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Special Natural Numbers (Posted on 2007-10-14) Difficulty: 3 of 5
Find all natural numbers which can be written in the form (a+b+c)2/(abc) where a,b,c are also natural numbers.

See The Solution Submitted by Praneeth    
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re: Some were discovered.... | Comment 2 of 8 |
(In reply to Some were discovered.... by Dej Mar)

I got the same set of values in my search, but some of the results are interesting:

I used a search limit of 1600>=a,b,c.  I found 87 sets with c>=b>=a>=1 which yeild an integer n.

The solution breakdown for n is:
n=1: 22;  n=2: 14;  n=3: 13;  n=4: 5;  n=5: 11 
n=6: 11;  n=7: 0;  n=8: 5;  n=9: 6;  n=10+: 0

More interesting were a few of the large finds:
a:12, b:18, c:150, n:1
a:12, b:150, c:1458, n:1
a:16, b:72, c:968, n:1
a:18, b:48, c:726, n:1
a:20, b:25, c:405, n:1
a:25, b:45, c:980, n:1

This suggests that there may be no limit to what values a,b,c can take to yield an integer.

  Posted by Brian Smith on 2007-10-15 11:38:07
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