Pressing the envelope, presumably the best scenario would be a simple proof of the Prime Number Theorem. After all, Wilson’s Theorem gives a necessary and sufficient condition, in terms of the Gamma Function, for a number to be a prime, and Stirling’s Formula specifies the asymptotic behaviour of the Gamma Function.
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Using Robbins' [1] form of Stirling's formula, $$\sqrt{2\pi}n^{n+1/2}\exp(-n+1/(12n+1))< n!< \sqrt{2\pi}n^{n+1/2}\exp(-n+1/(12n))$$ we get $$\left\lceil\sqrt{2\pi}(n-1)^{n-1/2}\exp(-n-1+1/(12n-11))\right\rceil$$ $$\le (n-1)!\le$$ $$\left\lfloor\sqrt{2\pi}(n-1)^{n-1/2}\exp(-n-1+1/(12n-12))\right\rfloor$$ which is accurate enough to distinguish prime from composite for $n\le8$. For larger numbers, the error bound is too large. This can be extended further using a modification of Wilson's theorem: for n > 9, $$\lfloor n/2\rfloor!\equiv0\pmod n$$ if and only if n is composite. This allows testing 10 through 15, plus (with some cleverness) 17. With tighter explicit bounds and high-precision evaluation, it might be possible to test as high as 100 with related methods: direct evaluation up to 25 and the 'divide by 4' variant of the above for n > 25. This is not so much 'using a cannon to swat a fly' (using methods more powerful than needed) as it is 'using the space station to swat a fly': the methods must be extremely powerful and accurate to do very little. [1] H. Robbins, "A Remark on Stirling's Formula." The American Mathematical Monthly 62 (1955), pp. 26-29. |
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