The codes listed as E3 and E4 on the BMW e46 will mean there is an error o2 sensor adaptation limit cylinder 1-3 (e3) and o2 sensor adaptation limit cylinder 4-6 (e4). This means the engine is running lean and some of the likely issues are bad O2 sensors, vacuum leak, or faulty EGR. The first thing You should do is do a smoke test which will allow You to see if there are any vacuum leaks within the exhauSTSystem. The next thing is to look to see after the smoke test if there are any vacuum leaks. If there is no smoke You will have to check the fuel pressure for any vacuum leaks and then replace the o2 sensors for good measure.
Kordd
October 25, 2020
I understand it to most likely be your intake boots need replacing, the fact that you are receiving BOTH E3/E4 means that your engine is sucking unmetered air so this = the O2 sensors reporting lean mixture and or malfunction. I have no idea how the fuel pump (1st answer you got) could be related to these codes
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