Prime Congruence: What Can You Expect of k?

[pedrommp]

Homework Statement

If p and q are prime numbers, p>q>2 , and 1+k*p divides q^n for some positive integer n. What can i expect of the values of k ? Does it works just for k=0 ?

Homework Equations

q^n = 1 (mod p)

The Attempt at a Solution

I know that k=0 works , and k=odd don't work cause 1+k*p would be even and q^n is odd ; if n= x*(p-1)+r , -1<r<(p-1) then q^n = q^(x*(p-1)+r) = q^(x*(p-1))*q^r = q^r (mod p) so i just need to check for -1<r<(p-1).

 

[mighty2000]

I would first try to understand the problem by breaking it down into smaller parts. First, I would start by reviewing the definitions of prime numbers and their properties. This would help me understand why p>q>2 is a necessary condition for the problem.

Next, I would look at the equation q^n = 1 (mod p) and try to understand its significance in relation to the problem. This equation tells us that q^n is congruent to 1 modulo p, which means that q^n leaves a remainder of 1 when divided by p. This could also be written as q^n = 1 + kp, where k is some integer.

Then, I would focus on the condition 1+k*p divides q^n. This means that 1+k*p is a factor of q^n, or in other words, q^n is a multiple of 1+k*p. This condition is necessary for the given statement to be true.

From here, I would try to come up with some examples to better understand the problem. For instance, if p=5 and q=3, we can see that 1+k*5 divides 3^n for some values of k. For k=0, we get 1*5 divides 3^n, which is true since 5 is a factor of 3^n. However, for k=1, we get 1+5*1=6, which does not divide any power of 3.

Based on this example, I would conclude that k=0 is a possible value for the problem, but it may not be the only one. I would then try to come up with a general solution for any prime numbers p and q.

Finally, I would use mathematical reasoning and proofs to show that k=0 is indeed the only possible value for the given conditions to be true. This would involve showing that for any other value of k, the condition 1+k*p does not divide q^n.

In conclusion, as a scientist, I would approach this problem by first understanding the definitions and properties involved, then using examples to gain insight, and finally using mathematical reasoning and proofs to come to a conclusion.