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Okay,

"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!

Moga
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    When you want to define something, always think about whether that thing is well-defined or not. – Eff Oct 14 '17 at 21:26
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    If you add three to your number, then there will be an infinitely long avalanche of carries resulting in ... ZERO. So your number is equal to $-3$. Assuming that you use the "usual" arithmetic. More seriously, I agree with Eff in that you should define your objects a lot more carefully. For example, what is the definition of a prime here? How is the arithmetic done? Also: $$\infty999\cdot3=\infty997,$$ so may be your number is not a prime after all? Just kidding, $\infty999=-1$, and being divisible by $-1$ doesn't count :-) – Jyrki Lahtonen Oct 14 '17 at 21:28
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    So you see that the problems you need to address are (at least) 1) how do you define the arithmetic of these infinitely long numbers? 2) how do you define prime? 3) how do you define larger/largest? Is the prime $\infty9998=-2$ larger than $\infty9997$. Is $\infty9999$ larger still? Is $1+\infty9999=0$ larger still? – Jyrki Lahtonen Oct 14 '17 at 21:36
  • Note: we have something like this quite well defined. These are 10-adic numbers (more general: p-adic). Then what you call $\infty999$ equals $-1$, just like one of the previous comments states. – Kamil Maciorowski Oct 14 '17 at 21:58
  • Thank you Jyrki!!! Yes I was thinking of the logic in the very same way you demonstrated! And indeed you would be correct that my largest possible is in fact incorrect!! LOL wow! But is does create an interesting look into such strings. This is why I bring it up! What IS the logic? – Moga Oct 16 '17 at 07:43
  • And in the same logic (∞x -1 > ∞x-2). – Moga Oct 16 '17 at 07:46
  • And Kamil, Is it the same thing? p-adic or just 'like'? ∞999 -3 = 2? – Moga Oct 16 '17 at 07:49

1 Answers1

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Infinity isn't a number, it's a concept. Having "an infinite" amount of $9$'s and then a $7$ wouldn't really help you at all. $\infty9997$ would be the same as just $\infty $.

Since at least the Greeks, and probably before that, we've known that there's an infinite amount of primes; with the normal definition of infinity this then implies that the "largest" prime is impossible to ascertain. Because we can always keep on adding $1$, until we get the next "largest" prime ad infinitum.

If you want your question to make any sense at all then you need to wholly redefine the concept of infinity and that would make basically everything relying on infinity behave in unpredictable ways. It's probably easier to just don't think of a specific "largest" prime, but instead of a concept of "the largest prime" which is just an extension of infinity.

Sam Anderson
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