CS 495 Special Topics in CS: Evolutionary Computation Fall 2002, Instructor: Jeffrey Horn

HW 3: "Generalized Fermat's Last Theorem"

Description:

Definition of Generalized Problem: x₀ⁿ + x₁ⁿ + x₂ⁿ+ . . . +x_k-1ⁿ = x_kⁿ

Find integers x_0, x_{1, ... ,} x_kand n (with all > 1) that satisfy the equation above.

Famous results:

For k = 2, n = 2 this is Pythagorem's theorem, with many known solutions (e.g., x₀= 3, x₁= 4, x₂= 5)

For k = 2, n > 2 this is Fermat's Last Conjecture, which Andrew Wiles recently proved to be a theorem, namely that there IS NO SOLUTION: x₀ⁿ + x₁ⁿ ¬= x₂ⁿfor n > 2 .

But what about other values of k and n?

Related Sites:

On a Generalized Fermat-Wiles Equation

Mark Dettinger's Conjecture

Computing Minimal Equal Sums Of Like Powers (a related but direction to go in!)

The Table: (summarizing MY current state of knowledge for this problem)

k \ n	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7 . . .
k = 2 (Fermat)	3² + 4² = 5² ^etc. (Pythagorem, B.C.)	No solution! (Wiles, 1993)	No solution! (Wiles, 1993)	No solution! (Wiles, 1993)	No solution! (Wiles, 1993)	No solution! (Wiles, 1993)
k = 3	2^2 + 3^2 + 6^2 = 7^2 Karl Haendler	3³ + 4³ + 5³ = 6³ ^{(classic result)}	95800⁴ + 217519⁴ + 414560⁴ = 422481⁴ (Norrie, 1911)	Paul Cornwell
k = 4	2^2 + 4^2 + 5^2 + 6^2 = 9^2 Karl Haendler	36^3 + 3^3 + 15^3 + 21^3 = 39^3 Derrick Cearfoss	30⁴ + 120⁴ + 272⁴ + 315⁴ = 353⁴ (Elkies and Frye, 1988)	27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵ (Lander and Parkin, 1966)	Nicola Makela
k = 5	2^2 + 4^2 + 6^2 + 7^2 + 8^2 = 13^2 Karl Haendler	1³ + 1³ + 2³ + 3³ + 3³ = 4³ Jim Cattron	12^4 + 24^4 + 42^4 + 18^4 + 27^4 = 45^4 Ryan Hallstrom	67^5 + 19^5 + 43^5 + 46^5 + 47^5 = 72^5 (Ryan Hallstrom) Jim Cattron
k = 6	2^2 + 4^2 + 5^2 + 6^2 + 8^2 + 12^2 = 17^2 Karl Haendler		1^4 + 2^4 + 2^4 + 2^4 + 2^4 + 2^4 = 81 = 3^4 Ryan Hallstrom	13^5 + 18^5 + 23^5 + 36^5 + 66^5 + 31^5 = 1350125107 = 67^5 (Ryan Hallstrom)	known open
k = 7 . . .	2^2 + 4^2 + 5^2 + 6^2 + 8^2 + 10^2 + 14^2 = 21^2 Karl Haendler		34^4 + 8^4 + 50^4 + 1^4 + 4^4 + 22^4 + 16^4 = 53^4 Ryan Hallstrom		Chaplin Cinelli	known open

Tasks:

Write "group code" for the optimization part of the problem. Distribute to all.
Write "group code" for the visualization of the problem. I want to see a window showing at least several items: Dan Cardin
1. the best solution so far, with the k integers shown in the full equation, and on the next line the computed values of the LHS and RHS, and then somewhere perhaps next to this line, I want to see the difference between LHS and RHS.
2. Usual stats: gen number, avg. fitness, best fitness, worst fitness, maybe median fitness.
3. A plot of best fitness over time.
4. All of this continuously updated.
Fill in the table above: I want to see names in each element of the table. Those folks will then decide how much research to do (i.e., see what is known already about that sub-problem!), and what kind of experimentation (e.g., emphasize re-discovery of known solutions, go for new solutions, etc.)