If m 4n 1 with n0 and m is prime then 3m12 1mod m Determine

If m =4^n +1 with n>0 and m is prime then 3^(m-1)/2 =-1(mod m)

Determine is, in general, true or false. Recall that a
universal statement is true if it is true for all possible cases while it is false if there is even one
counterexample. Be prepared to prove that your answer is correct by supplying a proof or
counterexample, whichever is appropriate

If m =4^n +1 with n>0 and m is prime then 3^(m-1)/2 =-1(mod m)

solution:

consider

m =4^n +1

m =(3+1)^(n) +1

m=(3^n +3^n-1 +........+3+1)+1

m=3k+2

Thus, 3 is not divisible by m
Using Fermat little’s theorem we have if a is not divisible by p and p is prime then

3^(m-1) =1(mod p)

3^(m-1) -1=pk where k is an integer

(3^(m-1)/2 -1) (3^(m-1)/2 +1)=pk

k=[(3^(m-1)/2 -1) (3^(m-1)/2 +1)]/p

since (3^(m-1)/2 -1) is not divisible by p

(3^(m-1)/2 +1) is divisible by p

3^(m-1)/2 =-1(mod m)

Could you check it for me please is it correct or not?

Solution

I think your solution is correct.