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The Euclidean Algorithm and Its Consequences
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algebra arithmetic assertion chapter coefficients common divisor common factor consider continued fraction expansion convergents deduce degp digits Diophantine equation divides divisible elements equation x2 equivalent Euclidean algorithm exactly example Fermat finite following theorem Gaussian integers Gaussian primes hence Hint implies ind9 inequality irrational number Lagrange's theorem mathematics mod 9 mod mn mod q modulo multiplication natural numbers nonnegative nonzero norm number theory obtain odd prime ordm partial quotients Pell equation Pell's equation period length pk-i pn/qn polynomial positive integer positive solution prime number primitive root Problems Proof properties prove quadratic residue rational integers rational number rational prime real number reduced residue system relatively prime representation residue classes residue system mod Section sequence Show simple continued fraction solvable solved square Suppose Theorem 1-1 theory of numbers tion true unique factorization theorem yy/d zero