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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.

Stefan4024
  • 36,357
  • 1
    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 http://math.stackexchange.com/questions/249735/generating-pythagorean-triples-for-a2b2-5c2/249778#249778 for an example of the method. – Will Jagy Sep 04 '13 at 01:58
  • I edited in a second formula that gives the primitive solutions with $c$ even and $a$ odd. – Will Jagy Sep 04 '13 at 18:35

1 Answers1

3

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

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

Will Jagy
  • 146,052
  • 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?

     – Stefan4024 Sep 04 '13 at 01:25
  • 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 http://en.wikipedia.org/wiki/Tree_of_Pythagorean_triples – Will Jagy Sep 04 '13 at 01:32
  • 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. – Stefan4024 Sep 04 '13 at 02:01
  • 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.

     – Stefan4024 Sep 04 '13 at 10:19
  • 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:

    http://math.stackexchange.com/questions/249735/generating-pythagorean-triples-for-a2b2-5c2/249778#249778

     – Stefan4024 Sep 04 '13 at 19:13
  • Buell gives the algorithm for composition including finding the values. The most concrete method is Dirichlet's "united" forms, Buell pages 55-57 especially. On page 57, formulas 4.2 and 4.3 do what you ask. One then would typically reverse some of the changes in variables that produced the united forms in the first place. Oh, here, a straightforward version on page 61, closer to what I did for your problem. – Will Jagy Sep 04 '13 at 19:34
  • How can I get that book? Is that book available to read online? – Stefan4024 Sep 04 '13 at 19:45
  • probably not. Available used, surely. For your apparent interests, worth the money. Half my answers on MSE are straight out of Buell, in particular my answers to your questions, with some side items out of Conway's little book. – Will Jagy Sep 04 '13 at 19:49
  • see pages 55-61 if it will show them http://books.google.com/books?id=qMa8-FDOR8YC&pg=PA55&lpg=PA55&dq=buell+binary+quadratic+forms+united+forms+composition&source=bl&ots=0I3vM88IQd&sig=zavTG6tQ8RPXNr0OtkWimsC1EwU&hl=en&sa=X&ei=so4nUqjuF4zyigKfnoHoAQ&ved=0CE8Q6AEwBA#v=onepage&q=buell%20binary%20quadratic%20forms%20united%20forms%20composition&f=false – Will Jagy Sep 04 '13 at 19:50
  • also used purchase http://www.amazon.com/gp/offer-listing/0387970371/ref=dp_olp_used?ie=UTF8&condition=used – Will Jagy Sep 04 '13 at 19:52
  • Sadly, but only pages 55-59 are available, but I can't see them, because the Loading thing never stops, while other available pages are OK. – Stefan4024 Sep 04 '13 at 20:08
  • let us continue this discussion in chat – Stefan4024 Sep 04 '13 at 20:13
  • So for the following example 7x^2 + 5y^2 = c^2. The LHS is quadratic form of $\langle7,0,5\rangle$ so the discriminant is -140. Also quadratic form of $\langle3,4,13 \rangle$ has discriminant of -140, so $c$ can be written as:

    $$c = 3u^2 + 4uv + 13v^2$$.

    Am I right?

     – Stefan4024 Sep 04 '13 at 21:39
  • NO. $7 x^2 + 5 y^2$ is not in the principal genus. There are no solutions to $7 a^2 + 5 b^2 = c^2.$ You will have better luck with this: $2 a^2 + 7 b^2 = c^2,$ where you may take $c = 3 u^2 + 2 u v + 5 v^2.$ BUY THE BOOK. – Will Jagy Sep 04 '13 at 22:34
  • For $2a^2 + 7b^2 = c^2$, I get the following form:

    $$c = 3u^2+2uv+5v^2$$ $$a = -u^2 + 4uv + 3v^2$$ $$b = u^2 + 2uv - v^2$$

    The only question is how to decide whether a quadratic form is in the principal genus?

    And I'm sorry but at the moment I can't afford this book, it's too expensive for me.

     – Stefan4024 Sep 04 '13 at 23:50
  • How can I find forms for $a$ and $b$, simular to one we have for $c$? Is there any fast way to generate? – Stefan4024 Sep 04 '13 at 23:55