The Representation of Large Numbers How High Can You Go? George T. Heineman WPI Computer Science Department 10-Sep-2004

Integer as Abstractions Fundamental units for calculation Common representations –Short [-32768, 32767] –Int[ , ] –Long[ , ] Suitable for “small computations”

Number Theorists need more Mersenne Primes (May 15 th 2004) –2 24,036, contains 7,235,733 digits How to determine primality? –Agrawal, Kayal, Saxena, 2002 –Paper every grad student should at least attempt to read! How can computations be performed over integers of 1,000,000 digits? How can one calculate pi to 1,000,000,000 digits? –Software Engineering comes to the rescue! –Abstraction vs. Representation

Necessary Requirements Standard algebraic operators (+, -, *, /) Power functions (a b ), Greatest Common Divisor (gcd), square root, modulo

Necessary Requirements Standard algebraic operators (+, -, *, /) Power functions (a b ), Greatest Common Divisor (gcd), square root, modulo

Primality

Big Picture The Gummitron 2004

Design Challenge Materials 1.One Pack of Sealed Poker Playing Cards 2.Scotch Tape 3.Fifteen Gummi TM Bears 1.Phase One: Build a continuous structure using only the materials provided 2.Phase Two: Place structure on desktop at least six inches away from the Gummitron 3.Phase Three: Over 15 second period, team launches up to fifteen Gummi Bears. Wait 10 seconds, then judge assesses score as shown in Equation 1. Note that Gummi Bears who are lower than 3 inches receive zero points 6 inches (x i,y i ) (0, 0) Equation 1 3 inches  (x i + 3 * y i ) Scoring Rules 1.Gummi TM Bears lower than 3 inches in height receive 0 points 2.Measurements will be made from the highest and/or rightmost point of the GummiBear. Gummitron