No 40

The Divided Sum

 

Write down the first three positive integers:

1 2 3

Keeping the order the same it is possible to divide them into two groups with the same total:

1 + 2 = 3

This is not possible with 1 2 3 4 or with 1 2 3 4 5 or....and further examples are extremely rare. So what is the next example? Check the following calculation:

1+2+3+4+5+6+7+8+9+10+11+12+13+14=15+16+17+18+19+20

The crucial numbers are 2, 14 ... and 3, 20 ... What are the next two numbers in these sequences? Look at these calculations:

6 x 14 - 2 + 2 = 84 and 6 x 20 - 3 + 2 = 119

Now check the calculation:

1+2+3+ ... +84 = 85+86+87+ ... +119

Is the next example given by:

6 x 84 - 14 + 2 = 492 and 6 x 119 - 20 + 2 = 696 ?

What have these amazing sums got to do with triangular numbers that are double other triangular numbers? And what have they to do with Pell's Equation?

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