Expressing "Every even number is the difference of two primes" in symbols

Question

Consider this conjecture:

"Every even number is the difference of two primes."

Express this statement in terms of quantifiers, variables, equality/inequality symbols, logical operators, and the following two predicates.

• $P(n)$: "$n$ is a prime number."
• $E(n)$: "$n$ is an even number."

My answer: $$∀x∃y,z>0( E(x) → (P(y)∧P(z)∧(y < z)∧(y ∧ x = z))).$$ I want it to say: "For every value of $x$, there exists at least two positive integers $y$ and $z$, such that if $x$ is an even number, that implies $y$, $z$ are prime numbers AND $y$ is smaller than $z$ AND $x + y$ is equivalent to $z$."

I am pretty sure I am wrong, especially with the part $(y ∧ x = z)$ because I do not know how to express the idea of "subtraction" or "difference" with the symbols available to me.

What am I missing?

Answer 1

Provided the addition operator, here's a conversion from natural language to a logical formula. I've put it step by step:

1. "Every even number is the difference between two primes".
2. "Every number, if it is even then it is the difference of two primes".
3. $\forall x$, "if $x$ is even, then it is the difference of two primes".
4. $\forall x, E(x) \rightarrow$ "$x$ is the difference of two primes".
5. $\forall x, E(x) \rightarrow$ "there are two prime numbers $y$ and $z$ such that $y - z = x$".
6. $\forall x, E(x) \rightarrow$ "there are two prime numbers $y$ and $z$ such that $y = x + z$".
7. $\forall x, E(x) \rightarrow$ "there are two numbers $y$ and $z$ such that they are prime and $y = x + z$".
8. $\forall x, E(x) \rightarrow \exists~y~z$, "such that they are prime and $y = x + z$".
9. $\forall x, E(x) \rightarrow \exists~y~z, P(y) \land P(z) $and $y = x + z$".
10. $\forall x, E(x) \rightarrow \exists~y~z, P(y) \land P(z) \land y = x + z$.