Question

Prove that either \(2 \cdot 1{0^{500}} + 15\,\, or\,\, 2\cdot1{0^{500}} + 16\) is not a perfect square. Is your proof constructive or non constructive?

Short answer

Both the numbers \({\bf{2 \cdot 1}}{{\bf{0}}^{{\bf{500}}}}{\bf{ + 15}}\) and \({\bf{2\cdot1}}{{\bf{0}}^{{\bf{500}}}}{\bf{ + 16}}\) cannot be a perfect square.

The proof used is non constructive.

Step-by-step solution

Introduction

Consider both the numbers to be perfect square.

\({\bf{2 \cdot 1}}{{\bf{0}}^{{\bf{500}}}}{\bf{ + 15}}\)and \({\bf{2\cdot1}}{{\bf{0}}^{{\bf{500}}}}{\bf{ + 16}}\)

Here \(x\) and \(y\) are distinct positive integral values.

Taking the subtraction and using nonconstructive method 

Consider the difference:

\(\begin{aligned}{}{y^2} - {x^2} = \left( {{{2.100}^{500}} + 16} \right) - \left( {{{2.100}^{500}} + 15} \right)\\\left( {y + x} \right)\left( {y - x} \right) = 1\end{aligned}\)

The positive factors of 1 are 1 itself, so \(y + x = 1\)and \(y - x = 1\).

This is not possible.

Hence, both the numbers cannot be a perfect square.

This proof is nonconstructive as constructive proof would have produced a factorization of one of the numbers and checked it for a perfect square.