Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
2026-03-25 22:30:55.1774477855
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Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.
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Generally, you can use the General formula. Solving a Diophantine equation of the form $x^2 = ay^2 + byz + cz^2$ with the constants $a, b, c$ given and $x,y,z$ positive integers
$$ax^2+bxy+cy^2=jz^2$$
For our case.
$$x^2+xy+y^2=3z^2$$
$$\sqrt{b^2+4a(j-c)}=3$$
$$\sqrt{j(a+b+c)}=3$$
There is a solution. The formula remains only to substitute.
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The solutions can be parametrized very nicely, three binary quadratic forms. The approach of Fricke and Klein (1897) guarantees a solution with a finite number of such parametrizations, here there is just one needed. I had it report just those with $xy > 0,$ each of these leads to two others with $xy < 0,$ those being $(-x,x+y)$ and $(x+y,-y).$ You can just negate both $x,y$ to get all positive. Negating all three leads, in general, to six solutions for each $r$ prime, $r \equiv 1 \pmod 6.$ More if $r$ is the product of such primes, compare $91 = 7 \cdot 13.$ $$ r = u^2 - uv + v^2, \; \; x = u^2 + 2uv - 2 v^2, \; \; y = -2u^2 + 2uv+ v^2 $$
Let's see, My program to find the coefficients for such parametrizations is in C++,
jagy@phobeusjunior:~$ ./homothety_indef 1 1 -3 0 0 1 0 9 0 0 -9 0 2
....
-2 -2 1 transposed -2 1 -1
1 -2 -2 transposed -2 -2 -1
-1 -1 -1 transposed 1 -2 -1
...... apparently I chose the next one
1 2 -2 transposed 1 -2 1
-2 2 1 transposed 2 2 -1
1 -1 1 transposed -2 1 1
r x y u v r
1 1 1 1 1 1 = 1
7 -2 -11 2 -1 7 = 7
13 22 1 4 3 13 = 13
19 -11 -26 3 -2 19 = 19
31 46 13 6 5 31 = 31
37 -26 -47 4 -3 37 = 37
43 61 22 7 6 43 = 43
49 -23 -71 5 -3 49 = 7^2
61 -47 -74 5 -4 61 = 61
67 109 13 9 7 67 = 67
73 97 46 9 8 73 = 73
79 -11 -131 7 -3 79 = 79
91 118 61 10 9 91 = 7 * 13
91 -74 -107 6 -5 91 = 7 * 13
97 -2 -167 8 -3 97 = 97
103 157 37 11 9 103 = 103
109 -71 -143 7 -5 109 = 109
127 -107 -146 7 -6 127 = 127
133 166 97 12 11 133 = 7 * 19
133 -23 -218 9 -4 133 = 7 * 19
139 229 22 13 10 139 = 139
151 -59 -227 9 -5 151 = 151
157 193 118 13 12 157 = 157
163 262 37 14 11 163 = 163
169 -146 -191 8 -7 169 = 13^2
181 313 1 15 11 181 = 181
193 -143 -239 9 -7 193 = 193
199 277 109 15 13 199 = 199
211 253 166 15 14 211 = 211
217 -191 -242 9 -8 217 = 7 * 31
217 334 73 16 13 217 = 7 * 31
223 -83 -338 11 -6 223 = 223
229 -26 -383 12 -5 229 = 229
241 286 193 16 15 241 = 241
247 -131 -347 11 -7 247 = 13 * 19
247 373 94 17 14 247 = 13 * 19
259 -11 -443 13 -5 259 = 7 * 37
259 349 157 17 15 259 = 7 * 37
271 -242 -299 10 -9 271 = 271
277 -122 -407 12 -7 277 = 277
283 -59 -458 13 -6 283 = 283
301 -239 -359 11 -9 301 = 7 * 43
301 481 73 19 15 301 = 7 * 43
307 358 253 18 17 307 = 307
313 457 142 19 16 313 = 313
331 -299 -362 11 -10 331 = 331
337 -167 -482 13 -8 337 = 337
343 397 286 19 18 343 = 7^3
349 502 169 20 17 349 = 349
361 601 46 21 16 361 = 19^2
367 -227 -491 13 -9 367 = 367
373 577 121 21 17 373 = 373
379 -83 -611 15 -7 379 = 379
397 -362 -431 12 -11 397 = 397
403 -218 -563 14 -9 403 = 13 * 31
403 517 277 21 19 403 = 13 * 31
409 -143 -626 15 -8 409 = 409
421 481 358 21 20 421 = 421
427 598 229 22 19 427 = 7 * 61
427 733 13 23 17 427 = 7 * 61
433 -359 -503 13 -11 433 = 433
439 709 94 23 18 439 = 439
457 -47 -767 17 -7 457 = 457
463 526 397 22 21 463 = 463
469 -431 -506 13 -12 469 = 7 * 67
469 649 262 23 20 469 = 7 * 67
481 -194 -719 16 -9 481 = 13 * 37
481 766 121 24 19 481 = 13 * 37
487 613 349 23 21 487 = 487
499 -26 -851 18 -7 499 = 499
511 -347 -659 15 -11 511 = 7 * 73
511 853 61 25 19 511 = 7 * 73
523 -179 -803 17 -9 523 = 523
541 793 241 25 21 541 = 541
547 -506 -587 14 -13 547 = 547
553 -338 -743 16 -11 553 = 7 * 79
553 622 481 24 23 553 = 7 * 79
559 -251 -818 17 -10 559 = 13 * 43
559 757 334 25 22 559 = 13 * 43
571 886 181 26 21 571 = 571
577 -71 -962 19 -8 577 = 577
589 1009 22 27 20 589 = 19 * 31
589 -503 -671 15 -13 589 = 19 * 31
601 673 526 25 24 601 = 601
607 814 373 26 23 607 = 607
613 -143 -983 19 -9 613 = 613
619 949 214 27 22 619 = 619
631 -587 -674 15 -14 631 = 631
637 -407 -842 17 -12 637 = 7^2 * 13
637 913 313 27 23 637 = 7^2 * 13
643 -314 -923 18 -11 643 = 643
661 -122 -1079 20 -9 661 = 661
673 -23 -1154 21 -8 673 = 673
679 -491 -851 17 -13 679 = 7 * 97
679 829 517 27 25 679 = 7 * 97
691 -299 -1019 19 -11 691 = 691
703 1117 181 29 23 703 = 19 * 37
703 781 622 27 26 703 = 19 * 37
709 934 457 28 25 709 = 709
721 1081 286 29 24 721 = 7 * 103
721 -674 -767 16 -15 721 = 7 * 103
727 -482 -947 18 -13 727 = 727
733 -383 -1034 19 -12 733 = 733
739 1222 109 30 23 739 = 739
751 -179 -1202 21 -10 751 = 751
757 838 673 28 27 757 = 757
763 -74 -1283 22 -9 763 = 7 * 109
763 997 502 29 26 763 = 7 * 109
769 -671 -863 17 -15 769 = 769
787 949 613 29 27 787 = 787
793 1297 142 31 24 793 = 13 * 61
793 -263 -1223 21 -11 793 = 13 * 61
811 1261 253 31 25 811 = 811
817 -47 -1391 23 -9 817 = 19 * 43
817 -767 -866 17 -16 817 = 19 * 43
823 -563 -1058 19 -14 823 = 823
829 -458 -1151 20 -13 829 = 829
853 1177 481 31 27 853 = 853
859 -131 -1418 23 -10 859 = 859
871 -659 -1067 19 -15 871 = 13 * 67
871 958 781 30 29 871 = 13 * 67
877 1129 598 31 28 877 = 877
883 -443 -1259 21 -13 883 = 883
889 1294 409 32 27 889 = 7 * 127
889 1489 97 33 25 889 = 7 * 127
907 1453 214 33 26 907 = 907
919 -866 -971 18 -17 919 = 919
931 1021 838 31 30 931 = 7^2 * 19
931 1606 13 34 25 931 = 7^2 * 19
937 1198 649 32 29 937 = 937
949 1369 454 33 28 949 = 13 * 73
949 -311 -1466 23 -12 949 = 13 * 73
961 -194 -1559 24 -11 961 = 31^2
967 1534 253 34 27 967 = 967
r x y u v r
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A few questions/answers that display this method
$x^2+y^2+z^2=5(xy+yz+zx)$ -- Is this all solutions?
Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution
The next one shows the English excerpt from Plesken that describes the central fact from F+K(1897):
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$$c\,{{x}^{2}}+dyx+g\,{{y}^{2}}=ab$$ $$\downarrow$$ $$g\,{{\left( psc+hkc+dks\right) }^{2}}+c\,{{\left( hpc-gks\right) }^{2}}+d\,\left( hpc-gks\right) \,\left( psc+hkc+dks\right) =$$ $$=c\,\left( {{h}^{2}}c+g\,{{s}^{2}}+dhs\right) \,\left( {{p}^{2}}c+dkp+g\,{{k}^{2}}\right) $$ $$\downarrow$$ $$c\,{{x}^{2}}+dyx+g\,{{y}^{2}}=a\,{{z}^{2}}$$ $$\downarrow$$ $$c\,{{\left( -cgp\,{{s}^{2}}-dgk\,{{s}^{2}}-2cghks+{{c}^{2}}\,{{h}^{2}}p\right) }^{2}}+$$ $$+g\,{{\left( cdp\,{{s}^{2}}-cgk\,{{s}^{2}}+{{d}^{2}}k\,{{s}^{2}}+2{{c}^{2}}hps+2cdhks+{{c}^{2}}\,{{h}^{2}}k\right) }^{2}}+$$ $$+d\,\left( cdp\,{{s}^{2}}-cgk\,{{s}^{2}}+{{d}^{2}}k\,{{s}^{2}}+2{{c}^{2}}hps+2cdhks+{{c}^{2}}\,{{h}^{2}}k\right)\cdot $$ $$\cdot\left( -cgp\,{{s}^{2}}-dgk\,{{s}^{2}}-2cghks+{{c}^{2}}\,{{h}^{2}}p\right)= $$ $$={{c}^{2}}\,\left( c\,{{p}^{2}}+dkp+g\,{{k}^{2}}\right) \,{{\left( g\,{{s}^{2}}+dhs+c\,{{h}^{2}}\right) }^{2}}$$
$$------------------------------$$
For $\;\;{{x}^{2}}+yx+{{y}^{2}}=3{{r}^{2}},\;\;$ $c=1,\;d=1,\;g=1,\;{{p}^{2}}+kp+{{k}^{2}}=3$
Two solutions:
$$r={{s}^{2}}+hs+{{h}^{2}},\;\;x=-{{s}^{2}}+2hs+2{{h}^{2}},\;\;y=2{{s}^{2}}+2hs-{{h}^{2}}$$
$$r={{s}^{2}}+hs+{{h}^{2}},\;\;x={{s}^{2}}-2hs-2{{h}^{2}},\;\;y={{s}^{2}}+4hs+{{h}^{2}}$$
There are infinitely many solutions to this Diophantine equation.
I change your variables. We have the Diophantine equation$$3(a^2+ab+b^2)=c^2$$Let's assume $c$ is not constant and find all possible solutions! $c=0$ gives $a=b=0$. W.L.O.G. suppose $c>0$. It's easy to see that $c=3d$ for some positive integer $d$. Equation becomes$$a^2+ab+b^2=3d^2$$For $a=b$ we get $a=b=\pm d$. W.L.O.G. suppose that $a>b$. Notice that$$a^2+ab+b^2 \equiv 0 \pmod 3$$It follows that $a^3 \equiv b^3 \pmod 3$ and $a \equiv b \pmod 3$. Let $a-b=3e$ for some positive integer $e$. Equation turns into$$b^2+3be+3e^2=d^2$$In order to get integer values for $b$ discriminant of this quadratic must be a perfect square, that is$$\Delta_{b}=-3e^2+4d^2=f^2$$For some integer $f$.
$f=0$ gives $\sqrt{3}=2\frac{d}{e}$, which is impossible. W.L.O.G. suppose that $f>0$. Therefore, we need to solve$$f^2+3e^2=4d^2$$In positive integers. Now this is of the form of extended Pythagorean equation $Ax^2+By^2=Cz^2$, which is widely studied. Even there are some questions here and here, which exactly discuss your particular case!
Once you find parametric forms of $f$, $e$ and $d$, you can substitute backwards and find parametric form of your original variables $a$, $b$ and $c$.