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I want to find solution in $\mathbb{Z}$ to the following quadratic Diophantene equation:

$$na^2 + kb^2 = c^2$$

where $n,k,a,b,c \in \mathbb{Z}$, $n,k > 0$ and $(n,k) = 1$

I know that for some this won't work for all values of $n$ and $k$ that satisfy the upper condition, but anyone know what will be the condition, so the Diophantene equation will have integer solution.

My second question is: "Let the equation have integer solution for some values of $n$ and $k$, is there infinite amount of them or their numbers is finite?"

And my third and last question is how to solve this type of equation. I'll give you one specific example:

$$19a^2 + 5b^2 = c^2$$

I don't know whether this equation has integer solution, but I hope it have because I found it in a book and it says: "Solve this equation in $\mathbb{Z}$". And if there is a infinite amount of solution is there any general formula to generate them?

I decided to try using congruence, because I suspect there is a "by-the-book" way to solve this, so we must make some restriction, but I didn't make a progress.

As I said I suspect there is a "by-the-book" way to solve this type of equation, but any method (e.g. congruence relations) would be helpful.

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  • 1
    $\begingroup$ You can do this yourself. Start with the rational point $\left( \frac{2}{11}, \frac{3}{11} \right)$ on the ellipse $19 x^2 + 5 y^2 = 1.$ draw lines of rational slope through that base point; see math.stackexchange.com/questions/249735/… for an example of the method. $\endgroup$ – Will Jagy Sep 4 '13 at 1:58
  • $\begingroup$ I edited in a second formula that gives the primitive solutions with $c$ even and $a$ odd. $\endgroup$ – Will Jagy Sep 4 '13 at 18:35
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For $$ 19 a^2 + 5 b^2 = c^2, $$ there are infinitely many.

Let $u,v$ be any integers, then take $$ a = 2 u^2 + 2 u v - 2 v^2, \; \; b = u^2 - 8 u v - 3 v^2, \; \; c = 9 u^2 + 4 u v + 11 v^2. $$ If $\gcd(u,v) = 1$ and $u,v$ are not both odd, then $a,b,c$ might be relatively prime.

To get both $a,b$ odd while $c$ is the one that comes out even, $$ a = u^2 + 4 u v - v^2, \; \; b = -5u^2 + 4 u v + 3 v^2, \; \; c = 12 u^2 - 2 u v + 8 v^2. $$

Those two recipes together give you all primitive solutions, that is $\gcd(a,b,c) = 1.$ To get other solutions, multiply through by a number. For example, these recipes do not give $a=14,b=21,c=77.$ However, they do give $a=2,b=3,c=11,$ then you multiply all by $7.$

The essential condition is that your form lie in the principal genus. The form $\langle 5,0,19 \rangle$ is indeed in the principal genus, of binary quadratic forms, for discriminant $-380.$ In the class group, it is the square of $\langle 9,4,11 \rangle$ and the square of $\langle 9,-4,11 \rangle.$ I learned from a large number of books; I recommend Binary Quadratic Forms by Duncan A. Buell, Binary Quadratic Forms by Buchmann and Vollmer, Primes of the Form $x^2 + n y^2$ by David A. Cox. Buell's book gives careful algorithms for composition, which is the topic here. Indeed, let's display that, it is how I solved the problem, $$ \langle 9,4,11 \rangle^2 = \langle 5,0,19 \rangle $$ in the group of equivalence classes of positive (primitive) forms of this discriminant.

I did not initially notice that there needed to be a way to generate primitive solutions with $a$ odd and $c$ even. I fiddled a bit, the result uses the "imprimitive" form $\langle 8,2,12 \rangle$ near the beginning of the output below. The class number is $8.$ There are four classes in the principal genus, the list below called "squares."

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./classGroup
Absolute value of discriminant? 
380
Impr 2 2 48
Impr 4 2 24
Impr 6 2 16
Impr 8 2 12
Impr 10 10 12
  class  number  8

 all  
( 1, 0, 95)
( 3, -2, 32)
( 3, 2, 32)
( 5, 0, 19)
( 8, -6, 13)
( 8, 6, 13)
( 9, -4, 11)
( 9, 4, 11)

 squares  
( 1, 0, 95)
( 5, 0, 19)
( 9, -4, 11)
( 9, 4, 11)


Discriminant       -380     h :    8     Squares :    4     
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Anyway, easy enough to print out just the PRIMITIVE solutions. I took a,b positive and up to 1100. I had to repeat the value of c as both the first and fourth column, to allow the computer to sort by size of c without extra work on my part.

19 a^2 + 5 b^2 = c^2

       c           a           b           c =  Factored
       8           1           3           8 = 2^3
       9           2           1           9 = 3^2
      11           2           3          11 = 11
      12           1           5          12 = 2^2 * 3
      39           2          17          39 = 3 * 13
      48          11           1          48 = 2^4 * 3
      52          11           9          52 = 2^2 * 13
      61           2          27          61 = 61
     101           2          45         101 = 101
     108          19          31         108 = 2^2 * 3^3
     111          22          25         111 = 3 * 37
     128          29           9         128 = 2^7
     132           1          59         132 = 2^2 * 3 * 11
     132          29          17         132 = 2^2 * 3 * 11
     139          22          45         139 = 139
     148          31          27         148 = 2^2 * 37
     149          22          51         149 = 149
     159           2          71         159 = 3 * 53
     169          38          15         169 = 13^2
     192          31          61         192 = 2^6 * 3
     201          22          79         201 = 3 * 67
     208           1          93         208 = 2^4 * 13
     212          41          51         212 = 2^2 * 53
     229          22          93         229 = 229
     268          61          15         268 = 2^2 * 67
     271          62           9         271 = 271
     288          11         127         288 = 2^5 * 3^2
     288          41         101         288 = 2^5 * 3^2
     291          38         107         291 = 3 * 97
     309          62          67         309 = 3 * 103
     311          58          81         311 = 311
     312          59          79         312 = 2^3 * 3 * 13
     312          61          73         312 = 2^3 * 3 * 13
     321          22         137         321 = 3 * 107
     339          58         101         339 = 3 * 113
     351          22         151         351 = 3^3 * 13
     352          19         153         352 = 2^5 * 11
     352          71          75         352 = 2^5 * 11
     359          82          15         359 = 359
     381          82          59         381 = 3 * 127
     388          89           3         388 = 2^2 * 97
     412          11         183         412 = 2^2 * 103
     428          59         153         428 = 2^2 * 107
     429          58         155         429 = 3 * 11 * 13
     429          62         149         429 = 3 * 11 * 13
     432          89          85         432 = 2^4 * 3^3
     452          11         201         452 = 2^2 * 113
     468         101          71         468 = 2^2 * 3^2 * 13
     468          71         157         468 = 2^2 * 3^2 * 13
     479          38         201         479 = 479
     481          58         183         481 = 13 * 37
     499          22         219         499 = 499
     501          82         157         501 = 3 * 167
     508          31         219         508 = 2^2 * 127
     519         118          31         519 = 3 * 173
     528         109         103         528 = 2^4 * 3 * 11
     528          79         179         528 = 2^4 * 3 * 11
     541         118          75         541 = 541
     571         122          93         571 = 571
     572         131          15         572 = 2^2 * 11 * 13
     572          41         243         572 = 2^2 * 11 * 13
     579          62         229         579 = 3 * 193
     592         101         177         592 = 2^4 * 37
     619         142           3         619 = 619
     631          62         255         631 = 631
     648          11         289         648 = 2^3 * 3^4
     648         131         137         648 = 2^3 * 3^4
     668         151          51         668 = 2^2 * 167
     669          58         277         669 = 3 * 223
     681         142         127         681 = 3 * 227
     689         158           9         689 = 13 * 53
     691           2         309         691 = 691
     692         109         225         692 = 2^2 * 173
     701         118         213         701 = 701
     719          82         279         719 = 719
     729         122         223         729 = 3^6
     732         121         227         732 = 2^2 * 3 * 61
     732         149         151         732 = 2^2 * 3 * 61
     761          58         321         761 = 761
     768          79         307         768 = 2^8 * 3
     769         158         153         769 = 769
     771         158         155         771 = 3 * 257
     772          79         309         772 = 2^2 * 193
     792         151         197         792 = 2^3 * 3^2 * 11
     792         181          31         792 = 2^3 * 3^2 * 11
     792          31         349         792 = 2^3 * 3^2 * 11
     792          59         335         792 = 2^3 * 3^2 * 11
     832         139         255         832 = 2^6 * 13
     848          29         375         848 = 2^4 * 53
     871         178         177         871 = 13 * 67
     879         122         313         879 = 3 * 293
     881          62         375         881 = 881
     888         199          85         888 = 2^3 * 3 * 37
     888          41         389         888 = 2^3 * 3 * 37
     891         202          61         891 = 3^4 * 11
     891          82         365         891 = 3^4 * 11
     892         199          93         892 = 2^2 * 223
     908         181         201         908 = 2^2 * 227
     921         142         305         921 = 3 * 307
     951         218          17         951 = 3 * 317
     968         149         321         968 = 2^3 * 11^2
     968         211         135         968 = 2^3 * 11^2
     972          29         431         972 = 2^2 * 3^5
     999         118         383         999 = 3^3 * 37
    1011         158         331        1011 = 3 * 337
    1028         209         213        1028 = 2^2 * 257
    1031          58         447        1031 = 1031
    1048         121         405        1048 = 2^3 * 131
    1048         239          51        1048 = 2^3 * 131
    1049         202         255        1049 = 1049
    1061         242          51        1061 = 1061
    1069         218         219        1069 = 1069
    1072          71         459        1072 = 2^4 * 67
    1089         122         425        1089 = 3^2 * 11^2
    1089           2         487        1089 = 3^2 * 11^2
    1119         242         167        1119 = 3 * 373
    1149         178         379        1149 = 3 * 383
    1172          79         501        1172 = 2^2 * 293
    1179         178         397        1179 = 3^2 * 131
    1179          58         515        1179 = 3^2 * 131
    1188         139         457        1188 = 2^2 * 3^3 * 11
    1188         191         379        1188 = 2^2 * 3^3 * 11
    1212         211         353        1212 = 2^2 * 3 * 101
    1212         239         277        1212 = 2^2 * 3 * 101
    1221         278          67        1221 = 3 * 11 * 37
    1221          38         541        1221 = 3 * 11 * 37
    1228          59         537        1228 = 2^2 * 307
    1248         229         335        1248 = 2^5 * 3 * 13
    1248         281         107        1248 = 2^5 * 3 * 13
    1259         262         237        1259 = 1259
    1261         242         309        1261 = 13 * 97
    1268         179         447        1268 = 2^2 * 317
    1272         209         397        1272 = 2^3 * 3 * 53
    1272         271         211        1272 = 2^3 * 3 * 53
    1279         278         183        1279 = 1279
    1299         298           5        1299 = 3 * 433
    1301         122         531        1301 = 1301
    1331         118         549        1331 = 11^3
    1332         121         547        1332 = 2^2 * 3^2 * 37
    1332          29         593        1332 = 2^2 * 3^2 * 37
    1339         142         531        1339 = 13 * 103
    1348          61         591        1348 = 2^2 * 337
    1369          82         591        1369 = 37^2
    1391         302         201        1391 = 13 * 107
    1431         142         577        1431 = 3^3 * 53
    1441         298         279        1441 = 11 * 131
    1441          62         633        1441 = 11 * 131
    1452         269         383        1452 = 2^2 * 3 * 11^2
    1452         331          73        1452 = 2^2 * 3 * 11^2
    1461         278         365        1461 = 3 * 487
    1469         218         501        1469 = 13 * 113
    1489         202         537        1489 = 1489
    1492         281         381        1492 = 2^2 * 373
    1499          22         669        1499 = 1499
    1528          29         681        1528 = 2^3 * 191
    1528         331         225        1528 = 2^3 * 191
    1532         349          81        1532 = 2^2 * 383
    1552         139         639        1552 = 2^4 * 97
    1569         358          73        1569 = 3 * 523
    1572         241         523        1572 = 2^2 * 3 * 131
    1572         359          67        1572 = 2^2 * 3 * 131
    1592          19         711        1592 = 2^3 * 199
    1592         341         255        1592 = 2^3 * 199
    1608         251         527        1608 = 2^3 * 3 * 67
    1608         349         233        1608 = 2^3 * 3 * 67
    1628         179         639        1628 = 2^2 * 11 * 37
    1628         271         501        1628 = 2^2 * 11 * 37
    1641         262         527        1641 = 3 * 547
    1648         361         219        1648 = 2^4 * 103
    1651         122         699        1651 = 13 * 127
    1668         379         103        1668 = 2^2 * 3 * 139
    1668          71         733        1668 = 2^2 * 3 * 139
    1669         382          51        1669 = 1669
    1689         358         289        1689 = 3 * 563
    1712         191         669        1712 = 2^4 * 107
    1719         242         607        1719 = 3^2 * 191
    1719         362         305        1719 = 3^2 * 191
    1721         178         687        1721 = 1721
    1728         109         743        1728 = 2^6 * 3^3
    1732         229         633        1732 = 2^2 * 433
    1749         178         701        1749 = 3 * 11 * 53
    1749         302         515        1749 = 3 * 11 * 53
    1759         202         681        1759 = 1759
    1788         101         775        1788 = 2^2 * 3 * 149
    1788         409          61        1788 = 2^2 * 3 * 149
    1791         262         617        1791 = 3^2 * 199
    1791         382         295        1791 = 3^2 * 199
    1808         359         405        1808 = 2^4 * 113
    1809           2         809        1809 = 3^3 * 67
    1811         358         411        1811 = 1811
    1821          38         811        1821 = 3 * 607
    1831         418          81        1831 = 1831
    1871         422         153        1871 = 1871
    1872         179         761        1872 = 2^4 * 3^2 * 13
    1872          31         835        1872 = 2^4 * 3^2 * 13
    1908         341         535        1908 = 2^2 * 3^2 * 53
    1908         431         149        1908 = 2^2 * 3^2 * 53
    1912         311         603        1912 = 2^3 * 239
    1912         409         309        1912 = 2^3 * 239
    1948         419         303        1948 = 2^2 * 487
    1949         382         453        1949 = 1949
    1952         379         465        1952 = 2^5 * 61
    1952         431         237        1952 = 2^5 * 61
    1961         298         657        1961 = 37 * 53
    1999         278         711        1999 = 1999
    2008         269         729        2008 = 2^3 * 251
    2008         451         183        2008 = 2^3 * 251
    2019         418         389        2019 = 3 * 673
    2028         191         827        2028 = 2^2 * 3 * 13^2
    2028         341         617        2028 = 2^2 * 3 * 13^2
    2031         458         167        2031 = 3 * 677
    2032         461         135        2032 = 2^4 * 127
    2048         121         885        2048 = 2^11
    2049         118         887        2049 = 3 * 683
    2069         458         243        2069 = 2069
    2092         319         699        2092 = 2^2 * 523
    2099         218         837        2099 = 2099
    2101         122         909        2101 = 11 * 191
    2101         482           3        2101 = 11 * 191
    2112         449         355        2112 = 2^6 * 3 * 11
    2112          61         937        2112 = 2^6 * 3 * 11
    2151         218         863        2151 = 3^2 * 239
    2151          22         961        2151 = 3^2 * 239
    2171         358         675        2171 = 13 * 167
    2188          89         963        2188 = 2^2 * 547
    2189         142         939        2189 = 11 * 199
    2189         502          27        2189 = 11 * 199
    2211         202         907        2211 = 3 * 11 * 67
    2211         398         613        2211 = 3 * 11 * 67
    2221         362         699        2221 = 2221
    2229         422         563        2229 = 3 * 743
    2239         398         633        2239 = 2239
    2249          82         993        2249 = 13 * 173
    2251         478         381        2251 = 2251
    2252         439         531        2252 = 2^2 * 563
    2259         302         821        2259 = 3^2 * 251
    2259          62        1003        2259 = 3^2 * 251
    2288         241         909        2288 = 2^4 * 11 * 13
    2288         389         687        2288 = 2^4 * 11 * 13
    2292         149         983        2292 = 2^2 * 3 * 191
    2292         451         527        2292 = 2^2 * 3 * 191
    2319         482         439        2319 = 3 * 773
    2328         319         835        2328 = 2^3 * 3 * 97
    2328         521         229        2328 = 2^3 * 3 * 97
    2332         251         921        2332 = 2^2 * 11 * 53
    2332         521         237        2332 = 2^2 * 11 * 53
    2361          22        1055        2361 = 3 * 787
    2368           1        1059        2368 = 2^6 * 37
    2379         538         179        2379 = 3 * 13 * 61
    2379         542         125        2379 = 3 * 13 * 61
    2381         482         501        2381 = 2381
    2388         139        1033        2388 = 2^2 * 3 * 199
    2388         461         577        2388 = 2^2 * 3 * 199
    2391         502         431        2391 = 3 * 797
    2412         241         971        2412 = 2^2 * 3^2 * 67
    2412         451         625        2412 = 2^2 * 3^2 * 67
    2428         421         711        2428 = 2^2 * 607
    2441         538         303        2441 = 2441
    2472         179        1049        2472 = 2^3 * 3 * 103
    2472         419         745        2472 = 2^3 * 3 * 103
    2479         158        1065        2479 = 37 * 67
    2481         262         985        2481 = 3 * 827
    2509         502         549        2509 = 13 * 193
    2531         362         885        2531 = 2531
    2568         251        1039        2568 = 2^3 * 3 * 107
    2568         589          25        2568 = 2^3 * 3 * 107
    2571         562         349        2571 = 3 * 857
    2589         302         997        2589 = 3 * 863
    2629         262        1059        2629 = 11 * 239
    2629         458         765        2629 = 11 * 239
    2631         382         911        2631 = 3 * 877
    2672         341         993        2672 = 2^4 * 167
    2708         449         837        2708 = 2^2 * 677
    2712         541         599        2712 = 2^3 * 3 * 113
    2721         622         103        2721 = 3 * 907
    2732         401         939        2732 = 2^2 * 683
    2748         311        1069        2748 = 2^2 * 3 * 229
    2748         619         233        2748 = 2^2 * 3 * 229
    2761         542         639        2761 = 11 * 251
    2768         599         411        2768 = 2^4 * 173
    2781         638           5        2781 = 3^3 * 103
    2792         619         321        2792 = 2^3 * 349
    2808         319        1091        2808 = 2^3 * 3^3 * 13
    2808         641         125        2808 = 2^3 * 3^3 * 13
    2809         638         177        2809 = 53^2
    2868         551         701        2868 = 2^2 * 3 * 239
    2868         649         211        2868 = 2^2 * 3 * 239
    2889         478         895        2889 = 3^3 * 107
    2899         622         459        2899 = 13 * 223
    2948         661         279        2948 = 2^2 * 11 * 67
    2949         418        1037        2949 = 3 * 983
    2969         638         465        2969 = 2969
    2972         641         453        2972 = 2^2 * 743
    2999         682         177        2999 = 2999
    3012         509         911        3012 = 2^2 * 3 * 251
    3012         691           1        3012 = 2^2 * 3 * 251
    3048         439        1061        3048 = 2^3 * 3 * 127
    3049         682         303        3049 = 3049
    3051         682         307        3051 = 3^3 * 113
    3072         659         487        3072 = 2^10 * 3
    3079         638         591        3079 = 3079
    3088         451        1065        3088 = 2^4 * 193
    3112         551         885        3112 = 2^3 * 389
    3121         538         921        3121 = 3121
    3141         638         653        3141 = 3^2 * 349
    3148         569         867        3148 = 2^2 * 787
    3159         482        1055        3159 = 3^5 * 13
    3172         491        1047        3172 = 2^2 * 13 * 61
    3201         622         761        3201 = 3 * 11 * 97
    3209         562         927        3209 = 3209
    3232         649         699        3232 = 2^5 * 101
    3232         701         471        3232 = 2^5 * 101
    3291         638         787        3291 = 3 * 1097
    3308         751         213        3308 = 2^2 * 827
    3328         691         633        3328 = 2^8 * 13
    3352         769           3        3352 = 2^3 * 419
    3392         649         837        3392 = 2^6 * 53
    3399         718         593        3399 = 3 * 11 * 103
    3428         631         915        3428 = 2^2 * 857
    3429         778         227        3429 = 3^3 * 127
    3432         751         461        3432 = 2^3 * 3 * 11 * 13
    3432         779         223        3432 = 2^3 * 3 * 11 * 13
    3459         662         853        3459 = 3 * 1153
    3492         599        1037        3492 = 2^2 * 3^2 * 97
    3501         682         827        3501 = 3^2 * 389
    3501         802          85        3501 = 3^2 * 389
    3531         758         557        3531 = 3 * 11 * 107
    3548         781         447        3548 = 2^2 * 887
    3551         778         471        3551 = 53 * 67
    3552         661         929        3552 = 2^5 * 3 * 37
    3632         671         963        3632 = 2^4 * 227
    3659         698         909        3659 = 3659
    3688         839         213        3688 = 2^3 * 461
    3691         838         237        3691 = 3691
    3708         769         709        3708 = 2^2 * 3^2 * 103
    3729         842         295        3729 = 3 * 11 * 113
    3732         811         535        3732 = 2^2 * 3 * 311
    3809         682        1065        3809 = 13 * 293
    3811         842         459        3811 = 37 * 103
    3839         878         135        3839 = 11 * 349
    3848         869         303        3848 = 2^3 * 13 * 37
    3849         722         991        3849 = 3 * 1283
    3852         839         541        3852 = 2^2 * 3^2 * 107
    3852         869         313        3852 = 2^2 * 3^2 * 107
    3861         802         733        3861 = 3^3 * 11 * 13
    3888         761         907        3888 = 2^4 * 3^5
    3928         869         465        3928 = 2^3 * 491
    3939         898         197        3939 = 3 * 13 * 101
    3939         902         107        3939 = 3 * 13 * 101
    3959         802         831        3959 = 37 * 107
    4008         919          59        4008 = 2^3 * 3 * 167
    4068         911         395        4068 = 2^2 * 3^2 * 113
    4101         878         659        4101 = 3 * 1367
    4111         902         537        4111 = 4111
    4149         778        1069        4149 = 3^2 * 461
    4152         781        1063        4152 = 2^3 * 3 * 173
    4181         958          93        4181 = 37 * 113
    4188         859         839        4188 = 2^2 * 3 * 349
    4191         818         985        4191 = 3 * 11 * 127
    4212         961         197        4212 = 2^2 * 3^4 * 13
    4241         902         711        4241 = 4241
    4268         929         603        4268 = 2^2 * 11 * 97
    4279         922         657        4279 = 11 * 389
    4281         982          31        4281 = 3 * 1427
    4329         838        1039        4329 = 3^2 * 13 * 37
    4392         899         887        4392 = 2^3 * 3^2 * 61
    4392         991         355        4392 = 2^3 * 3^2 * 61
    4409         998         321        4409 = 4409
    4448         961         669        4448 = 2^5 * 139
    4468        1021         177        4468 = 2^2 * 1117
    4481        1018         279        4481 = 4481
    4492         919         909        4492 = 2^2 * 1123
    4532        1039          75        4532 = 2^2 * 11 * 103
    4561         898        1047        4561 = 4561
    4609         998         681        4609 = 11 * 419
    4621         958         885        4621 = 4621
    4632        1049         331        4632 = 2^3 * 3 * 193
    4659        1018         635        4659 = 3 * 1553
    4699         982         867        4699 = 37 * 127
    4719        1082          71        4719 = 3 * 11^2 * 13
    4719         958         983        4719 = 3 * 11^2 * 13
    4752        1021         745        4752 = 2^4 * 3^3 * 11
    4768        1051         591        4768 = 2^5 * 149
    4772        1091         177        4772 = 2^2 * 1193
    4881         982        1049        4881 = 3 * 1627
    5088        1039        1037        5088 = 2^5 * 3 * 53
    5088        1091         809        5088 = 2^5 * 3 * 53
    5141        1082         915        5141 = 53 * 97
       c           a           b           c =  Factored

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

$\endgroup$
  • $\begingroup$ How did you obtain: $$c = 9u^2 + 4uv + 11v^2$$? This list is done by computer. But is there any algorithm that from one fundamental solution will generate a new one? $\endgroup$ – Stefan4024 Sep 4 '13 at 1:25
  • $\begingroup$ The ternary form $19 a^2 + 5 b^2 - c^2$ has an infinite automorphism group, just as the Pythagorean one $a^2 + b^2 - c^2$ does en.wikipedia.org/wiki/Tree_of_Pythagorean_triples $\endgroup$ – Will Jagy Sep 4 '13 at 1:32
  • $\begingroup$ But how do you obtainted for given ternary form? The link you gave me doesn't provide a infinite automorphism group for pythagorean triples with additional parametars. $\endgroup$ – Stefan4024 Sep 4 '13 at 2:01
  • $\begingroup$ But any way how to obtain the "formula" for generating a,b,c? I've tried the method you posted in the comment of the question and I come up with: $$a = 38m^2 + 30mn - 10n^2$$ $$b = 57m^2 - 76mn - 15n^2$$ $$c = 209m^2 + 55n^2$$ This one wokrs, but it misses a lot of solution that can be obtained using your formula. $\endgroup$ – Stefan4024 Sep 4 '13 at 10:19
  • $\begingroup$ Once you've obtained the from of $c$, how do you obtain the form for $a$ and $b$. Is there any simplier way to obtain those form, like here: math.stackexchange.com/questions/249735/… $\endgroup$ – Stefan4024 Sep 4 '13 at 19:13

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