by my calculations 2^{112} = 1 (mod 54410972897).
note that 112 divides both 560 and 544109728896.
I am slightly confused by the following:
Fact: 54410972897 is a prime. Set p = 54410972897.
560 is the least power of 2 such that 2^{560} = 1 (mod. p). [2^{280 }= -1 (mod. p).]
So, according to Euler 560 should divide p-1 but it does not.
Can anyone give a correction to this?