Can they check it?

I am not a lawyer....

You need to establish 2 things:

Continuous Residence:
You didn't do anything (like leave the country for more than 1 year (possibly more than 6 months, it depends)) since you got your green card

Physical Presence:
You have been in the US most of the time in the last 5 years.

You need to describe all of your trips on the N-400 (download it and read the questions) well before you meet with an interviewer. That includes both since you have been a GC and particularly in the last 5 years.

The interviewer can ask about anything on the form, or anything else for that matter. In addition to what's on the form, the various chunks of the immigration bureaucracy amass quite a bit of information about all the things you've done. As far as I know, you don't get to find out what that is.

I'm not sure about the 4 years + 1 day rule. I believe it relates to how the clock gets reset if you take a 6 month to 1 year trip (but, since the longest I've been out of the country since my GC is 16 days, I never really cared).
 
there buddy>>

“Continuous Residence” Example
• An applicant became a Permanent Resident on January 1, 1990.
• She lived in the United States for 3 years, then returned to her native country for 1 year and 3
months.
• She got a Re-entry Permit before leaving the United States so that she could keep her Permanent
Resident status.
• The applicant re-entered the United States with Permanent Resident status on April 1, 1994.
Question: When is the applicant eligible for naturalization?
Answer: On April 2, 1998, 4 years and 1 day after she returned to the United States The last 364
days the applic ant was out of the United States count toward her time as a PermanentResident in “continuous residence,” but the 3 years in the United States before leaving donot.

page 27...
 
Think I'll post my $0.02 now...

So the chances are, IO will only be interested in your last 5 yr travel history, which of course you will have declared on your N-400 application. No problem right? Well maybe. The issue is that USCIS has a duty to establish that your GC was obtained lawfully and held properly since its issue. Under most circumstances they rarely ask much more about this during interview as they usually already have the I-485 paperwork in front of them. In your case however, it sounds like there are significant issues with your travel dates prior to 2001, and if these are recorded in your passport, you might have a problem.

Any chance you have already, or might be about to, renew your passport?
 
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