Pilot light won't stay lit.

yes this would appear to me to be the classic symptoms of a bad thermistor , this little device proves that fire has lit (via heat) and keeps gas valve open , otherwise gas valve times out and shuts off (safety)

Check for error codes and service manual on following link:
http://waterheatertimer.org/How-to-troubleshoot-gas-water-heater.html#intellivent

Replace gas control valve
http://waterheatertimer.org/How-to-replace-Honeywell-gas-control-valve.html

Gene:

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Burners don't have anything to do with the problem of pilot not lighting. Is the problem that flame will not 'stay' lit? Or you can't even get a flame to ignite? If former: possible bad pilot generator or bad gas valve. If latter, possible bad gas valve or no fuel to ignite.

The first thing I would do is make sure the flame sensor is clean, this is a metal rod immersed in the burner flame on the opposite side of the igniter. If too much oxidation builds up on it, it will not allow the burner to stay lit. To clean, just turn off furnace and remove flame sensor and clean with steel wool and then reinstall. If this does not solve the problem there may be a problem with the gas valve.

Sounds like the igniter needs to be replaced. It is normal for a heater to run five or so minutes that means the thermostat is working. then restart when the temp calls for heat but if the igniter does not work it will not light up again.

I would first check the gas pressure to your furnace. Then make sure that burners are clean and watch flames when burner first ignites. I have seen burners that were dirty cause delayed ignition that would create a suction when they came on and extingush pilot and burners.
The next thing I would do is change gas valve. Thank you

Probably a bad solenoid in the gas valve. Higher voltage of a new thermocouple keeps it open longer but it still fails over a period of time. Look into a new gas valve for the water heater.

Hello. Begin by using a small brush to clean everything in that area. Then light the pilot and light the burner. If it still acts up, change the thermocouple. Regards, joe

MAKE SURE THE BURNER TUBE AND ORIFICES ARE CLEAN. MAKE SURE YOU HAVE THE PROPER LP PRESSURE.