Automating Elementary Number-Theoretic Proofs Using Gröbner Bases

  • John Harrison
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4603)


We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain number-theoretic predicates such as ‘divisible by’, ‘congruent’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings.


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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • John Harrison
    • 1
  1. 1.Intel Corporation, JF1-13, 2111 NE 25th Avenue, Hillsboro OR 97124USA

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