Proof: (Pigeon Hole Principle) from a Problem Solving Class

Click For Summary
SUMMARY

The discussion centers on proving the existence of integers a, b, and c, not all zero, with absolute values less than 10^6 such that the expression |a + sqrt(2)*b + cuberoot(3)*c| is less than 10^-11. The Pigeon Hole Principle is identified as a key method for approaching this proof. Participants express challenges in finding specific integer values for a, b, and c, while one user indicates progress towards a solution after considerable thought.

PREREQUISITES
  • Understanding of the Pigeon Hole Principle
  • Familiarity with irrational numbers such as sqrt(2) and cuberoot(3)
  • Basic knowledge of inequalities and absolute values
  • Experience with integer solutions in mathematical proofs
NEXT STEPS
  • Research the application of the Pigeon Hole Principle in number theory
  • Explore methods for approximating irrational numbers
  • Study techniques for solving inequalities involving multiple variables
  • Investigate previous mathematical problems similar to the one discussed
USEFUL FOR

Students in problem-solving classes, mathematicians interested in number theory, and anyone studying the application of the Pigeon Hole Principle in proofs.

rad0786
Messages
187
Reaction score
0

Homework Statement



Prove that THEIR EXISTS INTEGERS a,b,c NOT ALL 0 AND EACH OF ABSOLUTE VALUE <10^6 SUCH THAT
|a + sqrt(2)*b + cuberoot(3)*c| < 10^-11


Homework Equations




|a + sqrt(2)*b + cuberoot(3)*c| < 10^-11


The Attempt at a Solution



Well, I know that we have to use the pigeon hole principle, otherwise I am completely lost.

I did try to actually find values for a,b,c ... but no luck with that.

I am now attempting to rearrange the formula and solve for some variable in terms of another.

Has anybody seen a problem like this before? Does it have a name?
 
Physics news on Phys.org
Never mind people, after a lot of hard thinking, i got it (well...close to it!)
 

Similar threads

Proving convergence of rational sequence
  • · Replies 2 ·
Replies
2
Views
2K
Medium Hard continuity proof Tutorial Q6
  • · Replies 2 ·
Replies
2
Views
1K
Math Olympiad problem - Applying induction and the pigeon hole principle
  • · Replies 1 ·
Replies
1
Views
3K
Solving the Larsen Problem with Pigeon-Hole Principle
  • · Replies 5 ·
Replies
5
Views
3K
Solving for the Height and Width of St. Louis Arch: A Calculus Problem
  • · Replies 4 ·
Replies
4
Views
2K
Solve the given problem that involves probability
  • · Replies 1 ·
Replies
1
Views
2K
A discrete-time queue Markov Chain problem
  • · Replies 12 ·
Replies
12
Views
2K
MHB Solve Pigeon Hole Theory Problems
  • · Replies 1 ·
Replies
1
Views
3K
Combinatorics: looking for an alternative solution
  • · Replies 7 ·
Replies
7
Views
2K
Using Inner Product Properties to Solve Vector Problems
  • · Replies 4 ·
Replies
4
Views
2K