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