Is there a rather quick/easy way to check if a number in the form 10^n +x is prime or not, given that n and x are known.
I should also not that for our case n is mind-blowingly huge, and x is rather small, in magnitude of hundred.
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Do you mean any $n,x$? For example, if $n$ is not a power of two and $x$ is $1$, the primality of $10^n+x$ is simple to verify. – S.C.B. Mar 27 '17 at 23:58
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n is a fixed number, has around 1400 digit ending with 9087. x is on the other hand has many candidates, smallest is 7 and biggest is 999 and non of them is even. – TheDeltak Mar 28 '17 at 00:04
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there are no quick ways to confirm a big number is prime. Many ways to show a big number is not prime. – Will Jagy Mar 28 '17 at 00:04
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proving it is not prime just as good for us, any keyword or methods you can direct us to? – TheDeltak Mar 28 '17 at 00:06
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2If you can conclude that any particular number in the size range of $10^{10^{1400}}$ is prime, you will have broken the record for largest known prime by so large a margin it isn't even funny. "Quick/easy"? Don't think so. – hmakholm left over Monica Mar 28 '17 at 00:15
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1If $n$ is one less than a prime $p$ (other than 2 or 5) and $x$ is one less than a multiple of $p$, then by Fermat's little theorem $10^n + x$ is a multiple of $p$ and thus composite. There probably are other things you can choose to show the number is composite, but if the number is prime, you might need the help of distributed computing. – Robert Soupe Mar 28 '17 at 01:36