Proof using counterexample. HELP

Click For Summary
SUMMARY

The discussion focuses on proving the inequality \( \frac{5}{32n} - 4n \) for all \( n \geq 0 \) using a minimum counterexample approach. Participants suggest using induction and exploring simpler expressions for \( 3^{2n} \mod 5 \) to facilitate the proof. The conversation highlights the importance of identifying minimum counterexamples, with specific values like \( n = 0 \) and \( n = 2 \) being considered. The need for factoring expressions such as \( 9^n - 4^n \) is also emphasized as a potential strategy.

PREREQUISITES
  • Understanding of mathematical induction
  • Familiarity with modular arithmetic, specifically \( 3^{2n} \mod 5 \)
  • Basic knowledge of inequalities and counterexamples
  • Ability to factor algebraic expressions like \( 9^n - 4^n \)
NEXT STEPS
  • Study the principles of mathematical induction in depth
  • Research modular arithmetic techniques, focusing on \( 3^{2n} \mod 5 \)
  • Learn about constructing and identifying counterexamples in proofs
  • Practice factoring techniques for expressions such as \( 9^n - 4^n \)
USEFUL FOR

Students in mathematics, particularly those studying proofs and inequalities, as well as educators looking for examples of counterexample techniques in mathematical reasoning.

limegreen00
Messages
1
Reaction score
0
1. Prove by minimum counterexample that for all n>=0, 5/(32n)-4n)

2. Homework Equations : proof by induction?



3. I tried plugging in 0 for n because that would be the minimum counterexample since 5 can't divide 0. If it's not zero it might be 2 because that works as well. I'm not sure where to go from there or what to state if I am right.
 
Physics news on Phys.org
I would try to figure out a simpler way to write 3^(2n) mod 5. I.e. what's a simpler expression for its remainder after division by 5. Or just write it as 9^n-4^n and think about factoring.
 
Last edited:

Similar threads

Proving statements about matrices | Linear Algebra
  • · Replies 25 ·
Replies
25
Views
4K
What type of proof is this defined as?
  • · Replies 2 ·
Replies
2
Views
1K
Direct Proof that every zero of p(T) is an eigenvalue of T
  • · Replies 24 ·
Replies
24
Views
4K
Induction with binomial coefficient
  • · Replies 15 ·
Replies
15
Views
3K
Proving convergence of rational sequence
  • · Replies 2 ·
Replies
2
Views
2K
Proving piecewise function is k-differentiable
  • · Replies 5 ·
Replies
5
Views
2K
Discrete Math- Irrational numbers, proof or counterexample
  • · Replies 2 ·
Replies
2
Views
9K
High School What theorems are available when using modulo arithmetic?
  • · Replies 11 ·
Replies
11
Views
2K
Incredibly close to a modular arithmetic proof by minimum counterexample
  • · Replies 6 ·
Replies
6
Views
2K
Proof given ##x < y < z## and a twice differentiable function
  • · Replies 8 ·
Replies
8
Views
2K