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The NumbersWithNames Program

The NumbersWithNames Program. Simon Colton, University of Edinburgh, UK Louise Dennis, University of Nottingham, UK. Some Light Number Theory. Perfect Numbers: 6, 28, 496, … Equal to sum of proper divisors Are pernicious (prime num. 1s in binary) 6 = 110, 28 = 11100, 496 = 111110000

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The NumbersWithNames Program

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  1. The NumbersWithNames Program Simon Colton, University of Edinburgh, UK Louise Dennis, University of Nottingham, UK

  2. Some Light Number Theory • Perfect Numbers: 6, 28, 496, … • Equal to sum of proper divisors • Are pernicious (prime num. 1s in binary) • 6 = 110, 28 = 11100, 496 = 111110000 • Not refactorable numbers • Puzzle (Paul Zeitz) • Numbers of the form n(n+1)(n+2)(n+3) are never square numbers

  3. Neil Sloane’s Encylcopedia of Integer Sequences • Large database of sequences • E.g., Primes: 2, 3, 5, 7, 11, 13,… • Contains 60,000+ sequences (36 years) • Online: research.att.com/~njas/sequences • Identifies a sequence typed in • Uses tranformations to find matches • Think of these as equiv. conjectures

  4. NumbersWithNames Program Overview • Enhances EIS conjecture making • Uses a subset of 1000 sequences • Number types with names • Prime, square, odd, perfect, triangle,… • Four step process (given seq. S) • Invents related sequences S1, S2, … • Makes conjectures about S, S1, S2,… • Prunes uninteresting conjectures • Sorts conjectures by plausibility

  5. Inventing Related Sequences • User chooses sequence: 2, 3, 5, 7, 11, … • Add one and take one • E.g., 3, 4, 6, 8, 12, … (primes+1) • Monster-barring (Lakatos) • E.g, 3, 5, 7, 11, 13, … (primes except 2) • Difference sequence • 1, 2, 2, 4, … (difference between primes) • Other transformations + related seqs. • Combining sequence with ‘core’ seqs. • Conjunction and indexing, • e.g., palindrome primes, every second prime

  6. Making Conjectures • Given a sequence S • E.g, perfect numbers: 6, 28, 496, … • Finds all super and sub-sequences • E.g., perfect numbers are even • Finds all disjoint sequences • E.g., primes numbers are not perfect • Makes ‘moonshine’ conjectures • Large number in S and another seq. • E.g., 107374182 superperfect number

  7. Pruning • All weaker conjectures are pruned • E-perfect numbers are refactorables • E-perfect numbers are even refactorables • User supplies words for definition • Prune those which contain word • Prune those which don’t contain word • Keywords from Encyclopedia • Core, Nice, etc.

  8. Sorting using Plausibility • Plausibility measure of conjectures • Probability of it being a coincidence • E.g., odd refactorables are square • Squares numbers: • 1, 4, 9, …, 1849 (44 terms) • Odd refactorable numbers • 1,9,225,441,625,1089,1521,2025,… • P(n is square) = 44/1849 • P(conjecture occurs by chance) = (44/1849)7 = 4.3 x 10-12 • Unlikely to be a coincidence

  9. Demonstration • Type in your birthday numbers • NWN tells you about these numbers • Choose a sequence • Ask NWN to make conjectures • Very simple interface • Can also link to the online Encyclopedia.

  10. Results • Program available online: • machine-creativity.com/programs/nwn • Manual, Results, Project details • Developed over many months • Many proved theorems discovered • (n) is prime  (n) is prime • Perfects are nialpdrome+pernicious • Many theorems about refactorables • Congruent to 0,1,2 or 4 mod 8 • Odd refactorables are square numbers

  11. Sqrt(n)-rough numbers • Sqrt(n)-rough numbers (mail list) • Largest prime factor < n (e,g., 8) • 3n(n+1) are sqrt(n)-rough numbers • 6n(n+1) are sqrt(n)-rough numbers • Use Clam to plan a proof • 2 days to specify lemmas about <, n • Clam plans a proof • Level of a “pen and paper” proof

  12. Zeitz’s Problem • Hungarian maths competition • Multiply four consecutive numbers • n(n+1)(n+2)(n+3) • Never a square number • Add this sequence to NWN • Look for conjectures which help • Demonstration

  13. Conclusions • Newell and Simon predict in 1958: • Computer discover and prove theorem • Goal of our project (HR system also) • NWN program • Invents new seqs, makes conjectures • Prunes dull ones, sorts by plausibility • Makes interesting conjectures in N.T. • Possibility of proving them automatically

  14. Future Work • Add more transformations • Add more sequences • Strengthen link to Clam • Decision procedures • Encourage mathematicians to use it • Disappointing response so far • I’m not a research mathematician • machine-creativity.com/programs/nwn

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