I'm making a very simple program with c++ for linux usage, and I'd like to know if it is possible to make just one big binary containing all the dependencies that would work on any linux system.
If my understanding is correct, any compiler turns source code into machine instructions, but since there are often common parts of code that can be reused with different programs, most programs depend on another libraries.
However if I have the source code for all my dependencies, I should be able to compile a binary in a way that would not require anything from the system? Will I be able to run stuff compiled on 64bit system on a 32bit system?
In short: Maybe.
The longer answer is:
It depends. You can't, for example, run a 64-bit binary on a 32-bit system, that's just not even nearly possible. Yes, it's the same processor family, but there are twice as many registers in the 64-bit system, which also has twice as long registers. What's the 32-bit processor going to "give back" for the value of those bits and registers that doesn't exist in the hardware in the processor? It just plain won't work. Some of the instructions also completely change meaning, so the system really needs to be "right" for the compiled code, or it won't work - fortunately, Linux will check this and plain refuse if it's not right.
You can BUILD a 32-bit binary on a 64-bit system (assuming you have all the right libraries, etc, installed for both 64- and 32-bit, etc).
Similarly, if you try to run ARM code on an x86 processor, or MIPS code on an ARM processor, it simply has no chance of working, because the actual instructions are completely different (or they would be in breach of some patent/copyright or similar, because processor instruction sets contain portions that are "protected intellectual property" in nearly all cases - so designers have to make sure they do NOT do "the same as someone else's design"). Like for 32-bit and 64-bit, you simply won't get a chance to run the wrong binary here, it just won't work.
Sometimes, there are subtle differences, for example ARM code can be compiled with "hard" or "soft" floating point. If the code is compiled for hard float, and there isn't the right support in the OS, then it won't run the binary. Worse yet, if you compile on x86 for SSE instructions, and try to run on a non-SSE processor, the code will simply crash [unless you specifically build code to "check for SSE, and display error if not present"].
So, if you have a binary that passes the above criteria, the Linux system tends to change a tiny bit between releases, and different distributions have subtle "fixes" that change things. Most of the time, these are completely benign (they fix some obscure corner-case that someone found during testing, but the general, non-corner case behaviour is "normal"). However, if you go from Linux version 2.2 to Linux version 3.15, there will be some substantial differences between the two versions, and the binary from the old one may very well be incompatible with the newer (and almost certainly the other way around) - it's hard to know exactly which versions are and aren't compatible. Within releases that are close, then it should work OK as long as you are not specifically relying on some feature that is present in only one (after all, new things ARE added to the Linux kernel from time to time). Here the answer is "maybe".
Note that in the above is also your implementation of the C and C++ runtime, so if you have a "new" C or C++ runtime library that uses Linux kernel feature X, and try to run it on an older kernel, before feature X was implemented (or working correctly for the case the C or C++ runtime is trying to use it).
Static linking is indeed a good way to REDUCE the dependency of different releases. And a good way to make your binary huge, which may be preventing people from downloading it.
Making the code open source is a much better way to solve this problem, then you just distribute your source code and a list of "minimum requirements", and let other people deal with it needing to be recompiled.
In practice, it depends on "sufficiently simple". If you're using C++11, you'll quickly find that the C++11 libraries have dependencies on modern libc releases. In turn, those only ship with modern Linux distributions. I'm not aware of any "Long Term Support" Linux distribution which today (June 2014) ships with libc support for GCC 4.8
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