# Always prime

Is there a prime $$P$$ such that for any prime $$p$$ , $\frac{ (P-1)^p -1 }{ P-2 }$ is always prime?

Note by Takeda Shigenori
4 years, 8 months ago

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Another empirical relation i founded was that, 2pn+1=prime number or a number having its factors as prime,where p is a prime number,n is any odd natural number

- 4 years, 8 months ago

There cannot be such a function which always gives a prime. Its a well known fact. I think you can find the proof in Titu Andreescu's '104 Number Theory Problems'

- 4 years, 8 months ago

I think there isnt.let P-2=x.so P-1=1+x.now,finding the binomial expansion of(1+x)^p and putting it in our expression,it is found that our expression is always divisible by p.

- 4 years, 8 months ago

That is why i divided $$P-2$$ , or else it is always composite

- 4 years, 8 months ago