That will be the engine number. Under the seat there should be a plate or sticker with an MTD number. Having this will allow me to look up the wiring diagram so we don't miss anything.
Do you have a voltmeter? It's possible that your charging system isn't working for some reason. We need to check the voltage across the battery both running and non-running. With the engine off and switch on, you should have 12 or so volts at the battery. With the engine running, there should be a 1.5-3 volt increase.
Generally speaking, when there's a voltage drain, I tend to suspect the ignition switch. The simplest way to check for voltage drain through the ignition is to start off with a fully charged battery then put the voltmeter on it every few hours for a couple of days and watch for a drop.
As yours will let you mow 2 or 3 times over two or three weeks, I tend to think that there's a charging problem myself. There should be wires coming out from under the flywheel that go into a plug. I'll need the MTD numbers to be able to tell you how many and what color they are, but if you can check that they're in one piece and all connectors are connected that will help to narrow things down. If the wires are all in good shape, then other problems might be the ignition switch, flywheel magnets, stator assembly, rectifier and regulator.
If it is a charging problem and if you have or can get a voltmeter we can trace backwards until we find the trouble.