What is the Rare Number?

A rare number is a specific type of positive integer that possesses unique mathematical properties when combined with its reverse. To qualify as a rare number, a positive integer R must satisfy two conditions simultaneously: when R is added to its reverse (R1), and when R is subtracted from its reverse (R1), both resulting values must be perfect squares.

Understanding the Concept of a Rare Number

The definition hinges on two key components:

Original Number (R): Any positive integer.
Reverse Number (R1): The integer formed by writing the decimal digits of R in reverse order. For example, if R = 65, then R1 = 56. If R = 123, then R1 = 321. Leading zeros in R1 are dropped (e.g., reverse of 120 is 21).

For a number R to be classified as rare, the following two conditions must hold true:

Sum is a Perfect Square: The sum R + R1 must result in a perfect square (i.e., the square of an integer).
Difference is a Perfect Square: The absolute difference |R - R1| must also result in a perfect square. While the reference states R - R1, it's usually implied to be the positive difference to ensure a positive result for a square, or R is assumed to be greater than R1. For symmetry and clarity, we consider the absolute difference.

Examples of Rare Numbers

Let's illustrate the concept with an example to clarify the conditions.

Consider the number R = 65.

Step 1: Find R1.
- The reverse of 65 is 56. So, R1 = 56.
Step 2: Calculate R + R1.
- 65 + 56 = 121
- 121 is a perfect square, as 11 * 11 = 121.
Step 3: Calculate R - R1.
- 65 - 56 = 9
- 9 is a perfect square, as 3 * 3 = 9.

Since both R + R1 and R - R1 are perfect squares, 65 is indeed a rare number.

Here's a small table demonstrating the properties for R = 65:

Property	Value	Result	Perfect Square?
Original Number (R)	65
Reverse Number (R1)	56
R + R1	65 + 56	121	Yes (11²)
R - R1	65 - 56	9	Yes (3²)

This specific combination of properties makes rare numbers an interesting subject in recreational mathematics and number theory.