Find all sets of four natural numbers such that the square of each of them, when added to the sum of the other three, again yields a perfect square.

(In reply to computer exploration--is there a pattern to this? by Charlie)

Charlie asks, "Is there a pattern?"

Here is a copy of Charlie's given print-out for oly those results which begin with "1".   For the ensuing numerals I took the difference between the values for successive lines, eg 5-1 = 4.

In the next column I summed pairs of differences while in the latter I took the difference between successive sums.

1      1      1     1     Diff                   
1      5      5     5       4    Sum     
1      8      8     8       3      7          
1     16    16   16       8            Diff
1     21    21   21       5     13      6
1     33    33   33      12                    
1     40    40   40       7     19      6
1     56    56   56      16                    
1     65    65   65       9     25      6
1     85    85   85      20                    
1     96    96   96      11     31      6
1   120  120  120      24                    
1   133  133  133      13     37      6
1   161  161  161      28                    
1   176  176  176      15     43      6

The first thing that was obvious was a consistent difference of "6".

Now on closer examination of the first differences column it will be noted that the numbers show an odd-even alternation such that the odd numbers increment by 2 and the evens increment by 4.

Therefore the next values in columns 2,3 and 4 should be 176 +32 =  208.

So, yes, there is clearly a predictable pattern, but how do we get from the first being 1 to 6 (6  6  11  11) and 40 where the second column is different againt (40  57 96  96)?

Posted by brianjn on 2008-12-12 04:43:00