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a(mod arithmetic functions assume binary quadratic form Chapter class number classes of forms coefficient ring complete residue system complex numbers compute congruence f(x congruence x2 congruences modulo Definition denote Diophantine equation x2 distinct primes elements equivalent Euclid's lemma Euler Euler's criterion example Exercise Fermat's finite forms of discriminant Gauss Gaussian integers Gaussian primes gcd(a given greatest common divisor ideal implies infinite number integral module belonging l(mod Legendre symbol Moreover multiplicative nonzero norm number of integers number of solutions number theory odd prime ordinary integers Pell's equation perfect square polynomial congruences positive integer prime modules problem product of primes proof of Theorem properties Proposition quadratic field quadratic nonresidue quadratic residue rational integers rational numbers rational prime real number reduced residue system represented residue system modulo satisfies Section Show solution of x2 solvable solve square-free strictly similar Suppose unique factorization Wilson's theorem write