No. 5 is the third Prime number.
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There are several ways for such a number to be constructed: It can be the seventh power of a prime. It can be the third power of a prime multiplied by the third power of another prime. It can be the third power of a prime multiplied by the first power of two other primes. It can be the first power of three different primes.
The first prime number after 8 is 11
The third of June is prime
8 is NOT a prime number. It is divisible by 2, for example.
The prime number before 8 is 7.