"What is the Largest Possible Prime Number?"
First it appears this question is outside the bounds of good mathematical reasoning for two main reasons.
1) The concept of 'largest possible prime' is ill-defined. 2) The concept also appears to be ill-logical based on reason (1).
For the same reason there is 'no largest prime' there is also no number with an infinite amount of digits, yes? A “number” with an infinite number of digits is a natural number?
To set the question up properly I have been informed I must define 'what is prime' since the nature of the question seems to misunderstand the concept on a whole. Also, I will attempt to explain what I mean by 'Possible.'
I realize, as a concept, the 'largest possible prime' is not a natural number and therefore cannot be a prime number, however the same rules we use to generate primes can be abstracted to strings of numbers, infinite in length, despite there being no definition for such. What is such a string called? I cannot find an answer excepting the idea of infinite, which seems to bulldoze any distinction of pattern or lack thereof in an infinite string.
Reasonably an infinite strings of 2's is not different from an infinite string of 3's, 5's, or 17's, in terms of numbers ... or any other symbol, it is simply infinite. Despite this making sense it seems to leave something fundamental behind. I am sure I missed something here and there are strings of numbers infinite in length that assume some distinction based on their makeup and pattern.
It seems that such a string could not be defined at irrational either.
I am also realizing the depth of my lack of understand here but hopefully this still 'makes sense.'
Okay: as simple as I can:
1) Will a finite set of numbers (10^x) always have a greatest possible prime of (10^x)-3? 2) Can we extrapolated this logic to a set (10^∞) and say the largest possible irrational illogical prime is ∞97?
Or: Of all sets (10^x), is [(10^x)-3] the largest possible prime set.
Thank you for reading!
*** FIRST QUESTION ON HOLD Below *******************************************
[I am not a mathematician so please forgive my laymen question/topic.
This question, "What is the largest possible prime number?" is distinct and separate from the question "What is the largest prime number currently known?."
I understand the set of primes is infinite and that any 'largest' prime could be smaller then the next larger prime, but continuing in an illogical fashion (+Godel) I have come across a interesting idea. Let me explain.
If we could think about the largest possible prime as not real, ie.. not a real number.. but a theoretical number (not sure what the name is) then we could say maybe there is a 'not real prime' number with an infinite amount digits.
Further if we considered all 'not real primes' with infinite digits what would be the greatest of those?
I think the answer to this question is a number with an infinite amount of 9's ending in a 7.
Something like: ∞9997
The logic being ∞999 is not prime but ∞9997 could be and that would be the 'largest possible prime number.' I do have documentation to support the idea but it is lengthy to transcribe here.
I know the 'not real' quality of the answer will immediately annoy many but I am wondering what everyone thinks of the question and this obviation of normal thinking.
Thank you All! Thanks StackExhange!