Areas of primitive Heron Triangles that are perfect squares
I am trying to find a sequence (in ascending order) of areas of primitive Heron Triangles that are perfect squares. For instance 36, 900, 7056, 44100 etc. These have been found by searching through generated integer triangles. But I am not sure if there exists other perfect squares in between these values that are also areas of primitive Heron triangles. Because of the sparseness of these numbers a simple search seems very inefficient. Does anyone know of a method of generating these numbers?
Solution 1:
It appears that the recipe on Wikipedia needs additional work to make something that gives an obvious exhaustive list. I told it to find the gcd of the three sides, then divide through and report the primitive triangle, also its area. But also print the gcd, which gets very large. This is not yet satisfactory, although fast at what it does.
Meanwhile, I let $m$ go up to 1125 or so and just printed out the square roots of the areas that do come out squares, so it seems you have missed some small ones. However, I cannot be sure yet that i have all small ones either.
6 = 2 3 squared 36
30 = 2 3 5 squared 900
84 = 2^2 3 7 squared 7056
180 = 2^2 3^2 5 squared 32400
204 = 2^2 3 17 squared 41616
210 = 2 3 5 7 squared 44100
234 = 2 3^2 13 squared 54756
252 = 2^2 3^2 7 squared 63504
264 = 2^3 3 11 squared 69696
330 = 2 3 5 11 squared 108900
420 = 2^2 3 5 7 squared 176400
462 = 2 3 7 11 squared 213444
630 = 2 3^2 5 7 squared 396900
660 = 2^2 3 5 11 squared 435600
780 = 2^2 3 5 13 squared 608400
870 = 2 3 5 29 squared 756900
924 = 2^2 3 7 11 squared 853776
1020 = 2^2 3 5 17 squared 1040400
1092 = 2^2 3 7 13 squared 1192464
1170 = 2 3^2 5 13 squared 1368900
1386 = 2 3^2 7 11 squared 1920996
1554 = 2 3 7 37 squared 2414916
1560 = 2^3 3 5 13 squared 2433600
1638 = 2 3^2 7 13 squared 2683044
1716 = 2^2 3 11 13 squared 2944656
3876 = 2^2 3 17 19 squared 15023376
3990 = 2 3 5 7 19 squared 15920100
4158 = 2 3^3 7 11 squared 17288964
9030 = 2 3 5 7 43 squared 81540900
11220 = 2^2 3 5 11 17 squared 125888400
29070 = 2 3^2 5 17 19 squared 845064900
38610 = 2 3^3 5 11 13 squared 1490732100
38760 = 2^3 3 5 17 19 squared 1502337600
39270 = 2 3 5 7 11 17 squared 1542132900
53130 = 2 3 5 7 11 23 squared 2822796900
72930 = 2 3 5 11 13 17 squared 5318784900
80808 = 2^3 3 7 13 37 squared 6529932864
288666 = 2 3^2 7 29 79 squared 83328059556
jagy@phobeusjunior:~$
m: 6 n: 3 k: 4
sides gcd = 6
side a: 26 side b: 25 side c: 3
area 36
m: 8 n: 1 k: 2
sides gcd = 4
side a: 17 side b: 10 side c: 9
area 36
m: 40 n: 25 k: 26
sides gcd = 20
side a: 2845 side b: 2602 side c: 1053
area 1368900
m: 50 n: 48 k: 45
sides gcd = 150
side a: 1448 side b: 1443 side c: 245
area 176400
m: 60 n: 25 k: 36
sides gcd = 1020
side a: 120 side b: 113 side c: 17
area 900
m: 81 n: 36 k: 50
sides gcd = 468
side a: 697 side b: 657 side c: 104
area 32400
m: 81 n: 72 k: 50
sides gcd = 612
side a: 1066 side b: 1017 side c: 833
area 396900
m: 99 n: 22 k: 36
sides gcd = 198
side a: 1233 side b: 890 side c: 539
area 213444
m: 121 n: 4 k: 18
sides gcd = 20
side a: 2993 side b: 2057 side c: 1000
area 435600
m: 121 n: 9 k: 32
sides gcd = 65
side a: 2169 side b: 2057 side c: 130
area 69696
m: 121 n: 33 k: 63
sides gcd = 66
side a: 9305 side b: 9273 side c: 56
area 213444
m: 147 n: 72 k: 100
sides gcd = 1752
side a: 1299 side b: 1274 side c: 73
area 44100
m: 153 n: 136 k: 144
sides gcd = 1224
side a: 4905 side b: 4904 side c: 17
area 41616
m: 189 n: 7 k: 36
sides gcd = 189
side a: 1371 side b: 1345 side c: 28
area 7056
m: 196 n: 63 k: 111
sides gcd = 63
side a: 50737 side b: 50680 side c: 111
area 2414916
m: 216 n: 112 k: 147
sides gcd = 20664
side a: 370 side b: 357 side c: 41
area 7056
m: 224 n: 81 k: 98
sides gcd = 8540
side a: 567 side b: 424 side c: 305
area 63504
m: 225 n: 64 k: 80
sides gcd = 1600
side a: 2281 side b: 1476 side c: 1445
area 1040400
Solution 2:
I let my program reach $m=5000.$ My guess is that no primitive Heronian triangle has area 3600, although you can do it by doubling the edges of the one with area 900. Two types of output, one with the sides, the other factoring the square roots of the areas.
jagy@phobeusjunior:~$ grep NEW heron.txt
area 36 square root 6 NEW A: 26 B: 25 C: 3
area 1368900 square root 1170 NEW A: 2845 B: 2602 C: 1053
area 176400 square root 420 NEW A: 1448 B: 1443 C: 245
area 900 square root 30 NEW A: 120 B: 113 C: 17
area 32400 square root 180 NEW A: 697 B: 657 C: 104
area 396900 square root 630 NEW A: 1066 B: 1017 C: 833
area 213444 square root 462 NEW A: 1233 B: 890 C: 539
area 435600 square root 660 NEW A: 2993 B: 2057 C: 1000
area 69696 square root 264 NEW A: 2169 B: 2057 C: 130
area 44100 square root 210 NEW A: 1299 B: 1274 C: 73
area 41616 square root 204 NEW A: 4905 B: 4904 C: 17
area 7056 square root 84 NEW A: 1371 B: 1345 C: 28
area 2414916 square root 1554 NEW A: 50737 B: 50680 C: 111
area 63504 square root 252 NEW A: 567 B: 424 C: 305
area 1040400 square root 1020 NEW A: 2281 B: 1476 C: 1445
area 17288964 square root 4158 NEW A: 29354 B: 17545 C: 11907
area 1920996 square root 1386 NEW A: 17738 B: 17545 C: 291
area 845064900 square root 29070 NEW A: 89297 B: 58922 C: 38475
area 15023376 square root 3876 NEW A: 104257 B: 104040 C: 361
area 2433600 square root 1560 NEW A: 21605 B: 21152 C: 507
area 2683044 square root 1638 NEW A: 3370 B: 2873 C: 1869
area 853776 square root 924 NEW A: 3944 B: 2425 C: 1617
area 1502337600 square root 38760 NEW A: 148817 B: 127433 C: 30600
area 2944656 square root 1716 NEW A: 4035 B: 3232 C: 1859
area 608400 square root 780 NEW A: 2535 B: 2329 C: 544
area 15920100 square root 3990 NEW A: 37544 B: 37405 C: 861
area 54756 square root 234 NEW A: 8787 B: 8749 C: 40
area 2822796900 square root 53130 NEW A: 322921 B: 300174 C: 29095
area 108900 square root 330 NEW A: 1089 B: 641 C: 520
area 756900 square root 870 NEW A: 2307 B: 1898 C: 841
area 1542132900 square root 39270 NEW A: 93049 B: 74282 C: 42483
area 6529932864 square root 80808 NEW A: 426505 B: 363531 C: 71188
area 1192464 square root 1092 NEW A: 2785 B: 2548 C: 939
area 125888400 square root 11220 NEW A: 33625 B: 28712 C: 9537
area 83328059556 square root 288666 NEW A: 514385 B: 501903 C: 349496
area 1490732100 square root 38610 NEW A: 218378 B: 217811 C: 13689
area 81540900 square root 9030 NEW A: 78625 B: 78282 C: 2107
area 5318784900 square root 72930 NEW A: 261064 B: 237655 C: 48841
area 298116 square root 546 NEW A: 1183 B: 865 C: 696
area 3755844 square root 1938 NEW A: 6137 B: 5274 C: 1585
area 2822400 square root 1680 NEW A: 5811 B: 4285 C: 1904
area 1186113600 square root 34440 NEW A: 86921 B: 80441 C: 29520
area 512207424 square root 22632 NEW A: 64160 B: 51993 C: 21689
area 5731236 square root 2394 NEW A: 41497 B: 41210 C: 399
area 203918400 square root 14280 NEW A: 296881 B: 296436 C: 1445
area 67076100 square root 8190 NEW A: 38321 B: 36946 C: 3825
area 4600908900 square root 67830 NEW A: 143153 B: 127547 C: 72250
area 8421732900 square root 91770 NEW A: 314041 B: 301891 C: 56250
area 2375673507684 square root 1541322 NEW A: 4811210 B: 3761397 C: 1540081
area 8615552400 square root 92820 NEW A: 271449 B: 158161 C: 141610
area 925281492033600 square root 30418440 NEW A: 667317681 B: 667171609 C: 2777290
area 4161600 square root 2040 NEW A: 6936 B: 5689 C: 1825
area 183060900 square root 13530 NEW A: 406681 B: 405121 C: 1802
area 60735680756100 square root 7793310 NEW A: 19822952 B: 19311245 C: 6311053
area 254721600 square root 15960 NEW A: 183968 B: 183625 C: 2793
area 32216976 square root 5676 NEW A: 20857 B: 20339 C: 3180
area 2722500 square root 1650 NEW A: 12375 B: 12104 C: 521
area 630813456 square root 25116 NEW A: 119820 B: 110137 C: 14651
area 261468900 square root 16170 NEW A: 95489 B: 62218 C: 33957
area 1644729300900 square root 1282470 NEW A: 2543674 B: 1883921 C: 1746405
area 488852100 square root 22110 NEW A: 2407754 B: 2406129 C: 1675
area 2340900 square root 1530 NEW A: 5655 B: 3593 C: 2312
area 7452900 square root 2730 NEW A: 57322 B: 55133 C: 2205
area 335988900 square root 18330 NEW A: 114568 B: 59943 C: 55225
area 36072036 square root 6006 NEW A: 21601 B: 20735 C: 3528
area 19035720900 square root 137970 NEW A: 3127445 B: 3114566 C: 17739
area 1549944144648900 square root 39369330 NEW A: 88132184 B: 86969971 C: 36207405
area 38564100 square root 6210 NEW A: 16767 B: 13625 C: 6058
area 4769856 square root 2184 NEW A: 21623 B: 10985 C: 10656
area 499947984900 square root 707070 NEW A: 4268513 B: 4187393 C: 250120
area 30569025600 square root 174840 NEW A: 1140955 B: 1083521 C: 79524
area 44755094916 square root 211554 NEW A: 841317 B: 612170 C: 261121
area 4616610876900 square root 2148630 NEW A: 7975201 B: 6948378 C: 1613579
area 705600 square root 840 NEW A: 1904 B: 1285 C: 1131
area 768398400 square root 27720 NEW A: 75605 B: 68224 C: 22869
area 42928704 square root 6552 NEW A: 49825 B: 47979 C: 2548
area 5946087171600 square root 2438460 NEW A: 5089131 B: 4709536 C: 2541845
area 4281870096 square root 65436 NEW A: 171745 B: 102824 C: 95817
area 53233279500996 square root 7296114 NEW A: 299665416 B: 299546065 C: 374863
area 68018452128900 square root 8247330 NEW A: 44096521 B: 40677129 C: 4693000
area 77053284 square root 8778 NEW A: 61610 B: 43339 C: 18513
area 105267600 square root 10260 NEW A: 28729 B: 24245 C: 9234
area 294458169600 square root 542640 NEW A: 2509000 B: 2466713 C: 240737
area 837872945316 square root 915354 NEW A: 16248145 B: 16243656 C: 103247
area 5266404900 square root 72570 NEW A: 190250 B: 121329 C: 99179
area 23629838400 square root 153720 NEW A: 731536 B: 690605 C: 78141
area 1728896400 square root 41580 NEW A: 429529 B: 421619 C: 11340
area 334268985600 square root 578160 NEW A: 1053184 B: 955849 C: 719415
area 5205622500 square root 72150 NEW A: 160625 B: 125681 C: 83694
area 4410504014400 square root 2100120 NEW A: 3862875 B: 3590444 C: 2531281
area 18403635600 square root 135660 NEW A: 643897 B: 522228 C: 137275
area 144288144 square root 12012 NEW A: 29857 B: 23716 C: 12675
area 262175682585600 square root 16191840 NEW A: 81131008 B: 81049875 C: 6471917
area 1907942400 square root 43680 NEW A: 270529 B: 262081 C: 16640
area 63680400 square root 7980 NEW A: 162401 B: 159250 C: 3249
area 253895054400 square root 503880 NEW A: 1829081 B: 1328841 C: 597922
area 707435752464 square root 841092 NEW A: 1838745 B: 1791656 C: 800513
area 7609770756 square root 87234 NEW A: 283465 B: 253674 C: 64387
area 98711585856 square root 314184 NEW A: 912925 B: 894241 C: 220932
area 652178995776 square root 807576 NEW A: 5078975 B: 4715809 C: 450528
area 12531600 square root 3540 NEW A: 20761 B: 17405 C: 3606
area 40602384 square root 6372 NEW A: 62530 B: 59177 C: 3609
area 214726584996 square root 463386 NEW A: 1977049 B: 1590817 C: 456170
area 1129118760000 square root 1062600 NEW A: 4995848 B: 4917027 C: 462875
area 37779696900 square root 194370 NEW A: 2255591 B: 2230841 C: 41800
area 1098922500 square root 33150 NEW A: 217864 B: 204961 C: 16575
area 5336100 square root 2310 NEW A: 27843 B: 27608 C: 451
area 2039184 square root 1428 NEW A: 4760 B: 2897 C: 2169
area 736164 square root 858 NEW A: 4290 B: 4097 C: 401
area 214220865600 square root 462840 NEW A: 940068 B: 863537 C: 500395
area 66977856 square root 8184 NEW A: 33759 B: 30880 C: 5057
area 5946260544 square root 77112 NEW A: 522193 B: 517280 C: 23409
area 353816100 square root 18810 NEW A: 61534 B: 59691 C: 11875
area 3920400 square root 1980 NEW A: 4706 B: 4015 C: 1971
area 22230810000 square root 149100 NEW A: 531752 B: 385563 C: 176435
area 1976316553231524 square root 44455782 NEW A: 89352886 B: 84998657 C: 47309145
area 146313216 square root 12096 NEW A: 73385 B: 59392 C: 14679
area 773129147040000 square root 27805200 NEW A: 43077645 B: 42936269 C: 40857664
area 182305088784 square root 426972 NEW A: 18747170 B: 18742777 C: 19941
area 133402500 square root 11550 NEW A: 391017 B: 390922 C: 689
area 381655296 square root 19536 NEW A: 607539 B: 606505 C: 1628
area 451777556736 square root 672144 NEW A: 3197377 B: 2697889 C: 586850
area 761270210064 square root 872508 NEW A: 1793312 B: 1565785 C: 974169
area 3942584100 square root 62790 NEW A: 763921 B: 756171 C: 12950
area 40040010000 square root 200100 NEW A: 444889 B: 411361 C: 195000
area 3760142400 square root 61320 NEW A: 466192 B: 390025 C: 78183
area 42766498952100 square root 6539610 NEW A: 32024561 B: 30601480 C: 3082959
area 50979600 square root 7140 NEW A: 19025 B: 15912 C: 6713
area 45319824 square root 6732 NEW A: 16577 B: 13600 C: 6849
area 499651059600 square root 706860 NEW A: 8115833 B: 8085625 C: 127008
area 48024900 square root 6930 NEW A: 970250 B: 970249 C: 99
jagy@phobeusjunior:~$
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
report on square roots value of m is 5000
6 = 2 3 squared 36
30 = 2 3 5 squared 900
84 = 2^2 3 7 squared 7056
180 = 2^2 3^2 5 squared 32400
204 = 2^2 3 17 squared 41616
210 = 2 3 5 7 squared 44100
234 = 2 3^2 13 squared 54756
252 = 2^2 3^2 7 squared 63504
264 = 2^3 3 11 squared 69696
330 = 2 3 5 11 squared 108900
420 = 2^2 3 5 7 squared 176400
462 = 2 3 7 11 squared 213444
546 = 2 3 7 13 squared 298116
630 = 2 3^2 5 7 squared 396900
660 = 2^2 3 5 11 squared 435600
780 = 2^2 3 5 13 squared 608400
840 = 2^3 3 5 7 squared 705600
858 = 2 3 11 13 squared 736164
870 = 2 3 5 29 squared 756900
924 = 2^2 3 7 11 squared 853776
1020 = 2^2 3 5 17 squared 1040400
1092 = 2^2 3 7 13 squared 1192464
1170 = 2 3^2 5 13 squared 1368900
1386 = 2 3^2 7 11 squared 1920996
1428 = 2^2 3 7 17 squared 2039184
1530 = 2 3^2 5 17 squared 2340900
1554 = 2 3 7 37 squared 2414916
1560 = 2^3 3 5 13 squared 2433600
1638 = 2 3^2 7 13 squared 2683044
1650 = 2 3 5^2 11 squared 2722500
1680 = 2^4 3 5 7 squared 2822400
1716 = 2^2 3 11 13 squared 2944656
1938 = 2 3 17 19 squared 3755844
1980 = 2^2 3^2 5 11 squared 3920400
2040 = 2^3 3 5 17 squared 4161600
2184 = 2^3 3 7 13 squared 4769856
2310 = 2 3 5 7 11 squared 5336100
2394 = 2 3^2 7 19 squared 5731236
2730 = 2 3 5 7 13 squared 7452900
3540 = 2^2 3 5 59 squared 12531600
3876 = 2^2 3 17 19 squared 15023376
3990 = 2 3 5 7 19 squared 15920100
4158 = 2 3^3 7 11 squared 17288964
5676 = 2^2 3 11 43 squared 32216976
6006 = 2 3 7 11 13 squared 36072036
6210 = 2 3^3 5 23 squared 38564100
6372 = 2^2 3^3 59 squared 40602384
6552 = 2^3 3^2 7 13 squared 42928704
6732 = 2^2 3^2 11 17 squared 45319824
6930 = 2 3^2 5 7 11 squared 48024900
7140 = 2^2 3 5 7 17 squared 50979600
7980 = 2^2 3 5 7 19 squared 63680400
8184 = 2^3 3 11 31 squared 66977856
8190 = 2 3^2 5 7 13 squared 67076100
8778 = 2 3 7 11 19 squared 77053284
9030 = 2 3 5 7 43 squared 81540900
10260 = 2^2 3^3 5 19 squared 105267600
11220 = 2^2 3 5 11 17 squared 125888400
11550 = 2 3 5^2 7 11 squared 133402500
12012 = 2^2 3 7 11 13 squared 144288144
12096 = 2^6 3^3 7 squared 146313216
13530 = 2 3 5 11 41 squared 183060900
14280 = 2^3 3 5 7 17 squared 203918400
15960 = 2^3 3 5 7 19 squared 254721600
16170 = 2 3 5 7^2 11 squared 261468900
18330 = 2 3 5 13 47 squared 335988900
18810 = 2 3^2 5 11 19 squared 353816100
19536 = 2^4 3 11 37 squared 381655296
22110 = 2 3 5 11 67 squared 488852100
22632 = 2^3 3 23 41 squared 512207424
25116 = 2^2 3 7 13 23 squared 630813456
27720 = 2^3 3^2 5 7 11 squared 768398400
29070 = 2 3^2 5 17 19 squared 845064900
33150 = 2 3 5^2 13 17 squared 1098922500
34440 = 2^3 3 5 7 41 squared 1186113600
38610 = 2 3^3 5 11 13 squared 1490732100
38760 = 2^3 3 5 17 19 squared 1502337600
39270 = 2 3 5 7 11 17 squared 1542132900
41580 = 2^2 3^3 5 7 11 squared 1728896400
43680 = 2^5 3 5 7 13 squared 1907942400
53130 = 2 3 5 7 11 23 squared 2822796900
61320 = 2^3 3 5 7 73 squared 3760142400
62790 = 2 3 5 7 13 23 squared 3942584100
65436 = 2^2 3 7 19 41 squared 4281870096
67830 = 2 3 5 7 17 19 squared 4600908900
72150 = 2 3 5^2 13 37 squared 5205622500
72570 = 2 3 5 41 59 squared 5266404900
72930 = 2 3 5 11 13 17 squared 5318784900
77112 = 2^3 3^4 7 17 squared 5946260544
80808 = 2^3 3 7 13 37 squared 6529932864
87234 = 2 3 7 31 67 squared 7609770756
91770 = 2 3 5 7 19 23 squared 8421732900
92820 = 2^2 3 5 7 13 17 squared 8615552400
135660 = 2^2 3 5 7 17 19 squared 18403635600
137970 = 2 3^3 5 7 73 squared 19035720900
149100 = 2^2 3 5^2 7 71 squared 22230810000
153720 = 2^3 3^2 5 7 61 squared 23629838400
174840 = 2^3 3 5 31 47 squared 30569025600
194370 = 2 3 5 11 19 31 squared 37779696900
200100 = 2^2 3 5^2 23 29 squared 40040010000
211554 = 2 3^2 7 23 73 squared 44755094916
288666 = 2 3^2 7 29 79 squared 83328059556
314184 = 2^3 3 13 19 53 squared 98711585856
426972 = 2^2 3 7 13 17 23 squared 182305088784
462840 = 2^3 3 5 7 19 29 squared 214220865600
463386 = 2 3 7 11 17 59 squared 214726584996
503880 = 2^3 3 5 13 17 19 squared 253895054400
542640 = 2^4 3 5 7 17 19 squared 294458169600
578160 = 2^4 3^2 5 11 73 squared 334268985600
672144 = 2^4 3 11 19 67 squared 451777556736
706860 = 2^2 3^3 5 7 11 17 squared 499651059600
707070 = 2 3 5 7^2 13 37 squared 499947984900
807576 = 2^3 3 7 11 19 23 squared 652178995776
841092 = 2^2 3 7 17 19 31 squared 707435752464
872508 = 2^2 3 7 13 17 47 squared 761270210064
915354 = 2 3^3 11 23 67 squared 837872945316
1062600 = 2^3 3 5^2 7 11 23 squared 1129118760000
1282470 = 2 3 5 7 31 197 squared 1644729300900
1541322 = 2 3^3 17 23 73 squared 2375673507684
2100120 = 2^3 3 5 11 37 43 squared 4410504014400
2148630 = 2 3 5 11 17 383 squared 4616610876900
2438460 = 2^2 3^2 5 19 23 31 squared 5946087171600
6539610 = 2 3 5 7 11 19 149 squared 42766498952100
7296114 = 2 3 7 19 41 223 squared 53233279500996
7793310 = 2 3 5 7 17 37 59 squared 60735680756100
8247330 = 2 3^2 5 7 13 19 53 squared 68018452128900
16191840 = 2^5 3 5 7 61 79 squared 262175682585600
27805200 = 2^4 3 5^2 17 29 47 squared 773129147040000
30418440 = 2^3 3 5 13 17 31 37 squared 925281492033600
39369330 = 2 3^2 5 7 11 13 19 23 squared 1549944144648900
44455782 = 2 3 7 17 19 29 113 squared 1976316553231524
brief report over