Sequence of Squares Beginning and Ending with n 4s

Sequence

The following sequence:

\(\ds 484\)

\(=\)

\(\ds 22^2\)

\(\ds 44 \, 944\)

\(=\)

\(\ds 212^2\)

\(\ds 444 \, 171 \, 597 \, 444\)

\(=\)

\(\ds 666 \, 462^2\)

cannot be continued, as it is not possible for there to be a square number ending in $\ldots 4444$.

Proof

Let $n = 10000 k + 4444$.

We have:

\(\ds \frac n 4\)

\(=\)

\(\ds 2500 k + 1111\)

\(\ds \)

\(\equiv\)

\(\ds 3 \pmod 4\)

By Square Modulo 4, $\dfrac n 4$ cannot be a square number.

Therefore neither can $n = 4 \times \dfrac n 4$ be a square number.

$\blacksquare$

Historical Note

David Wells, in his $1997$ work Curious and Interesting Numbers, 2nd ed., attributes this result to Michel Criton, but gives no details.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $444,171,597,444$