Sequence of Squares Beginning and Ending with n 4s
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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): $444,171,597,444$