Radix

We will show that in the base 7 number system:

$41^2 = 2311$

By converting these numbers into the decimal system, we get:

$(4×7 + 1)^2 = 2 × 7^3+3×7^2+1×7+1$

$29^2 = 841$

$841 = 841$

$\text { Hence, our answer should be } \boxed {7}$

Remark:

We can obtain our answer 7 by trial and improvement.

However, for the sake of completeness we should rather solve the following equation (n is the base of the number sytem, therefore a positive integer, n ≥ 5 as we have 4 as the largest digit):

$(4n + 1)^2 = 2n^3 + 3n^2 + n + 1$

$16n^2 + 8n + 1 = 2n^3 + 3n^2 + n + 1$

$2n^3 - 13n^2 - 7n = 0$

$n(2n + 1)(n - 7) = 0$

$n_1 = 0, n_2 = -0.5 \text { (both are extraneous roots)}, \boxed { n_3 = 7 }$