No 45

Square Products Minus 1

What is special about the set of numbers 1, 3 and 8 ? Let's take all the possible products of pairs of these numbers:

1 x 3 = 3 = 42 - 1

1 x 8 = 8 = 32 - 1

3 x 8 = 24 = 52 - 1

They are all one less than a square. Can you find any other triples of whole numbers that have this property? Is it possible to find a set of four numbers with this property? Yes it is. We need only add the number 120 to 1, 3 and 8 :

1 x 3 = 3 = 42 - 1	1 x 120 = 120 = 112-1
1 x 8 = 8 = 32 - 1	3 x 120 = 360 = 192-1
3 x 8 = 24 = 52 - 1	8 x 120 = 960 = 312-1

Can you discover other sets of 4 whole numbers that have this property? Is it possible to find a set of five whole numbers with this property? Amazingly the answer is NO!! This was only proved in the year 1969.

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