Exploring Numbers That, When Added to Their Reversals, Result in a Perfect Square

Have you ever wondered which numbers, when added to their digit reversals, result in a perfect square? In this article, we will delve into the fascinating world of number theory to uncover such numbers. We will also dive into the methods to solve this intriguing problem and even provide some examples to solidify our understanding.

Defining the Problem

Let us denote a number ( n ) and its reversal ( r_n ). We aim to solve the equation:

[textcolor{blue}{n r_n k^2}]

where ( k ) is an integer.

Steps to Solve the Problem

1. Define the Number and Its Reversal

Consider a number ( n ) with ( d ) digits, represented as ( a_1 a_2 ldots a_d ) where ( a_1 a_2 ldots a_d ) are its digits. The reversal ( r_n ) would be ( a_d a_{d-1} ldots a_1 ).

2. Express the Numbers Mathematically

The value of ( n ) can be expressed as:

[textcolor{blue}{n a_1 cdot 10^{d-1} a_2 cdot 10^{d-2} ldots a_d} ]

Similarly, the reversal ( r_n ) would then be:

[textcolor{blue}{r_n a_d cdot 10^{d-1} a_{d-1} cdot 10^{d-2} ldots a_1} ]

3. Formulate the Equation

We need to check if:

[textcolor{blue}{n r_n k^2}]

Example Calculations

Let us explore some small integers and their reversals to find if their sum results in a perfect square:

n 1

( r1 1 quad Rightarrow quad 1 1 2 quad text{(not a perfect square)})

n 2

( r2 2 quad Rightarrow quad 2 2 4 2^2 quad text{(perfect square)})

n 3

( r3 3 quad Rightarrow quad 3 3 6 quad text{(not a perfect square)})

n 12

( r12 21 quad Rightarrow quad 12 21 33 quad text{(not a perfect square)})

n 21

( r21 12 quad Rightarrow quad 21 12 33 quad text{(not a perfect square)})

n 22

( r22 22 quad Rightarrow quad 22 22 44 quad text{(not a perfect square)})

n 30

( r30 03 3 quad Rightarrow quad 30 3 33 quad text{(not a perfect square)})

n 45

( r45 54 quad Rightarrow quad 45 54 99 quad text{(not a perfect square)})

Conclusion

From the calculations above, we found that:

( n 2 text{is a solution since} 2 2 4 text{which is} 2^2 .)

To find more solutions, one could automate this process with a program to iterate through integers, compute their reversals, and check for perfect squares. However, as of now, ( n 2 ) is a simple example that satisfies the condition.

If you would like to explore this further or need a specific range checked, please let me know!