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I tried to go all over the bases but it doesn't work well. I don't know how to present multiplying in general... I believe I somehow have to compute 11 times 11 in every base and convert it to decimal number and check if it bigger than 122. Any thoughts?

mathreadler
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Zap Sira
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    The bases should be $r>2$ because $2$ is a digit. We also must have $r^2+2r+1=(r+1)^2<r^2+2r+2$, which is equivalent to $0<1$. So all bases $r>2$ are good. – Nina Simone Jul 26 '17 at 10:35
  • The multiplication algorithm in any basis can be implemented by convolution with carry and the largest scalar resulting from the convolution $[1,1] * [1,1] = [1,2,1]$ is $2$ which won't overflow and cause carry for any $r>2$. But for $r=2$ we have $11_2 = 3$, $3^2 = 1001$ which we see has overflowed since it has more digits. – mathreadler Jul 26 '17 at 10:37

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If the basis is greater than two, then $$\begin{array}{ccc} &1&1\\ \times&1&1\\ \hline &1&1\\ 1&1\\ \hline 1&2&1 \end{array}$$

ajotatxe
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