# Perfect numbers/residue

by 1+1=1
Tags: numbers or residue, perfect
 Sci Advisor HW Helper P: 2,586 For your first question, read this (scroll down to equation 16 or so). As far as I can tell, it suggest that "every even perfect number is a triangular number" is false. For the second question, if n is prime, then r = n-1, otherwise r = 0, except in the case where n = 4, where r = 2. Now, why is this so? Well, if n is not prime, then n can be expressed as the product of two numbers less than n. Now, (n-1)! is the product of all numbers less than n, so in it somewhere will be those two numbers which multiply to n. For example, if n = ab for a,b < n and a $\neq$ b, then (n-1)! will be a x b x (all remaining numbers from 1 to n-1, excluding a and b). This makes (n-1)! a multiple of of n, since it is a multiple of ab, thus the residue (if it means the remainder) is zero. Now, in the case where we have a number like 9, which can be expressed as 3x3, we have the case where a = b. Now, for all n>4, $\sqrt{n} < n/2$. What good is this? Well, this tells us that for all n > 4, (n-1)! is still a multiple of n, because for example with 9, although we only have one 3 in (n-1)!, we have a 6. In general, if n>4 and n is square and cannot be expressed by two distinct numbers less than n (i.e. can only be expressed by two of the same number, namely it's square root), then if $\sqrt{n} = a$, then 2a < n, so (n-1)! = a x 2a x (all the other numbers less than n other than a and 2a) = $a^2$ x 2 x all the other numbers = n x 2 x all the other numbers. Again, (n-1)! is a multiple of n, so the residue is zero. Now, the only reason 4 is a big exception is because $\sqrt{4} = 4/2$, as opposed to $\sqrt{4} < 4/2$. In this case, a=2, but 2a is not less than n like we have with 9 and other such squares. Finally, it is obvious that if n is prime, (n-1)! will be no multiple of n. I don't, at the moment, have a proof that r = n-1 for prime n, I'll come back to it. --------------------- EDIT: For that last part, proving r = n-1 for prime n, check out the 'Prime Factorial Conjecture' thread. I think in that thread, a theorem is referred to which proves that.