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algebraic integer algebraic numbers angle of parallelism angle sum arbitrary chapter circle complex numbers conclusion congruent coordinates Corollary corresponding curvature curve Definition denumerable set determine digits divisible divisors elements equal equations of type equivalent Euclidean geometry example exists exponent F-axis Farey Series field follows geometry given hence indefinite integral infinite sets Lemma logarithm magic square mathematical induction mathematics means Miquel point multiplication natural logarithms natural numbers negative obtain pairs parallel perfect number perfect square perpendicular plane positive integers positive rational numbers positive real number postulates prime factors prime integers prime numbers problem projective geometry proof Prove Pythagorean triple rational operations reader relatively prime residue respect right angles right triangle satisfy set of natural set of numbers set of rational Show sides slope solution statement straight line Suppose Theorem theory of numbers tion variable X-axis zero