infinitudeofevenperfectnumbers
Infinitude of Even Perfect Numbers
There are an infinite number of Mersenne primes (2**p-1) and hence (even) perfect numbers.
Proof:
Suppose Mp = 2**p-1
hcf (Mp, n) = 1, for all n<p [by Lemma 1]
let p->99999… [Euclid]
=> Mp prime
=> there exist an infinite number of prime Mp
LEMMA 1
Any divisor of Mp is of the form 2kp+1
Proof:
Suppose q | Mp, i.e. 2**p == 1 [mod q]
Let 2 have order k modulo q, then k | p [by Lemma 2]
And hence k=p.
But, 2**(q-1) == 1 [mod q] [Fermat’s Little Theorem]
So, similarly, k | q-1
i.e. p | q-1
p and q must both be odd, so q = 2kp + 1
LEMMA 2
Definition:
The order of a modulo n is the smallest positive integer k, s.t. a**k == 1 [mod n]
Let a have order k mod n, then a**h == 1 [mod n] <=> k | h
Proof:
Suppose k | h, i.e. h = jk, then
a**k == 1 => a**h == 1 [mod n]
Conversely,
Suppose a**h == 1 [mod n]
h = qk + r [0 <= r < k]
a**h == a**(qk+r) == (a**k)**q.a**r
=> a**r == 1
=> r = 0; otherwise k is not the smallest integer such that a**k == 1
Therefore, h = qk, i.e. k | h