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Pythagorean triangle (Posted on 2014-03-10) Difficulty: 2 of 5
Find the smallest Pythagorean triangle, satisfying the following conditions:

1. its perimeter is a perfect square.
2. its area is a perfect cube.

See The Solution Submitted by Ady TZIDON    
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Solution The 21 smallest | Comment 1 of 2

DEFDBL A-Z
OPEN "pythagorean triangles.txt" FOR OUTPUT AS #2
FOR cr = 1 TO 9999
    area = cr * cr * cr
    prod = area * 2

    FOR a = 1 TO SQR(prod)
        b = prod / a
        IF b = INT(b) THEN
            hyp2 = a * a + b * b
            hyp = INT(SQR(hyp2) + .5)
            IF hyp * hyp = hyp2 THEN
                peri = a + b + hyp
                sr = INT(SQR(peri) + .5)
                IF sr * sr = peri THEN
                    PRINT USING "######## ######## ########    ######### ##### ############# #####"; a; b; hyp, peri; sr, area; cr
                    PRINT #2, USING "######## ######## ########    ######### ##### ############# #####"; a; b; hyp, peri; sr, area; cr
                END IF
            END IF
        END IF
    NEXT
NEXT cr

finds the following, with areas less than 10,000^3:

        legs       hypotenuse  perimeter  sqrt     area      cube root  
     144      192      240          576    24         13824    24
     150      360      390          900    30         27000    30
    9216    12288    15360        36864   192      56623104   384
    9600    23040    24960        57600   240     110592000   480
   12960    57600    59040       129600   360     373248000   720
   20808   176256   177480       374544   612    1833767424  1224
  113400   119070   164430       396900   630    6751269000  1890
  104976   139968   174960       419904   648    7346640384  1944
  103680   194400   220320       518400   720   10077696000  2160
   37026   610368   611490      1258884  1122   11299742784  2244
  109350   262440   284310       656100   810   14348907000  2430
  113256   301158   321750       736164   858   17053975224  2574
  127008   435456   453600      1016064  1008   27653197824  3024
  158400   784080   799920      1742400  1320   62099136000  3960
  191646  1238328  1253070      2683044  1638  118660303944  4914
  208568  1504926  1519310      3232804  1798  156939702984  5394
  589824   786432   983040      2359296  1536  231928233984  6144
  345744  1361367  1404585      3111696  1764  235342236024  6174
  259920  2462400  2476080      5198400  2280  320013504000  6840
  614400  1474560  1597440      3686400  1920  452984832000  7680
  363776  5086368  5099360     10549504  3248  925149302784  9744
 

so the smallest is  (144, 192, 240).


  Posted by Charlie on 2014-03-10 15:42:35
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