Problems & Puzzles: Puzzles

Puzzle 1022. x(n)=(n - 1)^n + n

Sebastián Martín Ruiz sent the following puzzle:

I  have checked for n=1 to 3000  that 

x(n)=(n - 1)^n + n

is always composite except for n = 2 and n = 3

 

Eric Snyder observed:
 
For n = 2 mod 3, n-1 = 1 mod 3, 1 mod 3 to any power is still 1 mod 3, +n = 2 mod 3 gives 0 mod 3, divisible by 3.
 
It appears that for n even and not = 2 mod 3, x(n) is divisible by n+1 (if n+1 is prime), or a factor of n+1 (if n+1 is composite), for "almost all"--there are a couple exceptions.
 
Q. Prove that x(n) is always composite for n>=4 or find a counterexample.

 


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