Use of PrimeQ in Solve

Question

I fail to use PrimeQ in Solve. Simple example:

Solve[PrimeQ[p] && 0 < p < 5, p, Integers]

yields {} rather than {{p -> 2}, {p -> 3}}.

How can this be fixed (for more complex problems)?

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There's the same issue with OddQrather than PrimeQ, but I know workarounds, like

Solve[p == 2 k + 1 && 0 < p < 5, {p, k}, Integers]

Answer 1

PrimeQ is a predicate that only ever evaluates to True or False. It does not indicate that $p$ should be a prime number. It shares this behavior with all other *Q predicates, like OddQ.

Instead, you could use either one of the following options:

Solve[{0 < p < 5, p ∈ Primes}, p]
Solve[0 < p < 5, p, Primes]

(* Out: {{p -> 2}, {p -> 3}} *)

To see what is happening in your attempt, take a look at the Trace results:

Solve[{PrimeQ[p], 0 < p < 5}, p] // Trace

As you can see, PrimeQ[p] is immediately evaluated to False because p is not manifestly a prime number. This behavior is also indicated in the documentation of PrimeQ.