Question

Let p be an odd prime. Show that the congruence x41(modp)x^{4} \equiv-1(\bmod p) has a solution if and only if p is of the form 8k + 1.

Solution

Verified
Step 1
1 of 3

Let pp is an odd prime and we have ϕ(p)=p1\phi(p) = p - 1 is even. We will write d=(4,p1)d = (4,p-1). If we look at theorem 9.17, we will know that x41x^4\equiv-1 (mod pp) has a solution if and only if (1)ϕ(p)d1(-1)\dfrac{\phi(p)}{d}\equiv1 (mod pp). Coming back from the order of 1-1 mod pp is 2, so we must have 2p1d2\vert\dfrac{p-1}{d}. So, there exist kk such that 2k=p1(p1,4)2k = \dfrac{p-1}{(p-1,4)}.

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Recommended textbook solutions

Elementary Number Theory and Its Application 6th Edition by Kenneth H. Rosen

Elementary Number Theory and Its Application

6th EditionISBN: 9780321500311Kenneth H. Rosen
1,873 solutions
Advanced Engineering Mathematics 10th Edition by Erwin Kreyszig

Advanced Engineering Mathematics

10th EditionISBN: 9780470458365 (8 more)Erwin Kreyszig
4,134 solutions
Mathematical Methods in the Physical Sciences 3rd Edition by Mary L. Boas

Mathematical Methods in the Physical Sciences

3rd EditionISBN: 9780471198260 (1 more)Mary L. Boas
3,355 solutions
Advanced Engineering Mathematics 6th Edition by Dennis G. Zill

Advanced Engineering Mathematics

6th EditionISBN: 9781284105902 (2 more)Dennis G. Zill
5,281 solutions

More related questions

1/2

1/3