20201014, 19:32  #1 
Nov 2016
11001_{2} Posts 
Factors for the differences between Mersenne numbers
It seems the differences between all Mersenne numbers have factors via:
a1 = Mersenne exponent a2 = next Mersenne exponent b = (a2a1)/2 (2^a1 * 3) * ((4^b  1)/3) For example: (2^261)  (2^121) = 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 
20201015, 09:53  #2 
Romulan Interpreter
Jun 2011
Thailand
2×23×199 Posts 
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.

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