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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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This is probably more like a long comment than an answer, but clearly some people have considered this earlier, because OEIS has sequences for them. Based on a simple search of $p<2000$ such that $(b+1)^p-b^p$ is prime, I found these references:

 b=1: A000043: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279,
 b=2: A057468: 2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061,
 b=3: A059801: 2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993,
 b=4: A059802: 3, 43, 59, 191, 223, 349, 563, 709, 743, 1663,
 b=5: A062572: 2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511,
 b=6: A062573: 2, 3, 7, 29, 41, 67, 1327, 1399,
 b=7: A062574: 7, 11, 17, 29, 31, 79, 113, 131, 139,
 b=8: A059803: 2, 7, 29, 31, 67, 149, 401,
 b=9: A062576: 2, 3, 7, 11, 19, 29, 401, 709,
b=10: A062577: 3, 5, 19, 311, 317, 1129,
...

The first column above (minimal $p$ values as a function of $b$) is A058013.

Somewhat surprisingly, I was not able to find any single OEIS entry which gives the entire table above. Maybe I will submit it.

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