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NumbersWithNames Program: Enhancing Number Theory Understanding

The NumbersWithNames Program, developed by Simon Colton from the University of Edinburgh and Louise Dennis from the University of Nottingham, explores perfect numbers, Neil Sloane’s Encyclopedia of Integer Sequences, and making conjectures related to number sequences. Through a four-step process, the program invents related sequences, prunes uninteresting conjectures, and sorts them by plausibility. Using various transformations, including monster-barring and difference sequences, it encourages users to make conjectures and demonstrate the likelihood of certain occurrences. The program aims to foster a deeper understanding of number theory through user interaction and exploration of diverse mathematical sequences.

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NumbersWithNames Program: Enhancing Number Theory Understanding

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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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