Find all positive integer solutions to the equation $17x^2 + 29y^2 = 41z^2$
Although I have got a simple instance $(3,8,7)$, I still don't know the form of general solution.
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2You already found one, then it has infinitely many via intersecting lines with rational slopes. – pisco Oct 29 '19 at 08:45
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1Not forgetting that the technique in pisco's comment gives you all the solutions. – Jyrki Lahtonen Oct 29 '19 at 08:50
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1https://math.stackexchange.com/questions/2773097/how-to-find-all-rational-solutions-of-x2-3y2-7/2788381#2788381 – individ Oct 29 '19 at 09:26
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@JyrkiLahtonen I put all the integer solutions, four Pythagorean Triples type recipes. Because I Care. – Will Jagy Oct 30 '19 at 01:06
4 Answers
3
All (primitive) solutions can be found from four Pythagorean Triple type recipes, along with absolute value signs when needed. Give me some time, I will also typeset the formulae.
Let's see. If you use coprime $(u,v)$ and get a triple from one of the recipes ABCD that is NOT primitive, just throw that one out. The, well, advantage of going into this much detail, compared with the general rational solution (of pisco and S. Dolan) is that we know the gcd of the resulting triple. To find all integral triples up to some bound, we just look at the (positive definite binary form) definitions of $z$ and deduce bounds from that.
A:
$$ x = 36 u^2 - 34 u v - 25 v^2 $$ $$ y = 14 u^2 + 46 u v - 12 v^2 $$ $$ z = 26 u^2 - 2 u v +19 v^2 $$
B:
$$ x = 76 u^2 - 62 u v - 3 v^2 $$ $$ y = 66 u^2 + 26 u v - 8 v^2 $$ $$ z = 74 u^2 - 10 u v +7 v^2 $$
C:
$$ x = 50 u^2 - 34 u v - 18 v^2 $$ $$ y = 24 u^2 + 46 u v - 7 v^2 $$ $$ z = 38 u^2 + 2 u v +13 v^2 $$
D: $$ x = 6 u^2 - 62 u v - 38 v^2 $$ $$ y = 16 u^2 + 26 u v - 33 v^2 $$ $$ z = 14 u^2 + 10 u v + 37 v^2 $$
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Tue Oct 29 13:25:20 PDT 2019
7 x,y,z 3 8 7 BRUTE FORCE SEARCH
7 x,y,z 3 8 7 u,v 0 1 B
7 x,y,z 3 8 7 u,v 0 -1 B
13 x,y,z 18 7 13 BRUTE FORCE SEARCH
13 x,y,z 18 7 13 u,v 0 1 C
13 x,y,z 18 7 13 u,v 0 -1 C
19 x,y,z 25 12 19 BRUTE FORCE SEARCH
19 x,y,z 25 12 19 u,v 0 1 A
19 x,y,z 25 12 19 u,v 0 -1 A
37 x,y,z 38 33 37 BRUTE FORCE SEARCH
37 x,y,z 38 33 37 u,v 0 1 D
37 x,y,z 38 33 37 u,v 0 -1 D
41 x,y,z 30 43 41 BRUTE FORCE SEARCH
41 x,y,z 30 43 41 u,v 1 -1 D
43 x,y,z 23 48 43 BRUTE FORCE SEARCH
43 x,y,z 23 48 43 u,v 1 1 A
47 x,y,z 45 44 47 BRUTE FORCE SEARCH
47 x,y,z 45 44 47 u,v 1 -1 A
49 x,y,z 66 29 49 BRUTE FORCE SEARCH
49 x,y,z 66 29 49 u,v 1 -1 C
53 x,y,z 2 63 53 BRUTE FORCE SEARCH
53 x,y,z 2 63 53 u,v 1 1 C
61 x,y,z 94 9 61 BRUTE FORCE SEARCH
61 x,y,z 94 9 61 u,v 1 1 D
71 x,y,z 11 84 71 BRUTE FORCE SEARCH
71 x,y,z 11 84 71 u,v 1 1 B
73 x,y,z 110 21 73 BRUTE FORCE SEARCH
73 x,y,z 110 21 73 u,v 2 -1 D
91 x,y,z 135 32 91 BRUTE FORCE SEARCH
91 x,y,z 135 32 91 u,v 1 -1 B
107 x,y,z 137 72 107 BRUTE FORCE SEARCH
107 x,y,z 137 72 107 u,v 1 3 B
113 x,y,z 138 83 113 BRUTE FORCE SEARCH
113 x,y,z 138 83 113 u,v 2 1 D
127 x,y,z 187 48 127 BRUTE FORCE SEARCH
127 x,y,z 187 48 127 u,v 2 -1 A
133 x,y,z 202 33 133 BRUTE FORCE SEARCH
133 x,y,z 202 33 133 u,v 3 -1 D
149 x,y,z 10 177 149 BRUTE FORCE SEARCH
149 x,y,z 10 177 149 u,v 1 -3 C
161 x,y,z 214 99 161 BRUTE FORCE SEARCH
161 x,y,z 214 99 161 u,v 1 3 C
161 x,y,z 250 3 161 BRUTE FORCE SEARCH
161 x,y,z 250 3 161 u,v 2 -1 C
167 x,y,z 235 84 167 BRUTE FORCE SEARCH
167 x,y,z 235 84 167 u,v 1 -3 B
169 x,y,z 114 181 169 BRUTE FORCE SEARCH
169 x,y,z 114 181 169 u,v 2 1 C
191 x,y,z 291 44 191 BRUTE FORCE SEARCH
191 x,y,z 291 44 191 u,v 1 3 A
193 x,y,z 170 189 193 BRUTE FORCE SEARCH
193 x,y,z 170 189 193 u,v 3 1 D
199 x,y,z 309 4 199 BRUTE FORCE SEARCH
199 x,y,z 309 4 199 u,v 1 5 B
247 x,y,z 197 252 247 BRUTE FORCE SEARCH
247 x,y,z 197 252 247 u,v 3 1 A
257 x,y,z 242 243 257 BRUTE FORCE SEARCH
257 x,y,z 242 243 257 u,v 2 -3 C
259 x,y,z 401 24 259 BRUTE FORCE SEARCH
259 x,y,z 401 24 259 u,v 3 -1 A
263 x,y,z 285 224 263 BRUTE FORCE SEARCH
263 x,y,z 285 224 263 u,v 2 3 A
281 x,y,z 166 309 281 BRUTE FORCE SEARCH
281 x,y,z 166 309 281 u,v 2 3 C
283 x,y,z 177 308 283 BRUTE FORCE SEARCH
283 x,y,z 177 308 283 u,v 2 1 B
299 x,y,z 311 264 299 BRUTE FORCE SEARCH
299 x,y,z 311 264 299 u,v 1 -5 B
299 x,y,z 95 348 299 BRUTE FORCE SEARCH
299 x,y,z 95 348 299 u,v 2 3 B
301 x,y,z 190 327 301 BRUTE FORCE SEARCH
301 x,y,z 190 327 301 u,v 4 1 D
317 x,y,z 150 359 317 BRUTE FORCE SEARCH
317 x,y,z 150 359 317 u,v 1 -3 D
329 x,y,z 54 389 329 BRUTE FORCE SEARCH
329 x,y,z 54 389 329 u,v 2 -3 D
337 x,y,z 422 237 337 BRUTE FORCE SEARCH
337 x,y,z 422 237 337 u,v 5 -1 D
347 x,y,z 505 144 347 BRUTE FORCE SEARCH
347 x,y,z 505 144 347 u,v 1 7 B
349 x,y,z 534 71 349 BRUTE FORCE SEARCH
349 x,y,z 534 71 349 u,v 3 -1 C
353 x,y,z 230 381 353 BRUTE FORCE SEARCH
353 x,y,z 230 381 353 u,v 1 -5 C
361 x,y,z 330 347 361 BRUTE FORCE SEARCH
361 x,y,z 330 347 361 u,v 3 1 C
371 x,y,z 391 324 371 BRUTE FORCE SEARCH
371 x,y,z 391 324 371 u,v 2 5 B
373 x,y,z 570 79 373 BRUTE FORCE SEARCH
373 x,y,z 570 79 373 u,v 1 5 C
419 x,y,z 649 36 419 BRUTE FORCE SEARCH
419 x,y,z 649 36 419 u,v 2 -3 B
427 x,y,z 415 396 427 BRUTE FORCE SEARCH
427 x,y,z 415 396 427 u,v 4 1 A
437 x,y,z 198 497 437 BRUTE FORCE SEARCH
437 x,y,z 198 497 437 u,v 5 1 D
437 x,y,z 498 353 437 BRUTE FORCE SEARCH
437 x,y,z 498 353 437 u,v 4 -3 D
443 x,y,z 687 28 443 BRUTE FORCE SEARCH
443 x,y,z 687 28 443 u,v 4 -1 A
449 x,y,z 690 77 449 BRUTE FORCE SEARCH
449 x,y,z 690 77 449 u,v 2 3 D
457 x,y,z 90 539 457 BRUTE FORCE SEARCH
457 x,y,z 90 539 457 u,v 2 -5 C
481 x,y,z 550 387 481 BRUTE FORCE SEARCH
481 x,y,z 550 387 481 u,v 6 -1 D
487 x,y,z 363 508 487 BRUTE FORCE SEARCH
487 x,y,z 363 508 487 u,v 1 -7 B
491 x,y,z 759 56 491 BRUTE FORCE SEARCH
491 x,y,z 759 56 491 u,v 1 5 A
497 x,y,z 590 381 497 BRUTE FORCE SEARCH
497 x,y,z 590 381 497 u,v 2 5 C
499 x,y,z 711 236 499 BRUTE FORCE SEARCH
499 x,y,z 711 236 499 u,v 2 7 B
511 x,y,z 419 516 511 BRUTE FORCE SEARCH
511 x,y,z 419 516 511 u,v 1 -5 A
559 x,y,z 821 216 559 BRUTE FORCE SEARCH
559 x,y,z 821 216 559 u,v 2 5 A
563 x,y,z 57 668 563 BRUTE FORCE SEARCH
563 x,y,z 57 668 563 u,v 4 3 A
571 x,y,z 849 196 571 BRUTE FORCE SEARCH
571 x,y,z 849 196 571 u,v 2 -5 B
599 x,y,z 141 704 599 BRUTE FORCE SEARCH
599 x,y,z 141 704 599 u,v 2 -5 A
601 x,y,z 194 699 601 BRUTE FORCE SEARCH
601 x,y,z 194 699 601 u,v 6 1 D
611 x,y,z 759 436 611 BRUTE FORCE SEARCH
611 x,y,z 759 436 611 u,v 4 -3 A
613 x,y,z 918 193 613 BRUTE FORCE SEARCH
613 x,y,z 918 193 613 u,v 4 -1 C
637 x,y,z 510 649 637 BRUTE FORCE SEARCH
637 x,y,z 510 649 637 u,v 3 -5 C
643 x,y,z 495 664 643 BRUTE FORCE SEARCH
643 x,y,z 495 664 643 u,v 3 1 B
653 x,y,z 690 569 653 BRUTE FORCE SEARCH
653 x,y,z 690 569 653 u,v 7 -1 D
659 x,y,z 705 568 659 BRUTE FORCE SEARCH
659 x,y,z 705 568 659 u,v 5 1 A
661 x,y,z 594 641 661 BRUTE FORCE SEARCH
661 x,y,z 594 641 661 u,v 1 -7 C
677 x,y,z 990 271 677 BRUTE FORCE SEARCH
677 x,y,z 990 271 677 u,v 4 3 D
679 x,y,z 1045 108 679 BRUTE FORCE SEARCH
679 x,y,z 1045 108 679 u,v 5 -1 A
679 x,y,z 811 516 679 BRUTE FORCE SEARCH
679 x,y,z 811 516 679 u,v 3 5 A
683 x,y,z 1055 84 683 BRUTE FORCE SEARCH
683 x,y,z 1055 84 683 u,v 2 9 B
689 x,y,z 1070 3 689 BRUTE FORCE SEARCH
689 x,y,z 1070 3 689 u,v 1 7 C
691 x,y,z 321 784 691 BRUTE FORCE SEARCH
691 x,y,z 321 784 691 u,v 3 5 B
701 x,y,z 1046 231 701 BRUTE FORCE SEARCH
701 x,y,z 1046 231 701 u,v 4 -3 C
703 x,y,z 867 508 703 BRUTE FORCE SEARCH
703 x,y,z 867 508 703 u,v 3 -1 B
739 x,y,z 209 864 739 BRUTE FORCE SEARCH
739 x,y,z 209 864 739 u,v 3 -5 A
749 x,y,z 230 873 749 BRUTE FORCE SEARCH
749 x,y,z 230 873 749 u,v 4 3 C
761 x,y,z 206 891 761 BRUTE FORCE SEARCH
761 x,y,z 206 891 761 u,v 2 -7 C
791 x,y,z 165 932 791 BRUTE FORCE SEARCH
791 x,y,z 165 932 791 u,v 5 3 A
793 x,y,z 178 933 793 BRUTE FORCE SEARCH
793 x,y,z 178 933 793 u,v 7 1 D
809 x,y,z 1254 59 809 BRUTE FORCE SEARCH
809 x,y,z 1254 59 809 u,v 7 -3 D
811 x,y,z 969 616 811 BRUTE FORCE SEARCH
811 x,y,z 969 616 811 u,v 1 11 B
817 x,y,z 1158 397 817 BRUTE FORCE SEARCH
817 x,y,z 1158 397 817 u,v 2 7 C
851 x,y,z 1185 448 851 BRUTE FORCE SEARCH
851 x,y,z 1185 448 851 u,v 5 -3 A
851 x,y,z 729 844 851 BRUTE FORCE SEARCH
851 x,y,z 729 844 851 u,v 4 5 A
853 x,y,z 842 783 853 BRUTE FORCE SEARCH
853 x,y,z 842 783 853 u,v 8 -1 D
881 x,y,z 306 1021 881 BRUTE FORCE SEARCH
881 x,y,z 306 1021 881 u,v 2 -5 D
889 x,y,z 634 939 889 BRUTE FORCE SEARCH
889 x,y,z 634 939 889 u,v 1 -5 D
893 x,y,z 1030 711 893 BRUTE FORCE SEARCH
893 x,y,z 1030 711 893 u,v 4 -5 C
923 x,y,z 1423 132 923 BRUTE FORCE SEARCH
923 x,y,z 1423 132 923 u,v 2 11 B
931 x,y,z 631 996 931 BRUTE FORCE SEARCH
931 x,y,z 631 996 931 u,v 4 -5 A
937 x,y,z 282 1093 937 BRUTE FORCE SEARCH
937 x,y,z 282 1093 937 u,v 3 -7 C
943 x,y,z 1067 768 943 BRUTE FORCE SEARCH
943 x,y,z 1067 768 943 u,v 6 1 A
943 x,y,z 1427 252 943 BRUTE FORCE SEARCH
943 x,y,z 1427 252 943 u,v 1 7 A
949 x,y,z 386 1089 949 BRUTE FORCE SEARCH
949 x,y,z 386 1089 949 u,v 4 -5 D
953 x,y,z 1402 363 953 BRUTE FORCE SEARCH
953 x,y,z 1402 363 953 u,v 5 -1 C
967 x,y,z 1475 216 967 BRUTE FORCE SEARCH
967 x,y,z 1475 216 967 u,v 6 -1 A
971 x,y,z 951 896 971 BRUTE FORCE SEARCH
971 x,y,z 951 896 971 u,v 1 -7 A
973 x,y,z 1062 823 973 BRUTE FORCE SEARCH
973 x,y,z 1062 823 973 u,v 5 1 C
973 x,y,z 330 1129 973 BRUTE FORCE SEARCH
973 x,y,z 330 1129 973 u,v 4 5 C
989 x,y,z 1254 679 989 BRUTE FORCE SEARCH
989 x,y,z 1254 679 989 u,v 1 5 D
989 x,y,z 1530 103 989 BRUTE FORCE SEARCH
989 x,y,z 1530 103 989 u,v 8 -3 D
991 x,y,z 1539 4 991 BRUTE FORCE SEARCH
991 x,y,z 1539 4 991 u,v 3 -5 B
1007 x,y,z 1557 112 1007 BRUTE FORCE SEARCH
1007 x,y,z 1557 112 1007 u,v 2 7 A
1013 x,y,z 150 1199 1013 BRUTE FORCE SEARCH
1013 x,y,z 150 1199 1013 u,v 8 1 D
1021 x,y,z 1146 839 1021 BRUTE FORCE SEARCH
1021 x,y,z 1146 839 1021 u,v 3 7 C
1031 x,y,z 395 1188 1031 BRUTE FORCE SEARCH
1031 x,y,z 395 1188 1031 u,v 1 -11 B
1043 x,y,z 1177 852 1043 BRUTE FORCE SEARCH
1043 x,y,z 1177 852 1043 u,v 2 -9 B
1063 x,y,z 605 1176 1063 BRUTE FORCE SEARCH
1063 x,y,z 605 1176 1063 u,v 2 -7 A
1081 x,y,z 1006 1029 1081 BRUTE FORCE SEARCH
1081 x,y,z 1006 1029 1081 u,v 9 -1 D
1081 x,y,z 1546 501 1081 BRUTE FORCE SEARCH
1081 x,y,z 1546 501 1081 u,v 2 5 D
1097 x,y,z 578 1227 1097 BRUTE FORCE SEARCH
1097 x,y,z 578 1227 1097 u,v 5 3 C
1109 x,y,z 1714 129 1109 BRUTE FORCE SEARCH
1109 x,y,z 1714 129 1109 u,v 1 9 C
1123 x,y,z 1615 504 1123 BRUTE FORCE SEARCH
1123 x,y,z 1615 504 1123 u,v 3 7 A
1127 x,y,z 1237 948 1127 BRUTE FORCE SEARCH
1127 x,y,z 1237 948 1127 u,v 1 13 B
1127 x,y,z 445 1296 1127 BRUTE FORCE SEARCH
1127 x,y,z 445 1296 1127 u,v 4 3 B
1129 x,y,z 1126 1029 1129 BRUTE FORCE SEARCH
1129 x,y,z 1126 1029 1129 u,v 6 -5 D
1151 x,y,z 965 1152 1151 BRUTE FORCE SEARCH
1151 x,y,z 965 1152 1151 u,v 4 1 B
1159 x,y,z 99 1376 1159 BRUTE FORCE SEARCH
1159 x,y,z 99 1376 1159 u,v 4 5 B
1169 x,y,z 646 1299 1169 BRUTE FORCE SEARCH
1169 x,y,z 646 1299 1169 u,v 2 -9 C
1183 x,y,z 1725 484 1183 BRUTE FORCE SEARCH
1183 x,y,z 1725 484 1183 u,v 3 11 B
1201 x,y,z 1826 291 1201 BRUTE FORCE SEARCH
1201 x,y,z 1826 291 1201 u,v 3 5 D
1219 x,y,z 1815 412 1219 BRUTE FORCE SEARCH
1219 x,y,z 1815 412 1219 u,v 2 13 B
1219 x,y,z 1839 344 1219 BRUTE FORCE SEARCH
1219 x,y,z 1839 344 1219 u,v 3 -7 B
1229 x,y,z 1350 1033 1229 BRUTE FORCE SEARCH
1229 x,y,z 1350 1033 1229 u,v 7 3 D
1231 x,y,z 1461 944 1231 BRUTE FORCE SEARCH
1231 x,y,z 1461 944 1231 u,v 4 -1 B
1261 x,y,z 110 1497 1261 BRUTE FORCE SEARCH
1261 x,y,z 110 1497 1261 u,v 9 1 D
1261 x,y,z 1514 951 1261 BRUTE FORCE SEARCH
1261 x,y,z 1514 951 1261 u,v 7 -5 D
1279 x,y,z 1501 996 1279 BRUTE FORCE SEARCH
1279 x,y,z 1501 996 1279 u,v 7 1 A
1291 x,y,z 1601 924 1291 BRUTE FORCE SEARCH
1291 x,y,z 1601 924 1291 u,v 4 7 A
1301 x,y,z 1034 1329 1301 BRUTE FORCE SEARCH
1301 x,y,z 1034 1329 1301 u,v 4 7 C
1307 x,y,z 1977 352 1307 BRUTE FORCE SEARCH
1307 x,y,z 1977 352 1307 u,v 7 -1 A
1337 x,y,z 1182 1307 1337 BRUTE FORCE SEARCH
1337 x,y,z 1182 1307 1337 u,v 10 -1 D
1349 x,y,z 2094 49 1349 BRUTE FORCE SEARCH
1349 x,y,z 2094 49 1349 u,v 4 5 D
1351 x,y,z 349 1584 1351 BRUTE FORCE SEARCH
1351 x,y,z 349 1584 1351 u,v 6 5 A
1367 x,y,z 1933 672 1367 BRUTE FORCE SEARCH
1367 x,y,z 1933 672 1367 u,v 4 -3 B
1369 x,y,z 1986 581 1369 BRUTE FORCE SEARCH
1369 x,y,z 1986 581 1369 u,v 6 -1 C
1387 x,y,z 375 1624 1387 BRUTE FORCE SEARCH
1387 x,y,z 375 1624 1387 u,v 1 -13 B
1391 x,y,z 1259 1344 1391 BRUTE FORCE SEARCH
1391 x,y,z 1259 1344 1391 u,v 4 9 B
1393 x,y,z 1578 1133 1393 BRUTE FORCE SEARCH
1393 x,y,z 1578 1133 1393 u,v 6 1 C
1403 x,y,z 303 1652 1403 BRUTE FORCE SEARCH
1403 x,y,z 303 1652 1403 u,v 4 -7 A
1403 x,y,z 825 1544 1403 BRUTE FORCE SEARCH
1403 x,y,z 825 1544 1403 u,v 7 3 A
1433 x,y,z 2118 523 1433 BRUTE FORCE SEARCH
1433 x,y,z 2118 523 1433 u,v 10 -3 D
1459 x,y,z 2241 256 1459 BRUTE FORCE SEARCH
1459 x,y,z 2241 256 1459 u,v 3 13 B
1469 x,y,z 1446 1351 1469 BRUTE FORCE SEARCH
1469 x,y,z 1446 1351 1469 u,v 8 3 D
1471 x,y,z 1691 1176 1471 BRUTE FORCE SEARCH
1471 x,y,z 1691 1176 1471 u,v 6 -5 A
1487 x,y,z 2253 388 1487 BRUTE FORCE SEARCH
1487 x,y,z 2253 388 1487 u,v 7 -3 A
1499 x,y,z 1529 1344 1499 BRUTE FORCE SEARCH
1499 x,y,z 1529 1344 1499 u,v 1 15 B
1511 x,y,z 1515 1372 1511 BRUTE FORCE SEARCH
1511 x,y,z 1515 1372 1511 u,v 5 7 A
1559 x,y,z 2381 336 1559 BRUTE FORCE SEARCH
1559 x,y,z 2381 336 1559 u,v 4 -5 B
1571 x,y,z 2231 756 1571 BRUTE FORCE SEARCH
1571 x,y,z 2231 756 1571 u,v 2 15 B
1583 x,y,z 1683 1372 1583 BRUTE FORCE SEARCH
1583 x,y,z 1683 1372 1583 u,v 1 -9 A
1589 x,y,z 1754 1329 1589 BRUTE FORCE SEARCH
1589 x,y,z 1754 1329 1589 u,v 1 -11 C
1589 x,y,z 566 1839 1589 BRUTE FORCE SEARCH
1589 x,y,z 566 1839 1589 u,v 4 -9 C
1591 x,y,z 1875 1232 1591 BRUTE FORCE SEARCH
1591 x,y,z 1875 1232 1591 u,v 4 11 B
1607 x,y,z 2493 88 1607 BRUTE FORCE SEARCH
1607 x,y,z 2493 88 1607 u,v 2 9 A
1609 x,y,z 2326 699 1609 BRUTE FORCE SEARCH
1609 x,y,z 2326 699 1609 u,v 9 -5 D
1621 x,y,z 1370 1617 1621 BRUTE FORCE SEARCH
1621 x,y,z 1370 1617 1621 u,v 11 -1 D
===========================================================
Will Jagy
- 139,541
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Your answer is quite good, but I have another question. Why four familys of solutions, not more or less? Is there any basis for this? Thanks! – Eufisky Oct 30 '19 at 01:38
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1@Eufisky it varies a good deal. It rises up a bit for each prime factor of the discriminant; any one of the recipes gives a multiple of every integer solution, then that common factor leads to another such recipe. Still not the whole story, but see https://math.stackexchange.com/questions/1972120/ternary-quadratic-forms/1976199#1976199 – Will Jagy Oct 30 '19 at 02:03
2
To expand on the above comments:-
You can use your answer to find all other solutions as follows.
Let $X=\frac{x}{z},Y=\frac{y}{z}$ then $17X^2 + 29Y^2 = 41$.
Define rationals $U$ and $T$ by $U=X-\frac{3}{7},UT=\frac{8}{7}-Y$ then $$17(U^2 + \frac{6}{7}U)+29(U^2T^2-\frac{16}{7}UT)=0$$ $U=0$ gives the initial solution and so $$17(U + \frac{6}{7})+29(UT^2-\frac{16}{7}T)=0$$ $$U=\frac{464T-102}{203T^2+119}$$
We can now express $X$ and $Y$ in terms of the rational parameter $T$. For any given $T$ you can just let $z$ be the l.c.m. of the denominators of $X$ and $Y$ and then $x,y,z$ will be an integer solution.
1
Above equation shown below:
$17x^2 + 29y^2 = 41z^2$ ----$(1)$
Equation $(1)$ has parametric solution & is shown below:
$x=(123k^2-574k+428)$
$y=2(164k^2-287k+99)$
$z=(287k^2-566k+322)$
For, $k=1$ we get:
$(x,y,z)=(23,48,43)$
Sam
- 11
0
Above equation shown below:
$$17x^2 + 29y^2 = 41z^2 \;\;\;\;(1)$$
"OP" commented on a general form of equation, $(1)$
So equation $(1)$ can be written as:
$$ax^2 + by^2 = cz^2\;\;\;\;\;\;\;\;\;\;(2)$$
Equation $(2)$ has parametric solution:
$$(x,y,z)=[(2)(p-2),(p+1),(3)(p-1)],$$
$$(a,b,c))=[(2p^2-2p+2),((p^2+2p-5),(p^2-2p+3)].$$
For, $p=4$, we get after removing common factors:
$$(x,y,z)=(4,5,9),$$
$$(a,b,c)=(26,19,11).$$
