Logic statements with predicate's

Question

I have this question:

Given $k,t$ and a prime number bigger than 1 ($p \geq 2$) and $k,t$ are naturals. Which of these predicates is the right for "p is a prime number":

1. $\forall k \forall t((p=kt) \wedge ((k=1) \vee (t=1)))$
2. $\forall k \forall t((k=1) \vee (t=1) \vee (p \neq kt))$
3. $\forall k \forall t(((k \neq 1) \vee (t \neq 1)) \rightarrow (p \neq kt))$
4. $\forall k \forall t(((k = 1) \vee (t=1)) \rightarrow (p \neq kt))$

I can find any $k,t$ that would contradict every option - so is there even a solution? For example for 2:

I can find $k=1$ and $t=13$ and $p = 13$ which is a prime but it says $p \neq 13$ ... what?! how..

I would appreciate your help! Thanks you very much.

Answer 1

2 is correct. Either one of $k$ and $t$ is $1$ or $p$ isn't prime.

Note that none of the predicates directly defines primes.

The questions asks which predicate is correct knowing $p$ is prime.

• if $k=1$ then (2) is correct regardless of everything else
• if $t=1$ then (2) is correct regardless of everything else
• if $p\ne kt$ then (2) is correct regardless of everything else

If none of $k$ and $t$ are $1$ then the last one is correct. Otherwise one of the first two ones is correct.