From: Wei Dai (weidai@eskimo.com)
Date: Sat Jun 08 2002 - 14:16:42 MDT
On Sat, Jun 08, 2002 at 09:01:37AM -0700, Lee Corbin wrote:
> What is the point of finding Mersenne primes? Or odd perfect
> numbers themselves?
I guess there the hope is that they may give us some deep insight into the
structure of natural numbers. But almost perfect odd numbers are
structually too simple and too easy to find.
Here the algorithm I used, I'll leave some spoiler space in case you want
to try to figure one out yourself. Hint: start with the formula you gave
spike66 for computing the sum of divisors.
Consider a square-free (not divisible by a square) number x whose sum of
divisors (counting 1 and itself as divisors) is less than 2 times itself.
This can be expressed as:
let y = (p_1+1)*(p_2+1)*...*(p_n+1) < 2*p_1*p_2*...*p_n = 2*x
where p_i are the prime factors, or
y / x < 2
Now you want to find another prime number p which will make y*(p+1) /
(x*p) as close to 2 as possible, but still less than it (so you can repeat
this process). Some simple algebra shows that the first prime after 1 /
(2*x/y - 1) satisfies this criteria. Repeating this process will let you
get as close to perfect oddness as you want (but you'll never reach it
this way).
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