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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?

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  • $\begingroup$ The accepted answer is incorrect for the stated problem with "r constant". For that problem there are only a finite number of solutions. Finding the solutions, similar to the Pythagorean case $a(x^2 + y^2) = r^2$, is a problem of integer factorization in the ring of integers of a quadratic (and cyclotomic) field. $\endgroup$ – zyx Jun 8 '16 at 18:05
  • $\begingroup$ I've edited the title with notation that ever so slightly conflicts with the one in the question, but better represents the actual material in the question and any solutions that might appear. Just "for your information" in case you prefer to reverse the edit. @qqq $\endgroup$ – zyx Jun 8 '16 at 18:20
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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 $c$.

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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  

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

A few questions/answers that display this method

$x^2+y^2+z^2=5(xy+yz+zx)$ -- Is this all solutions?

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

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):

Describe the rational points on $3x^2 + y^2 = 4$

Question about non-degenerate polynomials, and a proof

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  • $\begingroup$ Can you share your source code or provide a reference for the F&K method? $\endgroup$ – qqq Jun 10 '16 at 16:13
  • $\begingroup$ @qqq I put some links at the end. $\endgroup$ – Will Jagy Jun 10 '16 at 17:26
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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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$$c\,{{x}^{2}}+dyx+g\,{{y}^{2}}=ab$$ $$\downarrow$$ $$g\,{{\left( psc+hkc+dks\right) }^{2}}+c\,{{\left( hpc-gks\right) }^{2}}+d\,\left( hpc-gks\right) \,\left( psc+hkc+dks\right) =$$ $$=c\,\left( {{h}^{2}}c+g\,{{s}^{2}}+dhs\right) \,\left( {{p}^{2}}c+dkp+g\,{{k}^{2}}\right) $$ $$\downarrow$$ $$c\,{{x}^{2}}+dyx+g\,{{y}^{2}}=a\,{{z}^{2}}$$ $$\downarrow$$ $$c\,{{\left( -cgp\,{{s}^{2}}-dgk\,{{s}^{2}}-2cghks+{{c}^{2}}\,{{h}^{2}}p\right) }^{2}}+$$ $$+g\,{{\left( cdp\,{{s}^{2}}-cgk\,{{s}^{2}}+{{d}^{2}}k\,{{s}^{2}}+2{{c}^{2}}hps+2cdhks+{{c}^{2}}\,{{h}^{2}}k\right) }^{2}}+$$ $$+d\,\left( cdp\,{{s}^{2}}-cgk\,{{s}^{2}}+{{d}^{2}}k\,{{s}^{2}}+2{{c}^{2}}hps+2cdhks+{{c}^{2}}\,{{h}^{2}}k\right)\cdot $$ $$\cdot\left( -cgp\,{{s}^{2}}-dgk\,{{s}^{2}}-2cghks+{{c}^{2}}\,{{h}^{2}}p\right)= $$ $$={{c}^{2}}\,\left( c\,{{p}^{2}}+dkp+g\,{{k}^{2}}\right) \,{{\left( g\,{{s}^{2}}+dhs+c\,{{h}^{2}}\right) }^{2}}$$

$$------------------------------$$

For $\;\;{{x}^{2}}+yx+{{y}^{2}}=3{{r}^{2}},\;\;$ $c=1,\;d=1,\;g=1,\;{{p}^{2}}+kp+{{k}^{2}}=3$

Two solutions:

$$r={{s}^{2}}+hs+{{h}^{2}},\;\;x=-{{s}^{2}}+2hs+2{{h}^{2}},\;\;y=2{{s}^{2}}+2hs-{{h}^{2}}$$

$$r={{s}^{2}}+hs+{{h}^{2}},\;\;x={{s}^{2}}-2hs-2{{h}^{2}},\;\;y={{s}^{2}}+4hs+{{h}^{2}}$$

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