Definition:Rare Number

Definition

A rare number is a non-palindromic integer $n$ which has the property that $n + r$ and $n - r$ are both square, where $r$ is the reversal of $n$.

Sequence

The sequence of rare numbers begins:

$65, 621 \, 770, 281 \, 089 \, 082, 2 \, 022 \, 652 \, 202, 2 \, 042 \, 832 \, 002, 868 \, 591 \, 084 \, 757$

Examples

$65$ is a Rare Number

\(\ds 65 + 56\)

\(=\)

\(\ds 121\)

\(\ds \)

\(=\)

\(\ds 11^2\)

\(\ds 65 - 56\)

\(=\)

\(\ds 9\)

\(\ds \)

\(=\)

\(\ds 3^2\)

$621 \, 770$ is a Rare Number

\(\ds 621 \, 770 + 077 \, 126\)

\(=\)

\(\ds 698 \, 896\)

\(\ds \)

\(=\)

\(\ds 836^2\)

\(\ds 621 \, 770 - 077 \, 126\)

\(=\)

\(\ds 544 \, 644\)

\(\ds \)

\(=\)

\(\ds 738^2\)

Historical Note

The concept of a rare number appears to originate with Shyam Sunder Gupta, who has written some articles about them.