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How did Russell analyze the phrase “the greatest prime number” without treating it as the name of an object?

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Austrian postage stamp featuring a portrait of philosopher Ludwig Wittgenstein, with German text above and below, and a denomination of 5 schillings. — Philosophy of language An Austrian stamp depicting Ludwig Wittgenstein.

Bertrand Russell analyzed the phrase "the greatest prime number" by applying his theory of descriptions, which demonstrates that such phrases are not singular terms or names of objects, but rather quantificational expressions.1

When a phrase like "the greatest prime number" appears in a sentence, Russell argued that it should not be treated as referring to a specific entity. Instead, he analyzed the sentence as a complex conjunction of general statements. Specifically, the phrase is broken down into the following logical components:21
* There exists at least one number that is prime and the greatest.1
* There is at most one such number.1
* Any number that is prime and the greatest satisfies the remaining conditions of the statement.31

By analyzing the phrase this way, Russell showed that the sentence is meaningful even though no such number exists. The description does not function as a name, so it does not require a referent to be intelligible. This approach allows philosophers to avoid the error of positing "non-existent" objects to account for the meaning of such expressions, treating them instead as "logical fictions".12