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My understanding is that for any general formula to be accepted, there has to be a proof for all 'n', by construction or by negation.

Even if the formula can be tested for high values of 'n', we merely say it works up to the tested limit but not beyond.

Do we have some non trivial examples where a formula works till a certain high 'n', say n = 100 or 1000, but not beyond that?

I'm curious because we often have unproven formulae that work correctly for an arbitrary 'n', which one may test for; and neither counter examples are found, nor a proof. But I don't see famous cases where a counter example could disprove a 'promising' formula.


marked as duplicate by Rohan, Winther, user491874, hunter, Namaste Jan 27 '18 at 16:47

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    $\begingroup$ See this $\endgroup$ – Yash Jain Jan 27 '18 at 14:33
  • $\begingroup$ Thanks for the comments, I'm requesting moderators to close the questions $\endgroup$ – DS R Jan 27 '18 at 14:35
  • $\begingroup$ @DS R you can delete the question yourself. $\endgroup$ – Yanko Jan 27 '18 at 14:35
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    $\begingroup$ There are so many examples here. One can quite easily construct formulas that hold for $n=1,2,\ldots,N$ but fail for $N+1$ for any $N$. For example one can use $$\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kx)\,dx$$ that holds provided that $\displaystyle\sum_{k = 0}^{N}a_k \le 2\pi$. See e.g. arxiv.org/pdf/1105.3943v2.pdf $\endgroup$ – Winther Jan 27 '18 at 14:37

Yes,you may find many formulae which are true up to certain n and not true for other for integers larger than n. For example the statement $$(n-1)(n-2)(n-3)=0$$ is true for $n=1,2,3$ but is false for $ n>3.$


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