Question 1185809
<pre>
The products of two numbers among 3,8,21 are 24,63,168.
None of those products increased by 3,8, or 21 are squares.
So those products 24,63,168 must depend upon adding N to them to
give a square sum. Therefore, if there is any solution to the problem:

24+N = a<sup>2</sup> for some positive integer a.
63+N = b<sup>2</sup> for some positive integer b.
168+N = c<sup>2</sup> for some positive integer c.

Thus N = a<sup>2</sup>-24 = b<sup>2</sup>-63 = c<sup>2</sup>-168

That only has one solution, a=5, b=8, and c=13, which makes N=1

However, that means the product 21N is 21. But, no matter which of these:
3,8,21,1 we add to the product 21 gives a square for a sum.

Thus there is no possible positive integer N meeting the conditions
that the product of any two increased by 1 of the numbers 3,8,21,N is a square.

This proves there is no solution. Could there be a typo, perhaps in the number
21?

Edwin</pre>