I can see where that would be confusing. For 2[sup]n - 1[/sup] and 2[sup]n[/sup] - 1, the superscript represents exponentiation. For 10[sup]n - 2[/sup] and 1[sup]n - 1[/sup], the superscript represents concatenation.
Just curious.
The report refers to the first six perfect numbers, starting with 6.
Why isn’t 1 considered to be a perfect number?
The sum of the factors does not include the number itself. 6=3+2+1
Ah, completely missed that… Thanks.
Also, there’s no corresponding Mersenne prime, so if 1 were a perfect number, it would have to be a special case.
Perfect numbers are not defined by using the Mersenne primes. Odd perfect numbers, if they do exist, would not have Mersenne primes associated with them either.
Moreover, it does follow the 2[sup]n-1[/sup] * 2[sup]n[/sup]-1 formula, for n=1, counting 2[sup]1[/sup]-1 (1) as a degenerate quasi-prime.
I’m curious, and surprised that no one has asked yet: what was your joke?
It should have read “Descartes had studied pluperfect numbers”.
(1) in any unital ring cannot be a prime ideal because it is the entirety of the ring. Then R/(1) is the degenerate ring, which is not an integral domain. A prime ideal of R is one such that R/I is an integral domain.
The On-Line Encyclopedia of Integer Sequences link is no longer there (it didn’t work when I tried it before), but your list of the first nine perfect numbers is missing the 6th, 8,589,869,056
Hey, no fair trying to sneak a grammar joke into a math report!
Thanks for clearing that up!
:: goes off to consider a program that helps teach English to mathematicians ::
Makes perfect sense to me.
I didn’t say it wasn’t communication. I said it wasn’t English. 
Of course it’s not English. It’s jargon. You expect mathematicians to use strictly nontechnical vocabulary when discussing carefully-defined technical terms? Hell, why don’t you tell physicists to call quarks “thingies”?
That’s not fair. The name “quark” is essentially “thingie”
But that’s an interesting issue. Can you make jargon accessible to the layperson? Dex thinks so. There are a lot of advantages to it. As Feynman used to say, if you can’t present a concept in the terms of a freshman lecture, you really don’t understand it. Sometimes, just the exercise of trying to express it in simpler terms makes the concepts more clear. Your remark, which seemed to be aimed at John’s calling 1 a “quasi-prime”, was sorta the equivalent of saying, “most mathematicians don’t consider 1 to be a prime” but that is pretty much what John is saying to begin with.
There are a number of reasons that 1 is not considered a prime, traditionally, that if 1 were considered a prime, there would be a good many statements that now say “All primes…” that would have to be emended to “All primes except 1…” Remember, a certain amount of math can be regarded as always true, whether we like it or not, but some things, like the word “prime”, are defined for the convenience of the human mathematicians who use them.
At any rate, you can regard 1 as a kinda-sorta prime, and as a kinda-sorta perfect number, too, and it is interesting that the two mesh with each other, in that 2[sup]0[/sup]*(2[sup]1[/sup]-1) = 1
I get what you’re trying to say, but I feel I should point out that there’s nothing you can prove with this definition of “prime” that you can’t prove without it. My old logic textbook has a section on definitions that shows that in general, they’re all conveniences and don’t change anything.
This may “kinda-sorta” work for primes in the ring of integers, but as I just pointed out this doesn’t generalize. The definition of primes most people know is equivalent to the real definition only under certain conditions, and is more easily stated and understood.