This number is a prime.
When Samuel Yates started maintaining a list of the largest known prime numbers in 1979, M(44497) was the largest of all. It has 13395 decimal digits and was discovered to be prime by David Slowinski and Harry L. Nelson in April of that year. [Dobb]
The last term in the sequence which contains Mersenne prime exponents that are less than 50000 and starts with 4 (4253, 4423, 44497). Note that 4253 starts with one 4, and 4423 starts with two 4's, and 44497 starts with three 4's. [Jeong]
The Mersenne prime 2^44497-1 is a self-number. Note that among all 51 known Mersenne primes, only 7 of them are self, and the first three Mersenne primes 3, 7, and 31 are self-numbers. [Kontobesar]