Truth of Fundamental Theorem of Arithmetic beyond some large number

Question

Let $n$ be a ridiculously large number, e.g., $$\displaystyle23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23}}}}}}}}}}}}+5$$ which cannot be explicitly written down provided the size of the universe. Can a Prime factorization of $n$ still be possible? Does it enter the realms of philosophy or is it still a tangible mathematical concept?

It depends on whom you ask and what exactly you mean by possible. I consider the size of the number irrelevant. — Brian M. Scott, May 08 '13 at 23:15
DonAntonio's comment may be a bit confusing. It is not that it is an axiom that the number can be factored. Rather, we have a theorem stating that a factorization is possible, and the theorem applies to all numbers, regardless of their size. Of course, the proof of the theorem uses the basic axioms of arithmetic. — Andrés E. Caicedo, May 08 '13 at 23:18
One of the main points of mathematics is to prove that things always work. Of course it can be factored, it isn't infinitely large it is just very large. (It is divisible by 4 for a start) — John Marty, May 08 '13 at 23:19
consider $f(n)=n!$, $f^{n!^{n!}}(n)$ for $n = 9999999999$ is larger I think! — , May 08 '13 at 23:22
There are some ridiculously large numbers whose prime factorizations can be fully known (for instance, factorials of large numbers) thanks to various formulas, identities and algorithms. Whether any particular ridiculously large number you pick is among these is not always easy to see. — anon, May 08 '13 at 23:23
Congratulations on writing down a number which cannot be written down by the way ;) — Abel, May 08 '13 at 23:23
@CutieKrait It may be any number that exceeds the bounds of the universe and therefore becomes impossible to explicitly factorize into primes. — Maazul, May 08 '13 at 23:24
That's a non sequitor; just because a number is too big to be written in its decimal representation in our universe doesn't mean we can't explicitly factor it into primes. For a blaringly obvious family of examples, take $({\rm easy~to~factor~number})^{\rm ridiculously~large~number}$ — anon, May 08 '13 at 23:26
But Maazul, your question is not whether a super computer can factor the number or even whether I, Andres, can explicitly factor it. The answer to both questions is no. The question is whether a factorization is possible, which means precisely whether a factorization exists (in the mathematical sense of the word), and the answer is yes, and we have a theorem that tells us so. — Andrés E. Caicedo, May 08 '13 at 23:28
If your question is whether "an explicit factorization is possible", then I suggest you add that to the body of the question, together with a clarification of what you mean by "explicit", and what you mean by "possible", since it seems the issue is not about the mathematical existence of a factorization, but about something else which may well not be mathematical. — Andrés E. Caicedo, May 08 '13 at 23:31
So physically existing bounds do not limit the application of theorems such as FTA? — Maazul, May 08 '13 at 23:31
@Maazul: Banach proved there are 1000 distinct solid gold balls with radius 1 inside a solid gold ball with radius one. physics scientists never could extract them! — , May 08 '13 at 23:41
@user14111 Love the Russell quote. Recognize the thought but not the form of words, and am away from home library. So can I ask where it is from? :) — Peter Smith, May 08 '13 at 23:46
@CutieKrait That's true. But the nature of this non-physicality is a bit different in case of Banach-Tarski Theorem. It is a good example, nonetheless, of math implication of impossibility and non-intuition. — Maazul, May 08 '13 at 23:51
@user14111 Ooooof! I should have recognised it then!!! [hangs head in shame] — Peter Smith, May 09 '13 at 00:40
@user14111 "Pure logic, and pure mathematics (which is the same thing)" - incidentally, Poincare' disagreed with Russell on this point. See eg "Great Feuds in Mathematics" — alancalvitti, May 09 '13 at 00:48
@Mazul Can you factor $23^{1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$? — N. S., May 09 '13 at 04:02
@Maazul But if you try to write it down explicitly it is too big ;) — N. S., May 09 '13 at 04:10
But the question arises whether such factorization carries any physical "feel" or intuition. I think I've got this down now. — Maazul, May 09 '13 at 04:12

score 5 · Answer 1 · answered May 08 '13 at 23:21

5

Yes, there exists a unique prime factorization.

No, we probably won't ever know what it is.

answered May 08 '13 at 23:21

not all wrong

16,178
2
35
57

score 4 · Accepted Answer · answered May 09 '13 at 03:55

This position in philosophy of mathematics is known as ultrafinitism. Ultrafinitists do not accept the existence of arbitrarily large natural numbers. Among other things, I think the claim is that when we assume mathematical induction (which is basically all that is needed to prove unique prime factorization) we are making an unjustified assumption about how large numbers work based on our experiences with how small numbers work. An ultrafinitist might argue that there's no way to really know that this assumption is valid.

I should make it clear that ultrafinitism is something of a fringe position. Many mathematicians are happy with induction and in fact with much stronger assumptions like the axiom of choice. I myself am mildly agnostic about whether computations that can't be performed in the physical universe can reasonably be said to have outputs, but most of the time when I do mathematics I don't think about this kind of stuff.

Truth of Fundamental Theorem of Arithmetic beyond some large number

2 Answers2