Palindromic Squares

Algebra Level 2

A palindromic square is a square number that remains the same when its digits are reversed. For example, both:

$\begin{aligned} 11^2 & = 121 \\ 307^2 & = 94249 \end{aligned}$

are both palindromic squares. How many more palindromic squares are there?

finitely more (but more than zero) infinitely more no more

1 solution

Mark Hennings
Dec 17, 2020

The number $\big(10^n +1\big)^2 \; = \; 10^{2n} + 2\times 10^n + 1 \; = \; 1\overbrace{00...00}^{n-1}2\overbrace{00...00}^{n-1}1$ is a palindromic square for any integer $n \ge 1$ . Indeed $\big(10^n + 1\big)^3$ is always a palindromic cube, and $\big(10^n +1\big)^4$ is always a palindromic fourth power, but it is not known whether there are any palindromic fifth powers.

What if we exclude the digit $0$ from the square numbers? Are there still infinitely many? OEIS doesn't seem to have this subset as its own list; the largest zero-free square in the main list of palindromic squares is the square of $3036233455854775865623$ . What if we exclude $0$ from both the square and its square root?

Chris Lewis - 5 months, 3 weeks ago

I also thought in the same direction. What a telepathy! :)

A Former Brilliant Member - 5 months, 3 weeks ago

Palindromic Squares

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