Definition:Rare Number/Mistake

Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$621,770$

Mistake

$621,770 + 621,770 = 836^2$and $621,770 - 077,126$ (its reversal) $= 738^2$. The only other number with this property, less than $10^8$ is $65$: $65 + 65 = 11^2$ and $65 - 56$ (its reversal) $= 3^2$.

What is under discussion here is a rare number, but it has been misdescribed.

It is clear that $621 \, 770 + 621 \, 770$ is not $836^2$, and equally clearly $65 + 65$ is not equal to $11^2$.

What is meant is the number added to its reversal, not itself, is square:

$621 \, 770 + 077 \, 126 = 698 \, 896 = 836^2$

and:

$65 + 56 = 121 = 11^2$

The fact that there are no more such rare numbers less than $10^8$ is correct, however, as the next one is $281 \, 089 \, 082$:

\(\ds 281 \, 089 \, 082 + 280 \, 980 \, 182\)

\(=\)

\(\ds 562 \, 069 \, 264\)

\(\ds \)

\(=\)

\(\ds 23 \, 708^2\)

\(\ds 281 \, 089 \, 082 - 280 \, 980 \, 182\)

\(=\)

\(\ds 108 \, 900\)

\(\ds \)

\(=\)

\(\ds 330^2\)

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $621,770$