The Raspberry Pi 3 is built with a 64 bit chip. Yet the Raspberry Pi Foundation has only released Linux distros are 32 bit.
Can I install a 64-bit version of Debian on the RPi 3?
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Not unless you have a 64 bit version compiled for the ARM architecture. You can't install the AMD64 or x86-64 version as the architecture is different.
Quoting from the Pi Foundation's blog post announcing the Pi3:
At launch, we are using the same 32-bit Raspbian userland that we use on other Raspberry Pi devices; over the next few months we will investigate whether there is value in moving to 64-bit mode.
The switch to a 64 bit processor was primarily because of the gains it provided in clock speed.
Neither Debian 7 (wheezy) or Debian 8 (jessie) support the Raspberry Pi3 in AArch64 mode. Debian 9 (stretch) will very likely support the Pi3 in 64bit mode , though. Progress can be monitored here: https://wiki.debian.org/RaspberryPi3
If you can't wait for the official stretch release you can take a look at the pi64 image on GitHub. It is basically a Debian stretch with the raspberry kernel.
pi64 is an experimental 64-bit OS for the Raspberry Pi 3. It is based on Debian Stretch and backed by a 4.11 Linux kernel.
In case it helps - you can download and use free for a year (requires registration) the "Evaluation copy of SUSE Linux Enterprise Server 12 SP2 for Raspberry Pi" - here
They've been working on it for a few years, I was running an earlier version a year ago. But see https://wiki.debian.org/RaspberryPi3
It's not completely up to Debian release standards so they're still calling it a preview (unofficial, unsupported, etc.). It's Debian Buster.