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In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.
In this thesis, we seek to prove results about quadratic and cubic reciprocity in great detail. Although these results appear in many textbooks, the proofs often contain large gaps that may be difficult for the average reader to follow. To achieve this goal, we have built up to reciprocity theory from basic principles of algebra, and whenever possible, we have tried to prove the number theoretic results of reciprocity using ideas from group theory. This thesis could potentially serve as a reference for a student who desires to study quadratic or cubic reciprocity in more detail, or as a foundation for studying higher reciprocity laws.
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GOVERNMENT OF TAMILNADU A publication under Free Textbook Programme of Government of Tamil Nadu Department of School Education Untouchability is Inhuman and a Crime MATHEMATICS STANDARD TEN
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