Automated conjecture making in number theory using HR, Otter and Maple

Simon Colton

Research output: Contribution to journalArticleResearchpeer-review

10 Citations (Scopus)


One of the main applications of computational techniques to pure mathematics has been the use of computer algebra systems to perform calculations which mathematicians cannot perform by hand. Because the data is produced within the computer algebra system, this becomes an environment for the exploration of new functions and the data produced is often analysed in order to make conjectures empirically. We add some automation to this discovery process by using the HR theory formation system to make conjectures about Maple functions supplied by the user. HR forms theories by inventing concepts, making conjectures empirically which relate the concepts and appealing to third party theorem provers and model generators to prove/disprove the conjectures. It has been used with success in number theory, graph theory and various algebraic domains such as group theory and ring theory. Experience has shown that HR produces too many conjectures which can be easily proven from the definitions of the functions involved. Hence, we use the Otter theorem prover to discard any theorems which can be easily proven, leaving behind the more interesting ones which are empirically plausible but not easily provable. We describe the core functionality of HR which enables it to form a theory, and the additional functionality implemented in order for HR to work with Maple functions. We present two experiments where we have applied HR's theory formation in number theory. We discuss the modes of operation for the user and provide some of the results produced in this way. We hope to show that using HR, Otter and Maple in this fashion has much potential for the advancement of computer algebra systems.

Original languageEnglish
Pages (from-to)593-615
Number of pages23
JournalJournal of Symbolic Computation
Issue number5 SPEC. ISS.
Publication statusPublished - May 2005
Externally publishedYes


  • Automated mathematical discovery
  • Automated reasoning
  • Computer algebra systems
  • Integration of reasoning systems
  • Machine learning

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