Intermitten no start

My Name is Ken Smith and I have a 1996 Buick century with a 3.1.
The problem I am having is periodically the car doesn't start, and after 30 -40 mins of cycling the key on and off and checking fuel pressure and checking spark the car will start, and then it wont happen again for several months. The one thing I did notice is that when the problem exist, the fuel pump stays running. What I mean is when you turn the key on engine off you should hear the pump for 2 seconds then it should turn off .. well it doesn't shut off at this point. I have 40 psi of fuel pressure but the car wont run. Once the pump shuts off the car starts and I'm waiting for it to act up again. I'm not sure what i'm doing to get it to run but eventually it will run.
Thanks in advance for any suggestion on this problem

When the pump is running and the no-start condition is there, do you have Spark and Injector pulse? Does the pump, eventually, shut off by itself? Or, does it keep running until the key is turned off?

The fuel pump will run when the Fuel Pump relay is closed OR when the Oil Pressure switch is closed. If either one of these is stuck closed, the pump will keep running. The pump will also keep running if the Relay control wire, DK GRN/WHTon pin 8 of the PCM, maintains power.

Yes there is spark and the pump runs continuously when the key is on engine off. I have fuel pressure but The car will not start when this happens but every time after checking everything and cycling the key on and off several times the pump does what it is supposed to do..turns on for 2 seconds then stops and then the car starts right up and I don't have any problems for several weeks. I thought it was a relay getting stuck, so yesterday when it happened I went to the relay and it was continuously clicking, I swapped it out with another one but it was not the problem. I was wondering if the oil pressure switch could make it do this or do I have a short. I didn't check the injectors because I assumed that they were not firing because the car wouldn't start and didn't seem like it was even trying to start.

Also thanks for replying to this head scratcher

When you pulled the relay out did the fuel pump keep running, with NO relay in place?

While the relay was clicking as you say: Was the ignition on or off, was the key in the switch??

pump stopped when I pulled relay. relay was clicking with ignition on engine off. Just wondering why pump doesnt shut off after 2 seconds, and if and when that happens why the car wont start with spark and fuel pressure? is the signal from the pcm crossed up like short? I dont Know ...Thanks for any help you can provide

Low voltage and bad grounds can, both, cause weird things to happen. Check your powers and grounds on the PCM.

Pins 19 (pink) & 20 (Orange) of one connector are power . Pin 21 (Black/White) is ground.
On the other connector, Pin 21 (Black), Pin 61 (Black/White) are both grounds.

thanks for that info. I'm going to ask a couple of stupid question now. This happens every so often is this something I should check when it happens or will it show up now?. Also where is the PCM located on that vehicle ?

Ideally, these tests should be performed while the problem is present. But, perhaps, testing while the condition is NOT present will make it present itself. Who knows? Failing to find a fault when the problem is not present means nothing.

The PCM should be somewhere under/behind the dash on the passenger side.

Thanks hor the info I appreciate it