Is "Caesar is a prime number" true false or meaningless.

I'd say meaningless.

Quoting jospehus

logical positivism

Statements can still be meaningless without logical positivism being true.

Quoting jospehus

So the negations of both versions of the sentence would be "Caesar is not a prime number"(which seems sensible) and "Caesar is number that is not prime." (which is bonkers)

You were doing fine until here. That was not the negation.
The negation of the latter sentence is "Caesar is not a number that is prime".

Of course not. Caesar is a salad. A salad is composed of many different individual ingredients. Therefore it is composite, and not prime.

Glad I could help.

@noAxioms is correct.

Quoting jospehus

However when one affirms or denies a claim of this nature one affirms it the opposite claim.

If I say the negation of "Carmen is rich", that is "Carmen is not rich", do I affirm that "Carmen is poor"?

Your claim would work where there are only two options, either x or y. If it isn't x it's y. But this isn't the case. What would be the opposite of "prime number" anyway? Or simply, what are "opposites"? (If anyone has a suggestion, It would be interesting to know)

Back to the OP, in your example, you're referring to a specific person, Quoting jospehus

a human general

named Caesar, very famous, so there is already a lot of knowledge attached to him, like how he is human. Then, the property of "being a human" is incompatible with the property of "being a number". Therefore, because Caesar is a human, he cannot be a number , so Quoting jospehus

"Caesar is not a prime number"

nor any number.

Quoting Abaoaqu

the property of "being a human" is incompatible with the property of "being a number"

What a wonderful thought. In the old days people had phone numbers like MUrray Hill 5-9975. You'd dial the MU characters on your dial phone. In the 1960's the phone company started phasing out those exchanges. Many people objected at the dehumanization, erasing the history of the exchange names, the neighborhoods they represented, and replacing them with numbers. Some people understood what was coming.

In our contemporary AI society we are nothing but numbers, and not even individual numbers. We're datapoints in a huge "corpus" as they call a big pile of data. Data to be mined, sliced, diced, and statistically analyzed.

The property of being a human IS incompatible with being a number. I have always felt that. But society is going the other way. We are each numbers and our world is stumbling one innovation at a time into a monstrous cybertotalitarianism.

Quoting fishfry

In the old days people had phone numbers like MUrray Hill 5-9975.

I didn't know that, thank you for sharing.

Quoting fishfry

In our contemporary AI society we are nothing but numbers, and not even individual numbers. We're datapoints in a huge "corpus" as they call a big pile of data. Data to be mined, sliced, diced, and statistically analyzed.

Well, it's a question of simplicity. There are a lot of people who share the same name, and we don't want to confuse them, so it's just easier to use numbers that exist in an infinite amount. So I wouldn't say that Quoting fishfry

In our contemporary AI society we are nothing but numbers,

but rather "In our contemporary AI society we are represented by nothing but numbers". These numbers are linked to us but they're not us. They're basically substitutes for our names and while that isn't great, it's for simplicity's sake.