Analytic Number Theory

May 7th, 2015
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Lemma: If n is a positive integer,Theorem: If 2n + 1 is an odd prime, then n is a power of 2If n is a positive integer but not a power of 2, then n = rs where [pic], [pic]and s is odd. By the preceding lemma, for positive integer m,

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Analytic Number TheoryLemma: If n is a positive integer,[pic]proof:[pic][pic][pic]= an bn.Theorem: If 2n + 1 is an odd prime, then n is a power of 2.proof:If n is a positive integer but not a power of 2, then n = rs where [pic], [pic]and s is odd. By the preceding lemma, for positive integer m,[pic]where [pic]means "evenly divides". Substituting a = 2r, b = 1, and m = s and using that s is odd, [pic]and thus[pic]Because 1 < 2r + 1 < 2n + 1, it follows that 2n + 1 is not prime. Therefore, by contraposition n must be a power of 2.Well my thought is suppose n isn't a power of 2.Suppose it's 3.a^3 + 1No, that's not prime. if n is odd and n>=3, then I can factorize like this: (a + 1) (a^2 - a + 1)and both numbers are greater than one, so a^3+1 can't be prime.What if n is 6?a^6 + 1Well now I can just write that as (a^2)^3 + 1which we know will fac

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