# What are all possible positive integers $k$ such that $k=\frac{a^2+b^2+c^2}{bc+ca+ab}$ for some positive integers $a$, $b$, and $c$?

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.

• 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 $17,26,62$.
– zwim
Commented Jul 6, 2020 at 20:46
• @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. Commented Jul 6, 2020 at 20:59
• tio.run/##XZDLasQwDEX3/…
– zwim
Commented Jul 6, 2020 at 21:55
• Commented Jul 7, 2020 at 3:43
• mathoverflow.net/questions/225781/… Commented Jul 7, 2020 at 4:25

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

• Do you have a reference for why $k-1$ and $k+2$ are expressible in the form $u^2+3v^2$? Thanks! Commented Jul 7, 2020 at 2:25
• @Batominovski I put several answers at math.stackexchange.com/questions/1134075/… Commented Jul 7, 2020 at 2:29

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

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

[$$a,b,c$$]

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

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


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

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