Not sure about the exact number but it is approx 1086.
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4000 of them.
3439
If by "between" you mean "between but not including", then the answer is 9999 - 999 - 1 = 8999.If on the other hand, you mean to include both 9999 and 999, then the answer is 9999 - 999 + 1 = 9001.
There are infinite prime numbers as there is infinite numbers. You cannot limit the counting of primes.
16 prime numbers