Theorems on (x+1)^x +/- x^x

[cino hilliard]

Hi,

Fooling around with the  (x+1)^x +/- x^x sequences and Pari, I have come up 
with a bunch of
theorems, conjectures or (Arghh) falsehoods. Here are a few. Most of which I 
cannot prove
completely. Then again some one said "Trust but Verify" an well, I an 
applying the converse of that
theorem - "Verify but trust" :-)

Theorem 1
Let n = (x+1)^x - x^x
If x is prime and n is prime then n-1  is divisible by x.

Corollary 1
Let f(i) be the ith factor of n then
If n is not prime then  f(i) - 1 is divisible by x.
Corollary 2
If x is not prime then f(i)-1  is divisible by one or more factors of x.

Theorem 2
Let n = (x+1)^x + x^x. Here if x>1 is odd, 2x+1 is a divisor so n cannot be 
prime.
If x is prime  and factor f(i)>=x then x divides f(i)-1.

Corollary 1
If x is not prime then some  f(i)-1 is divisible by factors of  x.

I think these can be proved in part if we can prove

Theorem 3
If p is prime then
x^p - y^p  == 1 mod p

and

Theorem 4
if p is prime
x^(p-1) - y^(p-1) == 0 mod p

This one is easy from Fermat's Little theorem

x^(p-1)  == 1 mod p
y^(p-1)  == 1 mod p
subtracting we have
x^p-1 - y^(p-1) == 0 mod p

Theorem 5
for p > 1
x^p - y^p can be  prime if and only if p is prime.

This theorem is a take on an old friend of mine x^(p-1)/2 +/- y^(p-1)/2 is 
divisible by p (xy,p)=1
Now what about the divisors of  x^p - y^p?


Can someone prove these or get me started?

It seem that these Theorems could be used to some extent to help prove 
compositness of certain
numbers.

Cino
"Observation lends to analysis. Analysis lends to Law."

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