Mod and primes?!

Is it true that

1.if p is prime then n^(p squared)=n mod p for all natural numbers n??

2. n^7=n mod 35 for all natural numbers n

If so then what working should i show?and if not then what would be a counter-example??thanksss~~~

Quote:

---

Originally Posted by bryan06

2. n^7=n mod 35 for all natural numbers n

---

n = 2 is a counterexample.

-Dan

Quote:

---

Originally Posted by bryan06

1.if p is prime then n^(p squared)=n mod p for all natural numbers n??

---

This one is true by a version of Fermat's Little Theorem. We can state that
http://latex.codecogs.com/png.latex?...t{(mod p - 1)}

since p^2 - 1 is divisible by p - 1. So by Fermat's Little
http://latex.codecogs.com/png.latex?...\text{(mod p)}

-Dan

thankss~~still a bit confused though i researched on wikipedia what fermat little theorem is but still cant figure out how you derived the following:

Quote:

---

Originally Posted by topsquark

http://latex.codecogs.com/png.latex?...t{(mod p - 1)}

---

also how do you get from this line

Quote:

---

Originally Posted by topsquark

since p^2 - 1 is divisible by p - 1.

---

to this line

Quote:

---

Originally Posted by topsquark

So by Fermat's Little
http://latex.codecogs.com/png.latex?...\text{(mod p)}

---

thankss

Quote:

---

Originally Posted by topsquark

This one is true by a version of Fermat's Little Theorem. We can state that
http://latex.codecogs.com/png.latex?...t{(mod p - 1)}

since p^2 - 1 is divisible by p - 1. So by Fermat's Little
http://latex.codecogs.com/png.latex?...\text{(mod p)}

-Dan

---

Or just note that http://latex.codecogs.com/png.latex?...n\text{ mod }p