I should clarify my remark. There are 4 things that will cause a inability to maintain the pilot.
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1 Lack of gas
2Â Loose wiring in the curcuit
3Â no power from the thermopile
4Â open high temp safety limit.
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If the thermopile has a very high pilot flame, you can destroy it.  If the high limit has been tripping from a high temp setting, It will fail with repeated trips. Many are auto reset and can be re-lit after the fryer cools down, some have a manual reset red button that has to be pushed in to relight. But with the nature of a fryer, grease gets airborne and sometimes enters the high limit making the point contact have high resistance and not be able with such low voltage to carry enough current to hold in the gas solenoids. We are taking 750 millivolts here. It’s not uncommon for the high limit to also fail if the fryer constantly is working above 365F It’s just a Bordon tube pressure against a spring that activates a micro switch. Springs get weak with time. So as the machine ages, the set point will drop.