Are All Even Perfect Numbers Also Triangular Numbers?

[1+1=1]

i have two ?'s to ask yall. ok, i need to prove every even perfect number is a triangular number. the formula is t(n)= 1+2+... tn = (n(n+1))/2.

ok i know that to be a perfect number, it is sigma (a) which menas 2times a. for ex, sigma(6)=1+2+3+6=12. this is as far as i can get can anyone show me light for this?

find least residue for (n-1)! mod n for several n values and find a general rule.

alright, i know bty least residue means basically the remainder. it is in the form of a=bq + r, where r is the least residue. again, can anyone show me what I'm missing here for this problem?

please even if you are viewing this post, please say anything as to what you are thinking about the problem...

 

[Gokul43201]

2. Look at the residues for prime n.

Then look at today's thread titled 'Prime Factorial Conjecture' in this subforum.

 

[AKG]

You've asked these questions in another thread, where I've responded. In case you missed it, check [post=244396]my post[/post] along with the thread Gokul43201 suggested.

 

[robert Ihnot]

For the first question, regarding triangle numbers and perfect numbers; the two facts we need to know are the form of the even perfect numbers, the only kind ever found, and a way of relating a triangle number to a perfect number. The form of the perfect number is: (2^(p-1))((2^p) -1). In this case we must have (2^p) -1 is prime and this implies that p is also prime.

Now all that is necessary is to find an if and only if relationship between a triangle number, and something like a square, and see if that also holds for a perfect number.