Factors for the differences between Mersenne numbers

[a nicol]

It seems the differences between all Mersenne numbers have factors via:

a1 = Mersenne exponent
a2 = next Mersenne exponent
b = (a2-a1)/2

(2^a1 * 3) * ((4^b - 1)/3)

For example:

(2^26-1) - (2^12-1) = 67104768

(2^12 * 3) * ((4^7 - 1)/3) = 12288 * 5461 = 67104768

If ((4^b - 1)/3) mod 3 is not 0, these factors are in lowest terms ratio.

When ((4^b - 1)/3) mod 3 = 0, the lowest terms can be found by:

(2^a1):((4^b - 1)/9)

Is this always the case?


Some factors for the first differences of Mersenne primes - http://oeis.org/A139231

24 * 1 = 24
96 * 1 = 96
384 * 21 = 8064
24576 * 5 = 122880
393216 * 1 = 393216
1572864 * 1365 = 2146959360
6442450944 * 357913941 = 2305843007066210304
6917529027641081856 * 89478485 = 618970017336847128235868160
1856910058928070412348686336 * 87381 = 162258657859193720701440560726016

[LaurV]

Think about how mersenne numbers are represented in binary, and what's happen if you subtract one from another. hint: you get a mersenne number with an even exponent, multiplied by a power of two.