Question

Show that in \(Z_{p}\), if \(a^{i} \bmod p=1\), then \(a^{n} \bmod p=a^{n \bmod i} \bmod p\).

Short answer

In \\(\mathbb{Z}_{p}\\), if \\(a^{i} \equiv 1 \pmod{p}\\), then \\(a^{n} \equiv a^{n \bmod i} \pmod{p}\\).

Step-by-step solution

Understanding the Problem

We need to prove that in a modular arithmetic system with modulo a prime number, if raising any integer to a certain power gives 1, then any other power can be reduced mod that certain power without changing the modular result.

Identifying Properties

In \(\mathbb{Z}_{p}\), where \(p\) is prime, Fermat's Little Theorem states that \(a^{p-1} \equiv 1 \pmod{p}\) for any integer \(a\) not divisible by \(p\). Given \(a^{i} \equiv 1 \pmod{p}\), this means \(i\) is the order of \(a\).

Assume the Identity

Given \(a^{i} \equiv 1 \pmod{p}\), this implies any integer \(n\) can be expressed in terms of \(i\), that is, \(n = qi + r\), where \(0 \leq r < i\). So, \(a^{n} = a^{qi + r}\).

Applying Modulo Property

Since \(a^{i} \equiv 1 \pmod{p}\), it follows that \(a^{qi} \equiv (a^i)^q \equiv 1^q \equiv 1 \pmod{p}\) by properties of exponents. Thus, \(a^{n} = a^{qi+r} = a^{qi} \cdot a^{r} \equiv 1 \cdot a^{r} \equiv a^{r} \pmod{p}\).

Utilizing the Remainder

Since \(n = qi + r\) implies \(r = n \bmod i\), it follows that \(a^{n} \equiv a^{r} \equiv a^{n \bmod i} \pmod{p}\). Thus, we have shown that \(a^n \bmod p = a^{n \bmod i} \bmod p\).

Key concepts

Fermat's Little Theorem

Fermat's Little Theorem is a remarkable result in the field of number theory, particularly useful in modular arithmetic. It states that for any integer \( a \) not divisible by a prime \( p \), the following holds: \( a^{p-1} \equiv 1 \pmod{p} \).
This theorem is fundamental in understanding various properties of numbers in modular systems.
In our given problem, where we work with \( \mathbb{Z}_{p} \), this theorem provides a strong basis.

• If any integer \( a \) raised to the power of \( p-1 \) is congruent to 1 mod \( p \), it implies a cyclical pattern when higher powers of \( a \) are considered.
• This cyclical behavior is crucial because it underpins the concept of reducing powers in modular arithmetic, leading to results such as \( a^n \equiv a^{n \bmod i} \mod p \).

By using Fermat's Little Theorem, we recognize patterns in powers that help simplify calculations in modulo environments, establishing a foundation for the other concepts involved in our problem.

Order of an Element

The order of an element in modular arithmetic is the smallest positive integer \( i \) for which \( a^i \equiv 1 \pmod{p} \).
This concept captures the repeat cycle length of the powers of \( a \) that results in a remainder of 1 when divided by a prime number \( p \).

• In our problem, knowing \( a^i \equiv 1 \pmod{p} \) indicates that \( i \) is effectively the point at which the cyclic nature of \( a \)'s powers resets.
• Any exponentiation of \( a \) beyond this point can be adjusted back to this cycle without loss of generality or accuracy in calculations, such as finding \( a^n \equiv a^{n \bmod i} \mod p \).

Understanding the order helps determine how we can simplify expressions in modular systems by reducing exponents, leveraging the periodicity of powers within certain constraints. This gives us a powerful tool to approach the problem effectively.

Exponential Identity

The Exponential Identity principle in modular arithmetic is the idea that you can decompose exponents into smaller, more manageable parts to simplify calculations.
When given an expression involving powers, such as \( a^n \), this identity is crucial for breaking it down through modular arithmetic properties.

• For our exercise, \( n = qi + r \) is an expression that decomposes the exponent \( n \) into a quotient \( q \) and a remainder \( r \), with respect to the order \( i \) of \( a \).
• This allows us to express \( a^n \) as \( a^{qi+r} \) and simplifies further to \( a^r \), since \( a^{qi} \equiv 1 \pmod{p} \).
• The result, \( a^n \equiv a^{n \bmod i} \mod p \), shows how exponential identities help maintain the integrity of modular expressions through simplification.

This principle is a central tool in understanding how to handle higher powers in modular systems, allowing for precise results with reduced complexity, especially in proving identities like in our problem.