Definition:Rare Number
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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.