Consider a triangle with sides of length 5, 6, 7. If you square the area of that triangle, you get 216, a perfect cube.
Are there other triangles (not geometrically similar to the first triangle) with integral sides whose area squared is a perfect cube? Find one such triangle, or prove no others exist.
area^2 cube root
2 3 3 8 2
5 6 7 216 6
3 10 11 216 6
7 12 13 1728 12
6 14 16 1728 12
9 20 21 8000 20
10 16 24 3375 15
18 18 18 19683 27
16 24 24 32768 32
14 20 30 13824 24
7 27 30 8000 20
18 24 30 46656 36
11 30 31 27000 30
12 39 45 46656 36
13 42 43 74088 42
21 31 46 74088 42
30 32 38 216000 60
25 30 45 125000 50
18 45 45 157464 54
10 51 59 27000 30
14 51 61 74088 42
15 56 57 175616 56
40 48 56 884736 96
24 54 66 373248 72
30 57 63 729000 90
39 45 66 729000 90
30 50 70 421875 75
17 72 73 373248 72
9 73 80 46656 36
40 66 74 1728000 120
60 60 72 2985984 144
46 60 86 1728000 120
24 80 88 884736 96
46 51 95 216000 60
48 70 78 2744000 140
35 66 95 592704 84
55 70 75 3375000 150
51 65 84 2744000 140
19 90 91 729000 90
20 84 96 512000 80
10 94 96 216000 60
54 81 81 4251528 162
60 60 96 2985984 144
12 102 102 373248 72
33 78 105 729000 90
21 110 111 1331000 110
33 97 112 2299968 132
13 110 119 287496 66
But the above has some that are geometrically similar to others preceding. The following list prevents this:
2 3 3 8 2
5 6 7 216 6
3 10 11 216 6
7 12 13 1728 12
6 14 16 1728 12
9 20 21 8000 20
10 16 24 3375 15
18 18 18 19683 27
14 20 30 13824 24
7 27 30 8000 20
18 24 30 46656 36
11 30 31 27000 30
12 39 45 46656 36
13 42 43 74088 42
21 31 46 74088 42
30 32 38 216000 60
25 30 45 125000 50
18 45 45 157464 54
10 51 59 27000 30
14 51 61 74088 42
15 56 57 175616 56
24 54 66 373248 72
30 57 63 729000 90
39 45 66 729000 90
30 50 70 421875 75
17 72 73 373248 72
9 73 80 46656 36
40 66 74 1728000 120
60 60 72 2985984 144
46 60 86 1728000 120
46 51 95 216000 60
48 70 78 2744000 140
35 66 95 592704 84
55 70 75 3375000 150
51 65 84 2744000 140
19 90 91 729000 90
20 84 96 512000 80
10 94 96 216000 60
60 60 96 2985984 144
12 102 102 373248 72
33 78 105 729000 90
21 110 111 1331000 110
33 97 112 2299968 132
13 110 119 287496 66
35 105 110 3375000 150
17 109 124 216000 60
20 112 124 884736 96
11 123 130 287496 66
The program uses Heron's formula. It first does the first table, and then the second above.
DEFDBL A-Z
CLS
FOR t = 7 TO 1000
s = t / 2
FOR big = INT(t / 3) TO t / 2
FOR mid = INT((t - big) / 2) TO big
little = t - big - mid
IF little > 0 AND little <= mid AND mid <= big AND big < s THEN
sqAr = s * (s - big) * (s - mid) * (s - little)
cuRt = INT(sqAr ^ (1 / 3) + .5)
IF cuRt * cuRt * cuRt = sqAr THEN
PRINT USING "### ### ### ########## ###"; little; mid; big; sqAr; cuRt
ct = ct + 1
IF ct > 47 THEN GOTO nextPhase
END IF
END IF
NEXT
NEXT
NEXT
nextPhase:
ct = 0
DIM sol(100, 3)
FOR t = 7 TO 1000
s = t / 2
FOR big = INT(t / 3) TO t / 2
FOR mid = INT((t - big) / 2) TO big
little = t - big - mid
IF little > 0 AND little <= mid AND mid <= big AND big < s THEN
sqAr = s * (s - big) * (s - mid) * (s - little)
cuRt = INT(sqAr ^ (1 / 3) + .5)
IF cuRt * cuRt * cuRt = sqAr THEN
good = 1
FOR i = 1 TO ct
IF sol(i, 1) * mid = sol(i, 2) * little AND sol(i, 3) * mid = sol(i, 2) * big THEN good = 0: EXIT FOR
NEXT
IF good THEN
PRINT USING "### ### ### ########## ###"; little; mid; big; sqAr; cuRt
ct = ct + 1
sol(ct, 1) = little
sol(ct, 2) = mid
sol(ct, 3) = big
IF ct > 47 THEN END
END IF
END IF
END IF
NEXT
NEXT
NEXT
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Posted by Charlie
on 2007-03-21 15:57:32 |
computer list
