Got a new desktop computer in November: ASUS M32BF with AMD A10-7800 Radeon R7, 12 Compute Cores 4C+8Gx2, which came with Windows 10 installed.
My previous computer was about 8 years old, running Windows Vista - very very slowly. Just over a year ago, when I got fed up with Windows Vista, I decided to try Linux and installed Ubuntu 14.04 as a dual boot. The installation went very smoothly and Ubuntu ran well. But the computer didn't have a great deal of capacity, wasn't very fast, and I saw the new computer at a very good price in a Black Friday sale. I ran it with Windows 10 for about six weeks and then decided to install Linux as a dual boot. After trying Ubuntu 14.10, Ubuntu 15.10, and Linux Mint 17.3 and experiencing the same problem with all three, I found a report on one of the Linux Mint discussion sites from someone with a similar computer who had problems with Linux Mint 17.3 and so went back to 17.1. So I installed LM 17.1 and it works OK - but not completely. It crashes every so often.

Is there an incompatibility between the Linux distros that I have tried and my particular computer. And, if so, is there another distribution that would work better on my computer?

Let me describe what happened with the first three distros I tried, and what happens now with LM 17.1.

With each of the first three distros (Ubuntu 14.04, 15.10 and LM 17.3) I downloaded the iso file and burned a DVD with it. Then I ran the distro from the DVD and, when it ran OK, installed it onto a partition on my hard drive. Then I would reboot the computer and run the Linux distro from the grub. It would seem to run OK - but just once. When I shut down (at night, for example) and then rebooted the next day, I would get to the grub, select the Linux distro, and then the screen would go black and, after a couple of minutes, the monitor would behave as it does when it is turned on but the computer isn't on - i.e. the monitor would seem to be getting no signal from the computer. I could always get to the grub, and if I selected Windows 10 it would run just fine. I read all kinds of things on various Linux group sites and tried to do such things as make adjustments in the UEFI bios, make edits to the grub, and other tweaks that were suggested to people with what I thought were similar problems. Sometimes it would seem to work and I would get into Linux and it would seem to run well. But each time, when I shut down and then rebooted, I would have the same problem all over again. I had the same experience with Ubuntu 14.04, Ubuntu 15.10 and LM 17.3. But NOT with LM 17.1. It mostly runs fine, and I have had no trouble booting into it. But it's not completely problem free: it crashes occasionally.
I installed the backup program fwbackups. When I used it to run a backup, after seeming to be running for quite a few minutes, the monitor screen suddenly went black and then filled up with gobbledygook and then froze. I rebooted, and after a while tried to run fwbackups again. Same thing; it seemed to be running for while and then suddently went to black screen filled with gobbledygook. So I rebooted and uninstalled fwbackups. Since then it hasn't gone to black screen again, but occasionally it does simply freeze up and won't seem to unfreeze itself. Then I have no choice but to do a hardware shutdown and reboot. That reboot is slow, because after the hardward shutdown, it always wants to do a long check of some sort.

So ... does anyone know what my problem might be? Is it simply an incompatability between my particular hardware and the Linux distros I have tried. Are there other distros that might work better with this hardware? Are there other ways, I may not have tried, to overcome the problem with these distros?

Thanks in advance for any help someone can give.

Newer AMD boxes are very kernel sensitive. I have an older A10 and it would not even boot on the 3.19 kernel. On mine I'm running MX-15 with the 4.1 liquorix kernel and everything is working fine. I use the open source driver as I avoid proprietary drivers where possible. W10 and UEFI complicates matters but it's doable.

On any Buntu based system you can grab a kernel from here:

http://kernel.ubuntu.com/~kernel-ppa/mainline/

Mine likes the 4.1 version and the latest is 4.1.16. I suspect kernel issues are why 17.1 works and 17.2/17.3 does not...

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Thanks, NGIB, for your help. Ubuntu 14.04 and 15.10 and Linux Mint 17.3 all seem to install with kernel version 3.19 whereas LM 17.1 installs with kernel version 17.3 It seems that kernel version 3.19 is the culprit. I managed to update to Linux Mint 17.3 without updating the kernel and that seems to work fine (so far, at least). Then I updated the kernel to version 4.2.0-25, the most recent made available in the LM Update Manager and had the same black screen problem when trying to boot. So I managed to boot into the old version 3.13 and uninstalled version 4.2.0-25 and it seems to be working again.
It's too early to tell whether any of this has solved the problem of occasional freezing.
If you, or anyone, have any further thoughts about which updated kernels might work on my machine, I would be happy to hear. I know that you said your machine likes version 4.1, but the LM Update Manager doesn't make that available and I thought there might be a valid reason why, e.g. perhaps it isn't compatible with Mint Cinnamon.

[QUOTE=mgerson;5488950]Thanks, NGIB, for your help. Ubuntu 14.04 and 15.10 and Linux Mint 17.3 all seem to install with kernel version 3.19 whereas LM 17.1 installs with kernel version 17.3

Oops. That should have said ...whereas LM 17.1 installs with kernel version 3.13.

I have an A-10 5700 with ati radeon HD 7660D and I have Arch Linux using the kernel 4.3.3 everything is working good and I even play games on steam with high graphics no problem but I use the open source video driver because the proprietary doesn't work well and it's also not compatible with Xorg 1.18 I think.