A new generalized definition of Mersenne numbers is proposed of the form
, called global generalized Mersenne numbers and noted
with base
a and exponent
n positive integers. The properties are
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A new generalized definition of Mersenne numbers is proposed of the form
, called global generalized Mersenne numbers and noted
with base
a and exponent
n positive integers. The properties are investigated for prime
n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. It is found that for any
a,
is even and divisible by
n,
a and
for any prime
, and by
for any prime
. The remaining factor is a function of triangular numbers of
, specific for each prime
n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, showing first that
depending on the congruence of
; second, that
are divisible by 10 if
and, if
,
, depending on the congruence of
; third, that all factors
of
are of the form
such that
is either prime or the product of primes of the form
, with
natural integers; fourth, that for prime
, all
are periodically congruent to
depending on the congruence of
; and fifth, that the factors of a composite
are of the form
with
with
, 1, 2 or 3 depending on the congruences of
and of
. The potential use of generalized Mersenne primes in cryptography is shortly addressed.
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