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Unique Factorization and the
Congruences and the Ring
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algebraic number algorithm assertion best approximations called Cauchy sequence Chinese remainder theorem coefficients complete complex numbers computation congruence conjecture converges Corollary Deduce defined Diophantine equation Dirichlet elements equation x2 equivalent Euclidean domain Euler example fact Fermat finite following theorem function Gauss gives Hence Hint implies induction inequality infinitely many primes isomorphic Legendre symbol linear log log mathematics modp modulus multiplicative nonresidue nonzero number theory obtain odd prime p-adic Pell equation polynomial positive integers positive prime prime divisor prime number theorem primitive root Problem properties proved quadratic reciprocity quadratic residue quotients rational numbers real numbers reduced reduced residue system relatively prime representation residue classes residue classes mod result Ring Section Show smallest positive solutions of x2 solvable square square-free Suppose unique factorization values Verify zero