For each n∈{2,3...}, there's at most one prime followed by an n-th power. Proof. Take xⁿ-1=(x-1)(xⁿ⁻¹+xⁿ⁻²+...+1). It can't be a prime unless x-1=1 so x=2 & xⁿ=2ⁿ. But 2ⁿ-1 isn't always prime*. ■ *In fact, 2ⁿ-1 for n=2,...,1500 is prime only 1% of the time.)
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So by the same proof, 3 is the only prime followed by a square, because x²-1=(x+1)(x-1) which is composite unless x-1=1?
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And does that mean 31 is the only followed by a power of five, because x^5-1=(x-1)(x^4+x^3+x^2+x+1) which is composite unless x-1=1?
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We gonna need a YouTube video for this one!
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What does composite mean
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Composite means divisible by a number other than itself or 1. So 6 is composite because 6=2*3, but 7 isn’t composite (it’s prime) as the only divisors are 1 and 7.
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Very clever!
Thanks. Twitter will use this to make your timeline better. UndoUndo
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It could be the case that x²+x+1=1, so then x²+x=0 and either x=0 or x=-1, neither of which, when cubed and 1 subtracted, are prime.
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But -1 cubed is -1. If you subtract 1 you get -2 which is prime...
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