I need to convert some really big numbers from base-10 to base-8. All online converters I found cap the conversion at around 20 digits. Are there any downloadable programs with no digit limit? I'm running Windows 10, but Linux or Mac are possibilities if necessary to accomplish this.
Big Integer Base Converter
number-systems
Related Solutions
6/-2=-3 with remainder 0, not 1
-3/-2=2 with remainder 1 (remember that your remainders have to be positive, like the -5/-3 in the example)
2/-2=-1 with remainder 0
-1/-2=1 with remainder 1
1/-2=0 with remainder 1
giving $11010_{-2}=6_{10}$
For your question about largest and smallest numbers representable in $2n$ bits, the largest positive number has all the positive bits set, so you have $1+4+16+\ldots$ This is $\sum_{i=0}^{n-1}4^i=\frac{4^n-1}{3}$. The smallest number has all the negative bits set and is twice this large in absolute value, so it is $-\frac{2(4^n-1)}{3}$
Only a partial answer:
To prove the three digit pattern, I find it easiest to write it in terms of $b$, the lowest base, which has to be even and at least $6$. Then we have $$(\frac b2+1)b^2+(\frac b2+2)b+(\frac b2+1)\\= (\frac b2)(b+1)^2+(\frac b2+1)(b+1)+(\frac b2)\\= (\frac b2-1)(b+2)^2+(\frac b2+3)(b+2)+(\frac b2-1)\\= \frac{b^3}2+\frac {3b^2}2+\frac {5b}2+1$$ where the first three lines make the palindrome explicit in the three bases. I think finding this pattern is rather easy. If one did a computer search up to $1000$ one would find the first four numbers and the pattern is clear. The algebra to verify it is also not hard. We can prove that this pattern will never extend to a fourth base. If we try base $b-1$ we can divide the number by $(b-1)^2+1$ to find the first and third digit. We find it is $\frac b2+2$ as one might expect. The middle digit wants to be $\frac b2+6$ but the total is too high by $3$. Similarly if we try base $b+3$ we find the first and last digits are $\frac b2-2$, the closest middle digit is $\frac b2+8$, but we are $3$ too high again. These patterns are only established by $b=16$ for base $b-1$ and $b=12$ for $b+3$ but we can easily check the smaller numbers. This does not prove that there are no other examples for four successive bases. I think a similar analysis could be done for the five digit pattern but I haven't done it.
Best Answer
Python (the programming language and environment, you can download the IDLE environment on Windows, e.g.) has support for arbitrary precision integers (bigint) natively. It's not hard to convert bases using this (oct and hex are built-in functions). One can easily make a command line tool to do this (e.g. to treat numbers stored in a file if size is too large to write the numbers on the command line).
On MacOS and most Linuces python comes pre-installed. So nothing to do there.
It's free, easy to learn and requires no internet connection (as opposed to online tools). Check it out.