2
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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?

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1
1
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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
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5
  • $\begingroup$ It is interesting to note that the square root of these primitive Heron areas with perfect squares are themselves the areas of primitive Heron triangles. See OEIS A083875. Consequently these perfect squares are all 0 Mod 36. To ensure that there are no perfect squares less than 32400 it would be necessary to search all integer triangles whose maximum side < 32400 - rather daunting. $\endgroup$ – Frank M Jackson Sep 24 '14 at 22:59
  • $\begingroup$ Well, nothing easy about this; I ran it higher, it appears that the square roots (on the left) are always a power of 2 times a power of 3 times a squarefree number with other very small prime factors. I looked up the original formula, you are solving the Diophantine equation $$ 16 w^4 = 2(b^2c^2 + c^2 a^2 + a^2 b^2) - (a^4 + b^4 + c^4) $$ $\endgroup$ – Will Jagy Sep 24 '14 at 22:59
  • $\begingroup$ @FrankMJackson, you may have something about the square roots. I put in the outcomes of the run with $m \leq 1125,$ one of the new square roots is 330. $\endgroup$ – Will Jagy Sep 24 '14 at 23:05
  • $\begingroup$ I assume (due to its complexity) that there is no direct method of solving the Diophantine equation. Also if $w^2$ is the area of a Heron triangle can it be proved that $w$ is also the area of a Heron triangle? $\endgroup$ – Frank M Jackson Sep 26 '14 at 13:41
  • $\begingroup$ I have found another in the range of square roots - 546 {696,865,1183}. Also it is implying that this set of square roots contains all the areas of primitive Pythagorian triples (6,30,84,180,210,330,546,630) as a subset. Note that 60 and 504 which are also areas of PPT's are missing- see OEIS A024406. So I am searching to see if there exists primitive Heron triangles whose areas are $60^2$ or $504^2$. $\endgroup$ – Frank M Jackson Sep 28 '14 at 6:59
1
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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
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  • $\begingroup$ Your assumption about $60^2$ is correct. I have searched thru all integer triangles up to a max. size of 3600 with no result. The same applies to $504^2$. I have contacted PD Dr Sascha Kurz and he has searched his complete data base of Heron triangles up to max. side 6,000,000. He has identified 1312 square areas. His list is only exhaustive for the first 46 non-duplicated areas up to the value 6,000,000 i.e. $2394^2$. FYI your latest list is missing $714^2,882^2,990^2,1140^2,1158^2,1950^2,2142^2$. $\endgroup$ – Frank M Jackson Oct 1 '14 at 13:19
  • $\begingroup$ Also the square root of these primitive Heron areas with perfect squares are not always the areas of primitive Heron triangles. E.g. $858^2,870^2$. $\endgroup$ – Frank M Jackson Oct 1 '14 at 13:25
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In the system of equations: $$\left\{\begin{aligned}&x+y=a+b\\&xy=qab\end{aligned}\right.$$

Another solution can be written. $$a=p^2+2qps+(q^2-1)s^2$$ $$b=2s(p-(q-1)s)$$ $$x=2qs(p+(q-1)s)$$ $$y=p^2+2ps-(q^2-1)s^2$$

All three formulas derived me just describe all solutions of the system. I think the question can be considered closed.

Solutions can be written as: $$x=qs^2-qps$$ $$y=qps-p^2$$ $$a=ps-p^2$$ $$b=qs^2-ps$$

Or this: $$x=qp^2+qps$$ $$y=qps+(q-1)s^2$$ $$a=qp^2+(2q-1)ps+(q-1)s^2$$ $$b=ps$$

If you meant these triangles Gerona. The formula I have written in this thread. Problem Heron of Alexandria.

They are still stubbornly ignore. Seeking solutions by brute force.

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Well, the formula itself Geronova (Heronian) triangle.

$$S_g=\sqrt{(a+b+c)(a+b-c)(a-b+c)(b+c-a)}$$

If: $p,s,k,t$ -integers asked us. Then the solutions are.

$$a=(pt+ks)(k^2+t^2)ps$$

$$b=(pt-ks)((k^2+t^2)ps+(p^2+s^2)kt)$$

$$c=(pt+ks)(p^2+s^2)kt$$

$$S_g=4pskt(p^2t^2-k^2s^2)((k^2+t^2)ps+(p^2+s^2)kt)$$

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  • $\begingroup$ The problem when using the above formulas is the same as when using the standard Brahmagupta formulas to generate primitive Heron triangles. They both have to be divided by the GCD of the generated sides. So will never know when the generated list at any point (in time) is exhaustive. $\endgroup$ – Frank M Jackson Sep 26 '14 at 13:21
  • $\begingroup$ See Paper by Sacha Kurz $\endgroup$ – Frank M Jackson Sep 27 '14 at 18:55
  • $\begingroup$ @FrankMJackson yeah, I know. I wrote the other formulas. Himself received them. $\endgroup$ – individ Sep 28 '14 at 4:18
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An exhaustive list of the first 46 non-duplicated square areas of primitive Heronian triangles can be found at OEIS A248108. The list was generated by Sascha Kurz searching through lists of primitive Heronian triangles generated by him up to a maximum side length of 6,000,000.

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