I assume that when January 19, 2038 rolls around, there will be a plan for extending the time to more than four bytes. or is it the case that the four bytes are not UNIX time, or do not begin in 1970?

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It is a Unix timestamp, but let me correct a minor detail:

The Y2.038 problem applies to a signed 32 bit number. The Bitcoin timestamp is unsigned, so it can represent the date for twice as long. Therefore, it will become a problem in 136.1 years, or in the year 2106.

Somewhen before then, there will need to be a hard fork.

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