , 997 )   ( 133, 761736 , 997 )  p 1 q 15 r 13 s 29
    677  ( -43, 47 , 2033 )   ( 1408203, 47 , 2033 )  p 2 q 15 r 2 s 30
    685  ( -191, 283 , 595 )   ( 601621, 283 , 595 )  p 18 q 11 r 6 s 30
    689  ( 101, -87 , 677 )   ( 101, 536129 , 677 )  p 4 q 15 r 20 s 28
    701  ( -129, 161 , 671 )   ( 583361, 161 , 671 )  p 14 q 13 r 10 s 30
    722  ( -140, 163 , 1063 )   ( 885312, 163 , 1063 )  p 7 q 15 r 1 s 31
    725  ( -211, 323 , 615 )   ( 680261, 323 , 615 )  p 22 q 9 r 14 s 30
    730  ( 14, -13 , 365 )   ( 14, 276683 , 365 )  p 12 q 14 r 27 s 27
    770  ( -23, 25 , 1156 )   ( 909393, 25 , 1156 )  p 2 q 16 r 2 s 32
    773  ( -71, 85 , 451 )   ( 414399, 85 , 451 )  p 10 q 15 r 4 s 32
    785  ( -235, 653 , 369 )   ( 802505, 653 , 369 )  p 28 q 1 r 8 s 32
    794  ( -47, 54 , 391 )   ( 353377, 54 , 391 )  p 11 q 15 r 10 s 32
    830  ( -9, 10 , 103 )   ( 93799, 10 , 103 )  p 8 q 16 r 7 s 33
    842  ( 15, -14 , 421 )   ( 15, 367126 , 421 )  p 13 q 15 r 29 s 29
    845  ( -15, 19 , 73 )   ( 77755, 19 , 73 )  p 22 q 11 r 26 s 30
    869  ( -49, 53 , 2609 )   ( 2313327, 53 , 2609 )  p 2 q 17 r 2 s 34
    874  ( 41, -37 , 434 )   ( 41, 415187 , 434 )  p 3 q 17 r 15 s 33
    890  ( 97, -89 , 1330 )   ( 97, 1270119 , 1330 )  p 5 q 17 r 17 s 33
    901  ( 181, -149 , 871 )   ( 181, 948001 , 871 )  p 6 q 17 r 30 s 30
    917  ( -859, 1415 , 2201 )   ( 3316731, 1415 , 2201 )  p 26 q 9 r 14 s 34
    962  ( -65, 76 , 471 )   ( 526279, 76 , 471 )  p 14 q 16 r 13 s 35
    965  ( 245, -223 , 2879 )   ( 245, 3014883 , 2879 )  p 10 q 17 r 28 s 32
    973  ( -61, 155 , 101 )   ( 249149, 155 , 101 )  p 30 q 5 r 0 s 36
    974  ( -13, 14 , 731 )   ( 725643, 14 , 731 )  p 2 q 18 r 2 s 36
    989  ( -277, 411 , 857 )   ( 1254329, 411 , 857 )  p 22 q 13 r 8 s 36


Mon Jul  6 19:11:55 PDT 2020
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2
  • $\begingroup$ Do you have a reference for why $k-1$ and $k+2$ are expressible in the form $u^2+3v^2$? Thanks! $\endgroup$ Commented Jul 7, 2020 at 2:25
  • 2
    $\begingroup$ @Batominovski I put several answers at math.stackexchange.com/questions/1134075/… $\endgroup$
    – Will Jagy
    Commented Jul 7, 2020 at 2:29
3
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Question $2.$

$$\frac{a^2+b^2+c^2}{bc+ca+ab}=k\tag{1}$$
We can get a primitive parametric solution from a known solution below.

Let ${p,q,r}$ is a known solution for equation $(1)$.
Substitute $a=pt+m, b=qt+n, c=rt+s$ to equation $(1)$, then we get
$$t = \frac{-(-m^2+kmn+ksm+kns-s^2-n^2)}{-2nq-2mp+kmq+kpn+knr+kqs+ksp+krm-2sr}$$
Then we get a parametric solution.

$a = (-p+kr+kq)m^2+((-2q+kr)n+(-2r+kq)s)m+pn^2-pkns+ps^2$
$b = m^2q+((-2p+kr)n-kqs)m+(kr-q+kp)n^2+(-2r+kp)sn+qs^2$
$c = rm^2+(-knr+(-2p+kq)s)m+n^2r+(kp-2q)sn+(kp-r+kq)s^2$

$m,n,s$ are arbitrary.

Example:
$(k,p,q,r)=(5,3,5,41)$

$a = 227m^2-15ns+3s^2+3n^2+195mn-57sm$
$b = 5m^2-25sm+5s^2+215n^2+199mn-67ns$
$c = 41m^2-205mn-s^2+41n^2+5ns+19sm$

[$a,b,c$]

[$ 3, 5, 41$]
[$ 3, 5045, 1049$]
[$ 227, 5, 41$]
[$ 17, 5, 111$]
[$ 635, 3149, 17$]
[$ 545, 2901, 47$]
[$ 461, 2663, 75$]
[$ 383, 2435, 101$]
[$1277, 6375, 41$]
[$ 797, 5015, 201$]
[$ 593, 4395, 269$]
[$1361, 8517, 335$]
[$1223, 8105, 381$]
[$1091, 7703, 425$]
[$ 965, 7311, 467$]
[$ 731, 6557, 545$]
[$1739, 11933, 615$]
[$1427, 10965, 719$]
[$1139, 10037, 815$]
[$ 635, 111, 17$]
[$ 545, 59, 47$]
[$1623, 185, 131$]
[$3713, 635, 111$]
[$3491, 503, 185$]
[$3275, 381, 257$]
[$3065, 269, 327$]
[$2861, 167, 395$]
[$5393, 5, 1119$]
[$6653, 1335, 41$]
[$6065, 971, 237$]
[$5501, 647, 425$]
[$8643, 1175, 521$]
[$8301, 983, 635$]
[$7635, 629, 857$]
[$7311, 467, 965$]
[$10727, 75, 2141$]
[$12491, 1853, 615$]
[$11675, 1389, 887$]
[$10883, 965, 1151$]
[$11399, 2217, 125$]
[$11009, 1973, 255$]

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1
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this is a list giving just one positive solution for each legal $k < 5100.$ The method is that of my first answer, I just changed the int declarations in the C++ program to mpz_class, to allow for larger numbers.

Tue Jul  7 10:02:20 PDT 2020
      2  ( 1, 1 , 4 )  p 1 q 1 r 1 s 1
      5  ( -1, 5 , 17 )   ( 111, 5 , 17 )  p 2 q 1 r 2 s 2
     10  ( 2, -1 , 5 )   ( 2, 71 , 5 )  p 0 q 2 r 3 s 3
     14  ( -1, 2 , 11 )   ( 183, 2 , 11 )  p 2 q 2 r 2 s 4
     17  ( -13, 23 , 47 )   ( 1203, 23 , 47 )  p 4 q 1 r 4 s 4
     26  ( 3, -2 , 13 )   ( 3, 418 , 13 )  p 1 q 3 r 5 s 5
     29  ( -7, 11 , 89 )   ( 2907, 11 , 89 )  p 2 q 3 r 2 s 6
     37  ( -11, 19 , 31 )   ( 1861, 19 , 31 )  p 6 q 1 r 6 s 6
     50  ( -5, 7 , 76 )   ( 4155, 7 , 76 )  p 2 q 4 r 2 s 8
     62  ( -5, 7 , 22 )   ( 1803, 7 , 22 )  p 4 q 4 r 1 s 9
     65  ( -61, 107 , 155 )   ( 17091, 107 , 155 )  p 8 q 1 r 8 s 8
     74  ( 22, -17 , 109 )   ( 22, 9711 , 109 )  p 1 q 5 r 7 s 9
     77  ( -13, 17 , 233 )   ( 19263, 17 , 233 )  p 2 q 5 r 2 s 10
     82  ( 5, -4 , 41 )   ( 5, 3776 , 41 )  p 3 q 5 r 9 s 9
     98  ( -4, 5 , 29 )   ( 3336, 5 , 29 )  p 5 q 5 r 5 s 11
    101  ( -97, 173 , 233 )   ( 41103, 173 , 233 )  p 10 q 1 r 10 s 10
    109  ( -29, 43 , 97 )   ( 15289, 43 , 97 )  p 6 q 5 r 0 s 12
    110  ( -4, 5 , 83 )   ( 9684, 5 , 83 )  p 2 q 6 r 2 s 12
    122  ( 6, -5 , 61 )   ( 6, 8179 , 61 )  p 4 q 6 r 11 s 11
    125  ( -37, 59 , 105 )   ( 20537, 59 , 105 )  p 10 q 3 r 8 s 12
    145  ( 7, -5 , 19 )   ( 7, 3775 , 19 )  p 0 q 7 r 12 s 12
    149  ( -19, 23 , 449 )   ( 70347, 23 , 449 )  p 2 q 7 r 2 s 14
    170  ( -15, 19 , 82 )   ( 17185, 19 , 82 )  p 5 q 7 r 1 s 15
    173  ( -23, 31 , 97 )   ( 22167, 31 , 97 )  p 10 q 5 r 10 s 14
    190  ( 5, -4 , 23 )   ( 5, 5324 , 23 )  p 0 q 8 r 9 s 15
    194  ( -11, 13 , 292 )   ( 59181, 13 , 292 )  p 2 q 8 r 2 s 16
    197  ( -61, 159 , 101 )   ( 51281, 159 , 101 )  p 14 q 1 r 4 s 16
    209  ( -97, 119 , 611 )   ( 152667, 119 , 611 )  p 8 q 7 r 8 s 16
    226  ( 8, -7 , 113 )   ( 8, 27353 , 113 )  p 6 q 8 r 15 s 15
    242  ( 31, -24 , 115 )   ( 31, 35356 , 115 )  p 1 q 9 r 14 s 16
    245  ( -25, 29 , 737 )   ( 187695, 29 , 737 )  p 2 q 9 r 2 s 18
    257  ( 131, -109 , 755 )   ( 131, 227811 , 755 )  p 4 q 9 r 16 s 16
    269  ( -79, 123 , 227 )   ( 94229, 123 , 227 )  p 14 q 5 r 10 s 18
    290  ( 9, -8 , 145 )   ( 9, 44668 , 145 )  p 7 q 9 r 17 s 17
    302  ( -7, 8 , 227 )   ( 70977, 8 , 227 )  p 2 q 10 r 2 s 20
    305  ( -55, 69 , 293 )   ( 110465, 69 , 293 )  p 8 q 9 r 4 s 20
    314  ( 43, -38 , 469 )   ( 43, 160806 , 469 )  p 4 q 10 r 13 s 19
    325  ( -107, 199 , 235 )   ( 141157, 199 , 235 )  p 18 q 1 r 18 s 18
    334  ( -11, 13 , 82 )   ( 31741, 13 , 82 )  p 6 q 10 r 3 s 21
    362  ( 27, -23 , 178 )   ( 27, 74233 , 178 )  p 1 q 11 r 11 s 21
    365  ( -31, 35 , 1097 )   ( 413211, 35 , 1097 )  p 2 q 11 r 2 s 22
    398  ( -14, 19 , 55 )   ( 29466, 19 , 55 )  p 10 q 10 r 1 s 23
    401  ( -79, 101 , 381 )   ( 193361, 101 , 381 )  p 16 q 7 r 20 s 20
    410  ( -59, 67 , 610 )   ( 277629, 67 , 610 )  p 7 q 11 r 7 s 23
    434  ( -17, 19 , 652 )   ( 291231, 19 , 652 )  p 2 q 12 r 2 s 24
    437  ( -121, 179 , 381 )   ( 244841, 179 , 381 )  p 14 q 9 r 4 s 24
    442  ( -34, 41 , 215 )   ( 113186, 41 , 215 )  p 9 q 11 r 6 s 24
    469  ( -137, 211 , 397 )   ( 285289, 211 , 397 )  p 18 q 7 r 12 s 24
    482  ( -4, 5 , 21 )   ( 12536, 5 , 21 )  p 11 q 11 r 7 s 25
    485  ( -481, 905 , 1037 )   ( 942351, 905 , 1037 )  p 22 q 1 r 22 s 22
    497  ( -313, 407 , 1403 )   ( 899883, 407 , 1403 )  p 16 q 9 r 16 s 24
    509  ( -37, 41 , 1529 )   ( 799167, 41 , 1529 )  p 2 q 13 r 2 s 26
    514  ( 44, -37 , 251 )   ( 44, 151667 , 251 )  p 3 q 13 r 18 s 24
    530  ( 151, -125 , 772 )   ( 151, 489315 , 772 )  p 5 q 13 r 23 s 23
    554  ( -29, 33 , 274 )   ( 170107, 33 , 274 )  p 7 q 13 r 5 s 27
    557  ( -283, 347 , 1613 )   ( 1092003, 347 , 1613 )  p 14 q 11 r 14 s 26
    577  ( -191, 361 , 409 )   ( 444481, 361 , 409 )  p 24 q 1 r 24 s 24
    590  ( -10, 11 , 443 )   ( 267870, 11 , 443 )  p 2 q 14 r 2 s 28
    602  ( 61, -50 , 291 )   ( 61, 211954 , 291 )  p 4 q 14 r 23 s 25
    605  ( -81, 95 , 593 )   ( 416321, 95 , 593 )  p 10 q 13 r 8 s 28
    626  ( 13, -12 , 313 )   ( 13, 204088 , 313 )  p 11 q 13 r 25 s 25
    629  ( -511, 743 , 1661 )   ( 1512627, 743 , 1661 )  p 22 q 7 r 22 s 26
    674  ( 133, -116 , 997 )   ( 133, 761736 , 997 )  p 1 q 15 r 13 s 29
    677  ( -43, 47 , 2033 )   ( 1408203, 47 , 2033 )  p 2 q 15 r 2 s 30
    685  ( -191, 283 , 595 )   ( 601621, 283 , 595 )  p 18 q 11 r 6 s 30
    689  ( 101, -87 , 677 )   ( 101, 536129 , 677 )  p 4 q 15 r 20 s 28
    701  ( -129, 161 , 671 )   ( 583361, 161 , 671 )  p 14 q 13 r 10 s 30
    722  ( -140, 163 , 1063 )   ( 885312, 163 , 1063 )  p 7 q 15 r 1 s 31
    725  ( -211, 323 , 615 )   ( 680261, 323 , 615 )  p 22 q 9 r 14 s 30
    730  ( 14, -13 , 365 )   ( 14, 276683 , 365 )  p 12 q 14 r 27 s 27
    770  ( -23, 25 , 1156 )   ( 909393, 25 , 1156 )  p 2 q 16 r 2 s 32
    773  ( -71, 85 , 451 )   ( 414399, 85 , 451 )  p 10 q 15 r 4 s 32
    785  ( -235, 653 , 369 )   ( 802505, 653 , 369 )  p 28 q 1 r 8 s 32
    794  ( -47, 54 , 391 )   ( 353377, 54 , 391 )  p 11 q 15 r 10 s 32
    830  ( -9, 10 , 103 )   ( 93799, 10 , 103 )  p 8 q 16 r 7 s 33
    842  ( 15, -14 , 421 )   ( 15, 367126 , 421 )  p 13 q 15 r 29 s 29
    845  ( -15, 19 , 73 )   ( 77755, 19 , 73 )  p 22 q 11 r 26 s 30
    869  ( -49, 53 , 2609 )   ( 2313327, 53 , 2609 )  p 2 q 17 r 2 s 34
    874  ( 41, -37 , 434 )   ( 41, 415187 , 434 )  p 3 q 17 r 15 s 33
    890  ( 97, -89 , 1330 )   ( 97, 1270119 , 1330 )  p 5 q 17 r 17 s 33
    901  ( 181, -149 , 871 )   ( 181, 948001 , 871 )  p 6 q 17 r 30 s 30
    917  ( -859, 1415 , 2201 )   ( 3316731, 1415 , 2201 )  p 26 q 9 r 14 s 34
    962  ( -65, 76 , 471 )   ( 526279, 76 , 471 )  p 14 q 16 r 13 s 35
    965  ( 245, -223 , 2879 )   ( 245, 3014883 , 2879 )  p 10 q 17 r 28 s 32
    973  ( -61, 155 , 101 )   ( 249149, 155 , 101 )  p 30 q 5 r 0 s 36
    974  ( -13, 14 , 731 )   ( 725643, 14 , 731 )  p 2 q 18 r 2 s 36
    989  ( -277, 411 , 857 )   ( 1254329, 411 , 857 )  p 22 q 13 r 8 s 36
   1009  ( -107, 121 , 997 )   ( 1128169, 121 , 997 )  p 12 q 17 r 12 s 36
   1022  ( -5, 6 , 31 )   ( 37819, 6 , 31 )  p 16 q 16 r 14 s 36
   1025  ( -255, 353 , 929 )   ( 1314305, 353 , 929 )  p 28 q 9 r 32 s 32
   1034  ( -146, 163 , 1537 )   ( 1757946, 163 , 1537 )  p 8 q 18 r 5 s 37
   1037  ( -301, 459 , 881 )   ( 1389881, 459 , 881 )  p 26 q 11 r 16 s 36
   1070  ( -49, 53 , 800 )   ( 912759, 53 , 800 )  p 10 q 18 r 13 s 37
   1073  ( -113, 139 , 619 )   ( 813447, 139 , 619 )  p 20 q 15 r 20 s 36
   1085  ( -55, 59 , 3257 )   ( 3597915, 59 , 3257 )  p 2 q 19 r 2 s 38
   1090  ( 149, -115 , 512 )   ( 149, 720605 , 512 )  p 3 q 19 r 33 s 33
   1117  ( 127, -113 , 1105 )   ( 127, 1376257 , 1105 )  p 6 q 19 r 24 s 36
   1130  ( 82, -71 , 555 )   ( 82, 719881 , 555 )  p 7 q 19 r 29 s 35
   1154  ( -7, 9 , 32 )   ( 47321, 9 , 32 )  p 17 q 17 r 7 s 39
   1157  ( -337, 377 , 3437 )   ( 4413135, 377 , 3437 )  p 22 q 15 r 34 s 34
   1162  ( -43, 47 , 578 )   ( 726293, 47 , 578 )  p 9 q 19 r 9 s 39
   1169  ( -363, 593 , 941 )   ( 1793609, 593 , 941 )  p 32 q 7 r 28 s 36
   1198  ( 8, -7 , 59 )   ( 8, 80273 , 59 )  p 0 q 20 r 15 s 39
   1202  ( -29, 31 , 1804 )   ( 2205699, 31 , 1804 )  p 2 q 20 r 2 s 40
   1214  ( 12, -11 , 151 )   ( 12, 197893 , 151 )  p 4 q 20 r 17 s 39
   1226  ( -311, 379 , 1774 )   ( 2639889, 379 , 1774 )  p 16 q 18 r 10 s 40
   1229  ( -393, 671 , 953 )   ( 1996289, 671 , 953 )  p 34 q 5 r 32 s 36
   1250  ( -60, 67 , 619 )   ( 857560, 67 , 619 )  p 13 q 19 r 14 s 40
   1262  ( -36, 41 , 311 )   ( 444260, 41 , 311 )  p 8 q 20 r 1 s 41
   1265  ( -235, 293 , 1209 )   ( 1900265, 293 , 1209 )  p 20 q 17 r 16 s 40
   1297  ( -431, 829 , 901 )   ( 2244241, 829 , 901 )  p 36 q 1 r 36 s 36
   1298  ( 17, -16 , 389 )   ( 17, 527004 , 389 )  p 10 q 20 r 25 s 39
   1301  ( -429, 791 , 941 )   ( 2253761, 791 , 941 )  p 34 q 7 r 20 s 40
   1322  ( 343, -290 , 1933 )   ( 343, 3009162 , 1933 )  p 1 q 21 r 22 s 40
   1325  ( -61, 65 , 3977 )   ( 5355711, 65 , 3977 )  p 2 q 21 r 2 s 42
   1349  ( -961, 1301 , 3713 )   ( 6764847, 1301 , 3713 )  p 22 q 17 r 10 s 42
   1370  ( 10, -9 , 97 )   ( 10, 146599 , 97 )  p 7 q 21 r 26 s 40
   1394  ( -341, 412 , 2023 )   ( 3394731, 412 , 2023 )  p 14 q 20 r 5 s 43
   1397  ( -1279, 2045 , 3431 )   ( 7651251, 2045 , 3431 )  p 34 q 9 r 28 s 40
   1405  ( -191, 223 , 1375 )   ( 2245381, 223 , 1375 )  p 18 q 19 r 18 s 42
   1445  ( -465, 1109 , 803 )   ( 2763305, 1109 , 803 )  p 38 q 1 r 22 s 42
   1454  ( -16, 17 , 1091 )   ( 1611048, 17 , 1091 )  p 2 q 22 r 2 s 44
   1457  ( -1345, 2171 , 3551 )   ( 8338299, 2171 , 3551 )  p 28 q 15 r 4 s 44
   1469  ( -1351, 3593 , 2171 )   ( 8468667, 3593 , 2171 )  p 38 q 3 r 8 s 44
   1490  ( 151, -140 , 2227 )   ( 151, 3543360 , 2227 )  p 13 q 21 r 34 s 40
   1517  ( -19, 21 , 215 )   ( 358031, 21 , 215 )  p 14 q 21 r 16 s 44
   1522  ( -163, 212 , 713 )   ( 1408013, 212 , 713 )  p 18 q 20 r 3 s 45
   1550  ( -25, 27 , 386 )   ( 640175, 27 , 386 )  p 10 q 22 r 11 s 45
   1589  ( -67, 71 , 4769 )   ( 7690827, 71 , 4769 )  p 2 q 23 r 2 s 46
   1598  ( -10, 11 , 119 )   ( 207750, 11 , 119 )  p 20 q 20 r 29 s 43
   1601  ( 1283, -997 , 4523 )   ( 1283, 9296403 , 4523 )  p 4 q 23 r 40 s 40
   1610  ( 55, -51 , 802 )   ( 55, 1379821 , 802 )  p 5 q 23 r 19 s 45
   1649  ( 153, -139 , 1637 )   ( 153, 2951849 , 1637 )  p 8 q 23 r 28 s 44
   1682  ( 21, -20 , 841 )   ( 21, 1449904 , 841 )  p 19 q 21 r 41 s 41
   1685  ( 243, -211 , 1655 )   ( 243, 3198341 , 1655 )  p 10 q 23 r 38 s 42
   1729  ( -263, 313 , 1681 )   ( 3447889, 313 , 1681 )  p 12 q 23 r 0 s 48
   1730  ( -35, 37 , 2596 )   ( 4555125, 37 , 2596 )  p 2 q 24 r 2 s 48
   1742  ( 263, -217 , 1262 )   ( 263, 2656767 , 1262 )  p 4 q 24 r 34 s 44
   1745  ( -571, 1293 , 1025 )   ( 4045481, 1293 , 1025 )  p 40 q 7 r 8 s 48
   1754  ( -278, 313 , 2599 )   ( 5107926, 313 , 2599 )  p 13 q 23 r 10 s 48
   1757  ( -943, 1163 , 5057 )   ( 10929483, 1163 , 5057 )  p 26 q 19 r 26 s 46
   1765  ( -497, 739 , 1525 )   ( 3996457, 739 , 1525 )  p 30 q 17 r 12 s 48
   1790  ( 17, -16 , 335 )   ( 17, 630096 , 335 )  p 8 q 24 r 23 s 47
   1810  ( -73, 80 , 899 )   ( 1772063, 80 , 899 )  p 15 q 23 r 18 s 48
   1826  ( 103, -92 , 903 )   ( 103, 1837048 , 903 )  p 10 q 24 r 35 s 45
   1829  ( -529, 803 , 1557 )   ( 4316969, 803 , 1557 )  p 34 q 15 r 20 s 48
   1850  ( -111, 127 , 910 )   ( 1918561, 127 , 910 )  p 20 q 22 r 22 s 48
   1873  ( 13, -11 , 73 )   ( 13, 161089 , 73 )  p 0 q 25 r 24 s 48
   1874  ( 99, -89 , 928 )   ( 99, 1924687 , 928 )  p 1 q 25 r 17 s 49
   1877  ( -73, 77 , 5633 )   ( 10717743, 77 , 5633 )  p 2 q 25 r 2 s 50
   1898  ( 65, -58 , 563 )   ( 65, 1192002 , 563 )  p 5 q 25 r 26 s 48
   1934  ( -6, 7 , 43 )   ( 96706, 7 , 43 )  p 22 q 22 r 23 s 49
   1937  ( 1235, -1009 , 5591 )   ( 1235, 13222971 , 5591 )  p 8 q 25 r 44 s 44
   1949  ( -1549, 2213 , 5189 )   ( 14428047, 2213 , 5189 )  p 38 q 13 r 38 s 46
   1954  ( -91, 101 , 968 )   ( 2088917, 101 , 968 )  p 9 q 25 r 3 s 51
   1982  ( -32, 37 , 243 )   ( 554992, 37 , 243 )  p 16 q 24 r 11 s 51
   1985  ( -325, 347 , 5939 )   ( 12478035, 347 , 5939 )  p 20 q 23 r 32 s 48
   1994  ( -57, 61 , 994 )   ( 2103727, 61 , 994 )  p 11 q 25 r 13 s 51
   1997  ( -373, 465 , 1907 )   ( 4737257, 465 , 1907 )  p 26 q 21 r 22 s 50
   2026  ( 26, -19 , 71 )   ( 26, 196541 , 71 )  p 0 q 26 r 45 s 45
   2030  ( -19, 20 , 1523 )   ( 3132309, 20 , 1523 )  p 2 q 26 r 2 s 52
   2042  ( 226, -209 , 3049 )   ( 226, 6687759 , 3049 )  p 4 q 26 r 19 s 51
   2062  ( 31, -29 , 514 )   ( 31, 1123819 , 514 )  p 6 q 26 r 21 s 51
   2090  ( 487, -419 , 3070 )   ( 487, 7434549 , 3070 )  p 8 q 26 r 38 s 48
   2114  ( -31, 41 , 128 )   ( 357297, 41 , 128 )  p 23 q 23 r 5 s 53
   2117  ( -459, 599 , 1979 )   ( 5458085, 599 , 1979 )  p 38 q 15 r 46 s 46
   2129  ( -159, 173 , 2117 )   ( 4875569, 173 , 2117 )  p 16 q 25 r 20 s 52
   2162  ( -104, 109 , 3241 )   ( 7242804, 109 , 3241 )  p 17 q 25 r 29 s 51
   2170  ( 137, -121 , 1070 )   ( 137, 2619311 , 1070 )  p 12 q 26 r 42 s 48
   2189  ( -79, 83 , 6569 )   ( 14561307, 83 , 6569 )  p 2 q 27 r 2 s 54
   2197  ( -371, 451 , 2119 )   ( 5646661, 451 , 2119 )  p 18 q 25 r 6 s 54
   2210  ( 871, -680 , 3127 )   ( 871, 8836260 , 3127 )  p 5 q 27 r 47 s 47
   2222  ( 149, -136 , 1655 )   ( 149, 4008624 , 1655 )  p 14 q 26 r 41 s 49
   2237  ( -631, 939 , 1931 )   ( 6420821, 939 , 1931 )  p 34 q 19 r 14 s 54
   2269  ( -737, 1297 , 1711 )   ( 6825889, 1297 , 1711 )  p 42 q 13 r 18 s 54
   2282  ( -482, 565 , 3343 )   ( 8918538, 565 , 3343 )  p 16 q 26 r 7 s 55
   2285  ( 179, -165 , 2273 )   ( 179, 5602985 , 2273 )  p 10 q 27 r 32 s 52
   2305  ( -767, 1489 , 1585 )   ( 7086337, 1489 , 1585 )  p 48 q 1 r 48 s 48
   2309  ( -667, 1011 , 1967 )   ( 6876869, 1011 , 1967 )  p 38 q 17 r 22 s 54
   2354  ( -41, 43 , 3532 )   ( 8415591, 43 , 3532 )  p 2 q 28 r 2 s 56
   2357  ( -1483, 1913 , 6647 )   ( 20177403, 1913 , 6647 )  p 22 q 25 r 4 s 56
   2369  ( -583, 801 , 2153 )   ( 6998609, 801 , 2153 )  p 28 q 23 r 8 s 56
   2402  ( -632, 775 , 3463 )   ( 10180308, 775 , 3463 )  p 23 q 25 r 14 s 56
   2414  ( 221, -196 , 1787 )   ( 221, 4847508 , 1787 )  p 8 q 28 r 35 s 53
   2426  ( -107, 118 , 1203 )   ( 3204853, 118 , 1203 )  p 20 q 26 r 25 s 55
   2450  ( -269, 292 , 3655 )   ( 9670419, 292 , 3655 )  p 10 q 28 r 7 s 57
   2474  ( -86, 93 , 1231 )   ( 3275662, 93 , 1231 )  p 17 q 27 r 22 s 56
   2477  ( -1483, 1883 , 7037 )   ( 22096323, 1883 , 7037 )  p 34 q 21 r 34 s 54
   2494  ( -16, 17 , 311 )   ( 818048, 17 , 311 )  p 12 q 28 r 15 s 57
   2501  ( -2497, 4853 , 5153 )   ( 25027503, 4853 , 5153 )  p 50 q 1 r 50 s 50
   2510  ( -93, 110 , 611 )   ( 1809803, 110 , 611 )  p 22 q 26 r 17 s 57
   2522  ( 187, -162 , 1237 )   ( 187, 3591490 , 1237 )  p 1 q 29 r 26 s 56
   2525  ( -85, 89 , 7577 )   ( 19356735, 89 , 7577 )  p 2 q 29 r 2 s 58
   2549  ( -547, 711 , 2387 )   ( 7897349, 711 , 2387 )  p 26 q 25 r 10 s 58
   2570  ( 69, -65 , 1282 )   ( 69, 3472135 , 1282 )  p 7 q 29 r 23 s 57
   2573  ( -491, 1207 , 829 )   ( 5239119, 1207 , 829 )  p 50 q 5 r 14 s 58
   2602  ( 281, -229 , 1250 )   ( 281, 3983891 , 1250 )  p 9 q 29 r 51 s 51
   2609  ( -1417, 1751 , 7499 )   ( 24134667, 1751 , 7499 )  p 32 q 23 r 32 s 56
   2629  ( -653, 901 , 2383 )   ( 8634289, 901 , 2383 )  p 42 q 17 r 42 s 54
   2642  ( -239, 256 , 3949 )   ( 11109849, 256 , 3949 )  p 11 q 29 r 11 s 59
   2645  ( -877, 1869 , 1655 )   ( 9321857, 1869 , 1655 )  p 50 q 7 r 22 s 58
   2690  ( 124, -113 , 1335 )   ( 124, 3924823 , 1335 )  p 13 q 29 r 41 s 55
   2702  ( -22, 23 , 2027 )   ( 5539122, 23 , 2027 )  p 2 q 30 r 2 s 60
   2705  ( -2161, 7187 , 3095 )   ( 27814971, 7187 , 3095 )  p 52 q 1 r 4 s 60
   2714  ( 249, -209 , 1318 )   ( 249, 4253047 , 1318 )  p 4 q 30 r 38 s 56
   2717  ( -859, 965 , 8051 )   ( 24497331, 965 , 8051 )  p 14 q 29 r 8 s 60
   2750  ( -140, 163 , 1009 )   ( 3223140, 163 , 1009 )  p 20 q 28 r 14 s 60
   2765  ( -781, 1163 , 2385 )   ( 9811001, 1163 , 2385 )  p 38 q 21 r 16 s 60
   2798  ( -11, 12 , 139 )   ( 422509, 12 , 139 )  p 10 q 30 r 5 s 61
   2801  ( -2617, 4283 , 6743 )   ( 30886443, 4283 , 6743 )  p 44 q 17 r 20 s 60
   2810  ( 505, -449 , 4162 )   ( 505, 13114719 , 4162 )  p 17 q 29 r 53 s 53
   2834  ( -668, 799 , 4123 )   ( 13949616, 799 , 4123 )  p 22 q 28 r 13 s 61
   2845  ( -185, 199 , 2833 )   ( 8626225, 199 , 2833 )  p 18 q 29 r 24 s 60
   2882  ( 415, -377 , 4288 )   ( 415, 13554423 , 4288 )  p 1 q 31 r 19 s 61
   2885  ( -91, 95 , 8657 )   ( 25249611, 95 , 8657 )  p 2 q 31 r 2 s 62
   2897  ( 995, -889 , 8591 )   ( 995, 27771531 , 8591 )  p 4 q 31 r 28 s 60
   2917  ( 757, -593 , 2755 )   ( 757, 10245097 , 2755 )  p 6 q 31 r 54 s 54
   2926  ( -71, 79 , 724 )   ( 2349649, 79 , 724 )  p 24 q 28 r 30 s 60
   2954  ( 127, -122 , 4429 )   ( 127, 13458546 , 4429 )  p 16 q 30 r 37 s 59
   3002  ( 175, -167 , 4498 )   ( 175, 14028513 , 4498 )  p 11 q 31 r 29 s 61
   3005  ( -1021, 1157 , 8885 )   ( 30177231, 1157 , 8885 )  p 22 q 29 r 22 s 62
   3025  ( 205, -191 , 3013 )   ( 205, 9734641 , 3013 )  p 12 q 31 r 36 s 60
   3026  ( 28, -27 , 1513 )   ( 28, 4663093 , 1513 )  p 26 q 28 r 55 s 55
   3029  ( -859, 953 , 8999 )   ( 30145467, 953 , 8999 )  p 34 q 25 r 52 s 56
   3050  ( -71, 75 , 1522 )   ( 4870921, 75 , 1522 )  p 13 q 31 r 17 s 63
   3074  ( -47, 49 , 4612 )   ( 14327961, 49 , 4612 )  p 2 q 32 r 2 s 64
   3077  ( -361, 411 , 3029 )   ( 10585241, 411 , 3029 )  p 14 q 31 r 4 s 64
   3098  ( -45, 53 , 302 )   ( 1099835, 53 , 302 )  p 20 q 30 r 10 s 64
   3134  ( 19, -18 , 391 )   ( 19, 1284958 , 391 )  p 8 q 32 r 25 s 63
   3137  ( -757, 827 , 9347 )   ( 31916595, 827 , 9347 )  p 16 q 31 r 16 s 64
   3170  ( -440, 487 , 4711 )   ( 16478100, 487 , 4711 )  p 10 q 32 r 1 s 65
   3173  ( -623, 1117 , 1411 )   ( 8021967, 1117 , 1411 )  p 50 q 15 r 20 s 64
   3182  ( -412, 503 , 2297 )   ( 8910012, 503 , 2297 )  p 22 q 30 r 7 s 65
   3185  ( -615, 773 , 3029 )   ( 12109985, 773 , 3029 )  p 40 q 23 r 44 s 60
   3242  ( -167, 187 , 1602 )   ( 5800105, 187 , 1602 )  p 19 q 31 r 17 s 65
   3250  ( -175, 197 , 1604 )   ( 5853425, 197 , 1604 )  p 27 q 29 r 33 s 63
   3269  ( -97, 101 , 9809 )   ( 32395887, 101 , 9809 )  p 2 q 33 r 2 s 66
   3277  ( -1073, 2425 , 1927 )   ( 14262577, 2425 , 1927 )  p 54 q 11 r 6 s 66
   3314  ( 369, -299 , 1588 )   ( 369, 6485797 , 1588 )  p 7 q 33 r 53 s 59
   3317  ( -2659, 3815 , 8801 )   ( 41849931, 3815 , 8801 )  p 38 q 25 r 14 s 66
   3326  ( -22, 23 , 623 )   ( 2148618, 23 , 623 )  p 16 q 32 r 25 s 65
   3329  ( 851, -781 , 9923 )   ( 851, 35867427 , 9923 )  p 8 q 33 r 32 s 64
   3349  ( -947, 1411 , 2887 )   ( 14394949, 1411 , 2887 )  p 42 q 23 r 18 s 66
   3362  ( -7, 8 , 57 )   ( 218537, 8 , 57 )  p 29 q 29 r 34 s 64
   3365  ( 2207, -1795 , 9689 )   ( 2207, 40031835 , 9689 )  p 10 q 33 r 58 s 58
   3374  ( -523, 671 , 2384 )   ( 10308093, 671 , 2384 )  p 26 q 30 r 5 s 67
   3377  ( -2833, 4187 , 8783 )   ( 43802523, 4187 , 8783 )  p 52 q 15 r 52 s 60
   3389  ( -3187, 8099 , 5261 )   ( 45280227, 8099 , 5261 )  p 58 q 3 r 22 s 66
   3410  ( -260, 309 , 1657 )   ( 6704320, 309 , 1657 )  p 23 q 31 r 13 s 67
   3434  ( 109, -102 , 1711 )   ( 109, 6249982 , 1711 )  p 13 q 33 r 38 s 64
   3437  ( -991, 1499 , 2931 )   ( 15226901, 1499 , 2931 )  p 46 q 21 r 26 s 66
   3470  ( -25, 26 , 2603 )   ( 9122655, 26 , 2603 )  p 2 q 34 r 2 s 68
   3473  ( -617, 955 , 1747 )   ( 9384663, 955 , 1747 )  p 40 q 25 r 4 s 68
   3482  ( 502, -383 , 1623 )   ( 502, 7399633 , 1623 )  p 4 q 34 r 59 s 59
   3509  ( -1153, 2093 , 2571 )   ( 16367129, 2093 , 2571 )  p 58 q 7 r 50 s 62
   3530  ( 886, -755 , 5167 )   ( 886, 21367845 , 5167 )  p 8 q 34 r 47 s 63
   3557  ( -315, 347 , 3527 )   ( 13780133, 347 , 3527 )  p 26 q 31 r 34 s 66
   3569  ( -1159, 2037 , 2693 )   ( 16882529, 2037 , 2693 )  p 52 q 17 r 20 s 68
   3601  ( -1199, 2341 , 2461 )   ( 17293201, 2341 , 2461 )  p 60 q 1 r 60 s 60
   3629  ( -1987, 2459 , 10421 )   ( 46743507, 2459 , 10421 )  p 38 q 27 r 38 s 66
   3662  ( -39, 41 , 914 )   ( 3497249, 41 , 914 )  p 14 q 34 r 19 s 69
   3665  ( -211, 225 , 3653 )   ( 14213081, 225 , 3653 )  p 20 q 33 r 28 s 68
   3674  ( 313, -266 , 1791 )   ( 313, 7730362 , 1791 )  p 1 q 35 r 35 s 67
   3677  ( -103, 107 , 11033 )   ( 40961883, 107 , 11033 )  p 2 q 35 r 2 s 70
   3682  ( 137, -127 , 1832 )   ( 137, 7249985 , 1832 )  p 3 q 35 r 21 s 69
   3698  ( 236, -193 , 1067 )   ( 236, 4818687 , 1067 )  p 5 q 35 r 50 s 64
   3701  ( -149, 241 , 391 )   ( 2339181, 241 , 391 )  p 46 q 23 r 10 s 70
   3709  ( 577, -497 , 3631 )   ( 577, 15607969 , 3631 )  p 6 q 35 r 42 s 66
   3722  ( 46, -39 , 259 )   ( 46, 1135249 , 259 )  p 7 q 35 r 47 s 65
   3725  ( -1231, 2675 , 2283 )   ( 18469781, 2675 , 2283 )  p 58 q 11 r 14 s 70
   3749  ( 43, -41 , 1021 )   ( 43, 3988977 , 1021 )  p 22 q 33 r 52 s 64
   3754  ( 83, -79 , 1874 )   ( 83, 7346657 , 1874 )  p 9 q 35 r 27 s 69
   3790  ( 106, -95 , 937 )   ( 106, 3953065 , 937 )  p 18 q 34 r 57 s 63
   3794  ( -129, 139 , 1888 )   ( 7690567, 139 , 1888 )  p 11 q 35 r 7 s 71
   3845  ( -3535, 9407 , 5669 )   ( 57970755, 9407 , 5669 )  p 62 q 1 r 26 s 70
   3854  ( -163, 198 , 929 )   ( 4343621, 198 , 929 )  p 28 q 32 r 17 s 71
   3869  ( 33, -31 , 551 )   ( 33, 2259527 , 551 )  p 14 q 35 r 40 s 68
   3889  ( -1259, 2197 , 2953 )   ( 20029609, 2197 , 2953 )  p 48 q 23 r 0 s 72
   3890  ( -53, 55 , 5836 )   ( 22916043, 55 , 5836 )  p 2 q 36 r 2 s 72
   3898  ( -41, 46 , 385 )   ( 1680079, 46 , 385 )  p 15 q 35 r 6 s 72
   3905  ( -2653, 3515 , 10859 )   ( 56133123, 3515 , 10859 )  p 32 q 31 r 8 s 72
   3965  ( -745, 929 , 3783 )   ( 18683825, 929 , 3783 )  p 38 q 29 r 34 s 70
   3970  ( -355, 437 , 1904 )   ( 9294125, 437 , 1904 )  p 30 q 32 r 18 s 72
   3989  ( -1129, 1683 , 3437 )   ( 20424809, 1683 , 3437 )  p 46 q 25 r 20 s 72
   3997  ( 367, -335 , 3967 )   ( 367, 17323333 , 3967 )  p 18 q 35 r 54 s 66
   4034  ( 148, -143 , 6049 )   ( 148, 24998841 , 6049 )  p 19 q 35 r 43 s 69
   4037  ( -555, 647 , 3947 )   ( 18546533, 647 , 3947 )  p 34 q 31 r 38 s 70
   4085  ( -1177, 1779 , 3485 )   ( 21504617, 1779 , 3485 )  p 50 q 23 r 28 s 72
   4094  ( -13, 14 , 191 )   ( 839283, 14 , 191 )  p 32 q 32 r 50 s 68
   4097  ( -4093, 8003 , 8387 )   ( 67153923, 8003 , 8387 )  p 64 q 1 r 64 s 64
   4109  ( -109, 113 , 12329 )   ( 51124287, 113 , 12329 )  p 2 q 37 r 2 s 74
   4114  ( 236, -211 , 2033 )   ( 236, 9334877 , 2033 )  p 3 q 37 r 30 s 72
   4130  ( 376, -353 , 6175 )   ( 376, 27055983 , 6175 )  p 5 q 37 r 23 s 73
   4154  ( 319, -302 , 6217 )   ( 319, 27150846 , 6217 )  p 7 q 37 r 25 s 73
   4157  ( -529, 609 , 4079 )   ( 19488545, 609 , 4079 )  p 22 q 35 r 14 s 74
   4202  ( -146, 151 , 6301 )   ( 27111450, 151 , 6301 )  p 23 q 35 r 41 s 71
   4205  ( 1595, -1411 , 12437 )   ( 1595, 59005971 , 12437 )  p 10 q 37 r 46 s 70
   4226  ( -461, 499 , 6304 )   ( 28749939, 499 , 6304 )  p 11 q 37 r 5 s 75
   4229  ( -4171, 7583 , 9281 )   ( 71322027, 7583 , 9281 )  p 58 q 17 r 22 s 74
   4250  ( -386, 477 , 2035 )   ( 10676386, 477 , 2035 )  p 28 q 34 r 11 s 75
   4274  ( 283, -249 , 2104 )   ( 283, 10202287 , 2104 )  p 13 q 37 r 53 s 69
   4289  ( -463, 521 , 4233 )   ( 20390369, 521 , 4233 )  p 32 q 33 r 40 s 72
   4298  ( 71, -67 , 1286 )   ( 71, 5832453 , 1286 )  p 25 q 35 r 61 s 67
   4301  ( 1373, -1237 , 12773 )   ( 1373, 60843183 , 12773 )  p 14 q 37 r 50 s 70
   4330  ( 11, -10 , 113 )   ( 11, 536930 , 113 )  p 0 q 38 r 21 s 75
   4334  ( -28, 29 , 3251 )   ( 14215548, 29 , 3251 )  p 2 q 38 r 2 s 76
   4337  ( -4057, 10427 , 6647 )   ( 74053995, 10427 , 6647 )  p 64 q 9 r 4 s 76
   4349  ( -2287, 2801 , 12539 )   ( 66715947, 2801 , 12539 )  p 26 q 35 r 8 s 76
   4357  ( -1451, 2839 , 2971 )   ( 25315621, 2839 , 2971 )  p 66 q 1 r 66 s 66
   4373  ( -127, 173 , 479 )   ( 2851323, 173 , 479 )  p 50 q 25 r 44 s 72
   4402  ( -253, 287 , 2168 )   ( 10807163, 287 , 2168 )  p 27 q 35 r 27 s 75
   4430  ( 45, -43 , 1106 )   ( 45, 5098973 , 1106 )  p 10 q 38 r 29 s 75
   4465  ( -1355, 2149 , 3673 )   ( 25996585, 2149 , 3673 )  p 60 q 17 r 48 s 72
   4490  ( -446, 565 , 2127 )   ( 12087526, 565 , 2127 )  p 32 q 34 r 13 s 77
   4514  ( -248, 259 , 6763 )   ( 31697556, 259 , 6763 )  p 29 q 35 r 50 s 72
   4517  ( -1353, 2117 , 3755 )   ( 26525177, 2117 , 3755 )  p 62 q 15 r 58 s 70
   4526  ( 107, -103 , 3392 )   ( 107, 15836577 , 3392 )  p 14 q 38 r 35 s 75
   4562  ( 1105, -947 , 6688 )   ( 1105, 35552613 , 6688 )  p 1 q 39 r 37 s 75
   4565  ( -115, 119 , 13697 )   ( 63070155, 119 , 13697 )  p 2 q 39 r 2 s 78
   4573  ( -187, 239 , 863 )   ( 5039633, 239 , 863 )  p 30 q 35 r 6 s 78
   4589  ( -1867, 2171 , 13469 )   ( 71773827, 2171 , 13469 )  p 22 q 37 r 10 s 78
   4622  ( -29, 34 , 199 )   ( 1076955, 34 , 199 )  p 34 q 34 r 34 s 76
   4625  ( 3539, -2785 , 13127 )   ( 3539, 77083035 , 13127 )  p 8 q 39 r 68 s 68
   4637  ( -2743, 3467 , 13193 )   ( 77255163, 3467 , 13193 )  p 46 q 29 r 46 s 74
   4645  ( -1145, 1573 , 4219 )   ( 26904985, 1573 , 4219 )  p 42 q 31 r 18 s 78
   4682  ( -198, 217 , 2323 )   ( 11892478, 217 , 2323 )  p 11 q 39 r 1 s 79
   4685  ( -1327, 1979 , 4035 )   ( 28176917, 1979 , 4035 )  p 50 q 27 r 22 s 78
   4730  ( 82, -71 , 535 )   ( 82, 2918481 , 535 )  p 13 q 39 r 58 s 72
   4762  ( 35, -34 , 2381 )   ( 35, 11505026 , 2381 )  p 33 q 35 r 69 s 69
   4789  ( -1379, 2083 , 4087 )   ( 29549509, 2083 , 4087 )  p 54 q 25 r 30 s 78
   4802  ( -59, 61 , 7204 )   ( 34886589, 61 , 7204 )  p 2 q 40 r 2 s 80
   4814  ( 78, -73 , 1199 )   ( 78, 6147551 , 1199 )  p 4 q 40 r 23 s 79
   4817  ( -459, 509 , 4769 )   ( 25424585, 509 , 4769 )  p 16 q 39 r 8 s 80
   4850  ( -584, 637 , 7225 )   ( 38131284, 637 , 7225 )  p 17 q 39 r 14 s 80
   4862  ( 41, -36 , 299 )   ( 41, 1653116 , 299 )  p 8 q 40 r 46 s 76
   4865  ( -3421, 4595 , 13427 )   ( 87680451, 4595 , 13427 )  p 40 q 33 r 16 s 80
   4901  ( -1429, 4151 , 2181 )   ( 31034561, 4151 , 2181 )  p 70 q 1 r 20 s 80
   4910  ( -109, 120 , 1217 )   ( 6564779, 120 , 1217 )  p 32 q 36 r 43 s 77
   4922  ( -1037, 1213 , 7210 )   ( 41459043, 1213 , 7210 )  p 19 q 39 r 1 s 81
   4925  ( -4645, 7697 , 11729 )   ( 95677695, 7697 , 11729 )  p 62 q 19 r 38 s 78
   4942  ( -74, 79 , 1231 )   ( 6474094, 79 , 1231 )  p 12 q 40 r 9 s 81
   4949  ( -3721, 5153 , 13421 )   ( 91926447, 5153 , 13421 )  p 58 q 23 r 58 s 74
   4994  ( -332, 349 , 7477 )   ( 39083376, 349 , 7477 )  p 14 q 40 r 17 s 81
   4997  ( -3583, 4853 , 13727 )   ( 92847843, 4853 , 13727 )  p 46 q 31 r 28 s 80
   5005  ( -377, 409 , 4975 )   ( 26947297, 409 , 4975 )  p 30 q 37 r 42 s 78
   5009  ( -1243, 1713 , 4541 )   ( 31327529, 1713 , 4541 )  p 56 q 25 r 52 s 76
   5042  ( 817, -603 , 2308 )   ( 817, 15756853 , 2308 )  p 1 q 41 r 71 s 71
   5045  ( -121, 125 , 15137 )   ( 76996911, 125 , 15137 )  p 2 q 41 r 2 s 82
   5054  ( -23, 24 , 631 )   ( 3310393, 24 , 631 )  p 16 q 40 r 23 s 81
   5057  ( 521, -471 , 5009 )   ( 521, 27965681 , 5009 )  p 4 q 41 r 32 s 80
   5090  ( 652, -599 , 7585 )   ( 652, 41926929 , 7585 )  p 7 q 41 r 34 s 80
Tue Jul  7 10:02:21 PDT 2020

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

$\endgroup$
1
$\begingroup$

Well, this may be discussed in terms of Vieta Jumping. However, for each legal $k,$ all primitive solutions may be found by a finite number of Pythagorean Triple type parametizations. I did 5090, the largest $k$ in the answer with just $k$ and solutions:

The first of several matrices below means $$ x = 1837 u^2 + 4226 uv + 1549 v^2 \; , \; \; y = 1549 u^2 - 1128 uv -840 v^2 \; , \; \; z = -840 u^2 -552uv + 1837v^2 $$ which solves $$ x^2 + y^2 + z^2 = 5090 ( yz + zx +xy). $$ To get primitive solutions we are taking $u,v$ coprime. Then, if $\gcd(x,y,z) > 1$ we discard that triple.

   1837   4226   1549
   1549  -1128   -840
   -840   -552   1837

   1897   4208   1480
   1480  -1248   -831
   -831   -414   1897

   2085   4098   1237
   1237  -1624   -776
   -776     72   2085

   2319   3786    847
    847  -2092   -620
   -620    852   2319

   2355   3708    772
    772  -2164   -581
   -581   1002   2355

   2449   3426    537
    537  -2352   -440
   -440   1472   2449

   2455   3402    519
    519  -2364   -428
   -428   1508   2455

   2539   2796    132
    132  -2532   -125
   -125   2282   2539
$\endgroup$

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