As your CPU is getting hot anyway, this indicates that Prime pushed the FPU in the CPU very hard and overheated the CPU, causing the crash. Therefore your CPU is definitely overheating and causing the instability in your system. Maybe think about getting a better heatsink. Also what case cooling do you have? And have you measured the temperature of the inside of the case?
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Its the std case of an HP. The case fan is std medium size 8cm or what ever it is. sometimes it sounds like a jet plane taking off!
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8cm fans are tools of the devil. There are aftermarket coolers available that have much better heatsinks than the standard ones, as well as either 9.2cm or 12cm fan. Bigger fans generate better airflow at lower fan speeds, as well as making a lower-pitched sound, so they're much quieter. Most even come with a voltage regulator, which allows you to adjust the fan speed manually.
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I've not seen inside an HP case for some years, but when I used to have to upgrade or repair ones in a company I worked for years ago there were horribly designed with restricted air flow.

Where is the case fan? Front or rear? For the best air flow and cooling the best setup is to have a fan at the lowest point at the front of the case, drawing air into the case, and the other fan level or above the CPU at the rear extracting air out. This will draw the cooler outside air into the case at the lowest point, and the higher rear fan will pull this air over everything inside the case and extract the hottest air out of the rear. This will also create the correct pressure within the case.

If only one fan is used to extract air from the case, then it has to be drawing the air from somewhere, and without another fan drawing air into the case it will be drawn into the case from the nearest holes in the case to the fan extracting air, and so won't be very effective compared to having a two fan setup.

You mentioned the 80mm fan in your case is very noisy. This is a big problem with these smaller fans. Using the much larger 120mm fans is a much better solution. They don't have to spin as fast to draw in much more air and are therefore much quieter. Having one of these larger fans at the front and one at the rear is a great cooling setup for any case. And if your case only has 80mm mounting points you can buy cheap mounting kits to mount 120mm fans in a 80mm location.

The other cooling issue will always be the CPU heatsink and fan. One thing to make sure is that the fan is mounted the right way around on the heatsink. It should be drawing air into the heatsink, not away from it. This is because to cool properly the fan should be drawing air into the heatsink and across the fins to dissipate as much heat as possible. If you find the fan is the wrong way around and is actually blowing air up out of the heatsink then it is normally easy to just unscrew and turn the fan around.

Failing that, investing in a bigger copper heatsink with a larger quieter fan is the next option.
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I'm glad you pointed that out as I've just checked mine and found i had 3 errors. 1 I sorted out with regedit (Not advised) as there was no other way of uninstalling a faulty driver, 1 I sorted out with services and the other was a problem with ATI driver software which they can't be bothered to fix so i complained to them (ATI's response to the problem was to go back to an old, even more unstable driver). Now everything seems to be slightly tweked.
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It is definitely worth checking the system logs from time to time just to make sure everything is working correctly.

It is annoying regarding graphics drivers. I recently installed the game Timeshift, and it complained the graphics drivers I had installed were not new enough to run the game properly as they may cause some instability and crash the game. So I checked for updates and I am using the latest drivers. I then discovered that the game is actually looking for the latest beta drivers which have their own warning that they might not be completely stable. Great game development there, telling customers to install beta drivers or risk crashes.

Luckily I don't normally have any issues with graphics drivers or games these days, and I much say that I haven't since the early Windows 2000 days. Things have improved greatly with standardised support. Who remembers the nightmare days of Windows 9x where games had to directly support your graphics card manufacturer, or even worse, back in the DOS days when a game actually had to support the model of card.

Regarding ATi cards, a couple of years ago I was with all other gamers and a big supporter of them, with their cards offering better drivers and performance than others. But now they have fallen badly behind. Their drivers are less stable and the performance can't match other cards. nVidia cards are really the only cards to buy at the moment. My 7800GT hasn't had any issues and the drivers have always been stable. I am tempted to upgrade to a 8800 GTX in the new year though as some of the latest games are starting to push the 7800GT beyond it's limits (I have to run DiRT in 1024x768 with some of the settings on medium, otherwise the controls update about once per second and you can't steer!).
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