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If we think of primes of the form $a^n-b^n,$ where $a,b,n$ are positive natural numbers and $a>b$,

$(a-b)\mid (a^n-b^n)$, so $a-b$ must be $1$

and $n$ must be prime else $(a^r-b^r)\mid (a^n-b^n)$ where $r>1$ and $r\mid n$

So, $a^n-b^n$ reduces to $(b+1)^p-b^p$ where $p$ is prime.

If $b=1,(b+1)^p-b^p=2^p-1$ which is well-known Mersenne number.

My questions are :

(1) Why $2$ was chosen for the Mersenne numbers? Is it for the ease of calculation or something else?

(2) Is there any known development on the generalized version i.e., $(b+1)^p-b^p$

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I think that Mersenne chose to investigate primes of the form $M_p=2^p-1$ and not the other way around. – draks ... Oct 5 '12 at 7:14
People have looked at $(b+1)^p-b^p$ and more generally at $(a^p-b^p)/(a-b)$. Some results are tabulated in Riesel's book on primality testing and factorization. – Gerry Myerson Oct 5 '12 at 7:25
@draks: "the other way around"?? As in "the primes of the form $2^p-1$ chose to investigate Mersenne"? – Marc van Leeuwen Oct 5 '12 at 7:37
@marc yes, it's not us studying them, they are alive... – draks ... Oct 5 '12 at 7:38
up vote 1 down vote accepted

The numbers $$u_n={a^n-b^n\over a-b}$$ are said to be a Lucas sequence. You'll find a lot of links if you just type that term into the internet, starting with Wikipedia.

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