Prove whether a specific return map R_4(n) equals the identity map for elements in a subset of integers modulo a prime p | Step-by-Step Solution

What We're Solving:

We need to investigate whether a specific return map R₄(n) acts as the identity map on a subset W₄ of integers modulo a prime p. Essentially, we're checking if R₄(n) = n for all elements n in W₄.

The Approach:

This is a beautiful problem that combines modular arithmetic with function analysis! Our strategy will be to:

• First understand what the map R₄ and set W₄ represent
• Examine the algebraic structure we're working within
• Test whether the mapping preserves elements (identity property)
• Use properties of modular arithmetic and prime numbers to our advantage

Step-by-Step Solution:

Step 1: Define our mathematical objects Since the specific definitions of R₄(n) and W₄ aren't provided, here's the general approach:

• Identify what operation R₄ performs on elements
• Determine which integers modulo p belong to W₄
• Note that we're working modulo a prime p (this gives us a field structure!)

Step 2: Set up the identity condition For R₄ to be the identity map on W₄, we need:

• R₄(n) ≡ n (mod p) for every n ∈ W₄
• This must hold for ALL elements in W₄, not just some

Step 3: Use properties of modular arithmetic Working modulo a prime p gives us powerful tools:

• Every non-zero element has a multiplicative inverse
• Fermat's Little Theorem: aᵖ⁻¹ ≡ 1 (mod p) for a ≢ 0 (mod p)
• The structure forms a field ℤ/pℤ

Step 4: Analyze the mapping algebraically

• Apply R₄ to a general element n ∈ W₄
• Use the field properties to simplify
• Check if the result equals n modulo p

Step 5: Consider special cases

• What happens when n = 0?
• What about when n is a quadratic residue modulo p?
• Are there any elements where R₄(n) ≠ n?

The Answer:

Without the specific definitions of R₄ and W₄, I can't give you the final conclusion, but your proof should either:

• Show that R₄(n) ≡ n (mod p) for all n ∈ W₄ (proving it IS the identity)
• Find a counterexample where R₄(n) ≢ n (mod p) for some n ∈ W₄ (proving it's NOT the identity)

Your conclusion should be a clear statement: "R₄ is/is not the identity map on W₄" followed by your mathematical justification.

Memory Tip:

Remember "PRIME = POWER"! When working modulo a prime, you have the full power of field theory at your disposal. Use Fermat's Little Theorem and the fact that every non-zero element has an inverse - these are your secret weapons in modular arithmetic problems!

Keep going - you're tackling a sophisticated problem that beautifully connects abstract algebra concepts! 🌟