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Rational and irrational numbers
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0(x log Bertrand's Postulate Chapter character modulo Chinese Remainder Theorem completes the proof congruent modulo consider constant Corollary d=l d defined degx divide divisible elementary symmetric function equation equivalent exactly exists expression finite number follows form 4m greatest common divisor Hence induction infinite integral coefficients inversion formula least common multiple Lemma linear congruence log p log log q log2 log2x mn<x monomial multiple root n<x n n=l n number of primes number of solutions O(log obtain odd number odd perfect numbers odd prime polynomial congruence polynomial with integral positive integer pq<x Prime Number Theorem primitive root modulo prove quadratic residue reduced residue system relatively prime residue system modulo result root of f(x root of unity sequence solutions modulo Subcase sufficiently large symmetric polynomial Theorem 3.1 unique factorization values write yields zero