Question about congruences and orders

• Ch1ronTL34
In summary, the conversation discusses the relationship between an odd prime number p, an integer a, and an integer t. The order of a, mod p^a, is denoted as ord(p^a)a, and the congruence relation is represented by the double equal sign (==). The speaker provides a table of examples to show that if p is an odd prime and ord(p^a)a=2t, then a^t == -1 mod p^a. They also mention that p must divide t in all examples, and suggest squaring a^t in order to find a proof.
Ch1ronTL34
The question is:

Show that if p is an odd prime and ord(p^a)a=2t, then
a^t== -1 mod p^a

First, I used ord(p^a)a to mean "order of a, mod p^a" and the == sign means congruent.

So first, I tried a few examples. Let p=3, a=2
Since ord(9)2=6, then t=3 and:
2^3 == -1 mod 9 TRUE

I continued with different values of p and a. Here is a table(sorry it looks weird):

p--a--t--p^a
3--2--3--9
5--2--10--25
11--2--55--121
13--2--78--169
17--2--68--289

It seems that p|t in all of my examples but I'm stuck...THANKS!

If we take a^t and square it what do we get? Now, since a^t is not 1, since 2t is the order of a, what do you need to show?

It is a known fact that for any odd prime p, the order of p modulo p^a is p^(a-1)(p-1). This means that for any positive integer k such that p^a|k, the order of k modulo p^a is also p^(a-1)(p-1).

In this case, we are given that ord(p^a)a=2t. This means that the order of p^a modulo p^a is 2t. Since p^a|p^a, this means that the order of p^a modulo p^a is also 2t.

By the definition of order, this means that p^(2t)≡ 1 (mod p^a).

Now, we can use the fact that ord(p^a)a=2t to rewrite this as (p^(a-1)(p-1))^2t≡ 1 (mod p^a).

By the properties of congruence, this can be rewritten as p^(2t(a-1)(p-1))≡ 1 (mod p^a).

Since p is an odd prime, we can rewrite p^(2t(a-1)(p-1)) as (p^(a-1))^2t(p-1)^2t.

Using the fact that p^(a-1)≡ 1 (mod p^a), we can rewrite this as (1)^2t(p-1)^2t≡ 1 (mod p^a).

Simplifying, we get (p-1)^2t≡ 1 (mod p^a).

Now, we can use the fact that p is an odd prime to rewrite this as (-1)^2t≡ 1 (mod p^a).

Finally, simplifying, we get (-1)^t≡ 1 (mod p^a).

This is equivalent to saying that (-1)^t-1 is divisible by p^a, which is the same as saying that (-1)^t≡ -1 (mod p^a).

Therefore, we have shown that a^t≡ -1 (mod p^a) when p is an odd prime and ord(p^a)a=2t.

1. What is the definition of congruence?

Congruence refers to the relationship between two objects or shapes that have the same size and shape. In other words, they are identical in terms of their measurements and angles.

2. How do you write a congruence statement?

A congruence statement is written in the form of "object A is congruent to object B" and is denoted by the symbol ≅. For example, if two triangles have the same measurements and angles, the congruence statement would be written as "triangle ABC is congruent to triangle DEF" or "ΔABC ≅ ΔDEF".

3. What is the difference between congruence and similarity?

While congruence refers to two objects being identical in terms of size and shape, similarity refers to two objects having the same shape but possibly different sizes. Similar objects have the same angles, but their measurements may be different.

4. What is the order of congruence?

The order of congruence refers to the number of times an object can be rotated or reflected to match another object. For example, a square has an order of 4 because it can be rotated 4 times to match itself. However, a rectangle has an order of 2 because it can only be rotated twice to match itself.

5. How are congruence and orders related?

Congruence and orders are related in the sense that the order of congruence determines how many times an object can be rotated or reflected to match another object. The greater the order of congruence, the more times an object can be rotated or reflected to match another object, and the more identical they are in terms of size and shape.

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