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I am trying to find a theorem that is valid only for very large numbers.

Example:

There are numbers which have more than 100 distinct factors.

Above theorem satisfies this condition, but it is a trivial one. What I would like is not a trivial one just for the purpose of being for huge numbers. I mean the huge number part should be a side effect rather than main focus.

I want to use it to show that even if you use brute force, trying up to hundred digit numbers, it may still not be successful proof.

Thank you

xycf7
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    The asymptotically-fastest known algirithm for integer multiplication is only faster than other methods when the numbers to be multiplied are larger than around $10^{10^{10^{38}}}$. Similar examples are given at: https://en.wikipedia.org/wiki/Galactic_algorithm – MJD Aug 22 '21 at 09:27
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    This is, in some sense, the same as this – Arctic Char Aug 22 '21 at 09:42
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    My favourite would be : There are positive integers $x$ with $\pi(x)>li(x)$ – Peter Aug 22 '21 at 09:48
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    Also related: Examples of patterns that eventually fail – MJD Aug 22 '21 at 10:09
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    This one is from Math Overflow: Examples of patterns that eventually fail – MJD Aug 22 '21 at 13:46
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    @xycf7 Are the linked posts close enough to what you wanted? If you're looking for something different, could you please amend your question to explain in what way it should be different? – MJD Aug 22 '21 at 17:24
  • @MJD they are indeed good examples. Unfortunately no one entered them as answers, so I cannot accept them. Thank you all for the information. – xycf7 Aug 22 '21 at 21:45
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    Sure, I'm glad we could help. I've suggested that we close your question as being a duplicate of one of the earlier ones. – MJD Aug 22 '21 at 21:50
  • The questions are not duplicates. The earlier one calls for plausible conjectures where there is at least one counterexample, but the smallest one is huge. Perhaps the set of counterexamples is unbounded. This question calls for statements where the set of counterexamples is bounded above and the largest one is huge. Perhaps the smallest one is small. – Rosie F May 26 '24 at 09:33

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