In standard English, is a verb, not a noun, so it does not take an article itself . However, the choice of article for the noun following it (e.g., "There exists a ..." vs. "There exists the ...") depends on whether the object is being introduced for the first time or is already known. Indefinite vs. Definite Articles
: Use this when referring to a specific, unique, or previously identified entity. For example, "The puncture exists in the tire" is used because the specific puncture has been found. "a" or "the" when used with "exists" and superlative Exists
: Use this when introducing something new to the conversation. In mathematical logic, "There exists a number In standard English, is a verb, not a