Perfectly capable of googling it myself. I was interested in opinions from the board.
I would think yes they is an infinite number of them. Even though prime numbers may get further apart there will always be another one. They may be ridiculously large, but always another. RPerfectly capable of googling it myself. I was interested in opinions from the board.
I would think yes they is an infinite number of them. Even thoughprimeperfect numbers may get further apart they will always be another one. They may be ridiculously large, but always another. R
In order to prove a finite number, would you have to show that perfect numbers don't occur above a certain threshold?Probably just a finite number. They are very rare, much more spread out than the primes and the primes have some seriously large gaps. For instance, the number of primes less than a billion is a lot, maybe on the order of hundreds of thousands of them (I am not going to bother to look it up) but there's only like five perfect numbers in that span. We will probably solve this problem within a generation or two.
But there is an infinite number of billions. And a billion is no closer to Infiniti than 1. So just because a number seems so impossible to even comprehend, it is still nothing compared to what infinite is.Probably just a finite number. They are very rare, much more spread out than the primes and the primes have some seriously large gaps. For instance, the number of primes less than a billion is a lot, maybe on the order of hundreds of thousands of them (I am not going to bother to look it up) but there's only like five perfect numbers in that span. We will probably solve this problem within a generation or two.
Sure. How many even primes are there? Just one. How many sets of triple primes are there? Just one set (3,5,7). How many factorials do not end in zero? Only five (0! through 4!) I have a pretty good feel for infinity and my intuition tells me that the perfect numbers are drying up as you go out to infinity, on orders of magnitude more than the way the primes thin out. And to the other poster, yes, we could show that the number of perfect numbers is finite by establishing a largest one. Not an easy task. It's been unsolved for centuries but new facts about primes are discovered now and then. One of those innovations will take down this problem in a few decades.How can their be a finite amount of something that goes on to infinity?
If numbers do not end, How can something inside of that end?
It's a billion closerBut there is an infinite number of billions. And a billion is no closer to Infiniti than 1. So just because a number seems so impossible to even comprehend, it is still nothing compared to what infinite is.
IIRC perfect numbers are related in some way to Mersenne primes, and proving that there are infinitely many of one kind would imply that there are infinitely many of the other.Probably just a finite number. They are very rare, much more spread out than the primes and the primes have some seriously large gaps. For instance, the number of primes less than a billion is a lot, maybe on the order of hundreds of thousands of them (I am not going to bother to look it up) but there's only like five perfect numbers in that span. We will probably solve this problem within a generation or two.
It depends. There is negative infinity. Numbers can also be infinitely less than 1.It's a billion closer
I didn't want to come across as arrogant but I'll expound. Bachelors and Masters in pure mathematics. Extensively studied Cantor's theory of transfinite numbers, specialized in analysis in grad school, with central ideas in Lesbesgues measure which is mostly about measuring the sizes of infinite sets. So, I feel pretty good with my grasp of the concept, yes.
Good times.I didn't want to come across as arrogant but I'll expound. Bachelors and Masters in pure mathematics. Extensively studied Cantor's theory of transfinite numbers, specialized in analysis in grad school, with central ideas in Lesbesgues measure which is mostly about measuring the sizes of infinite sets. So, I feel pretty good with my grasp of the concept, yes.
Establishing the largest perfect number is limited by our computing power, so thatSure. How many even primes are there? Just one. How many sets of triple primes are there? Just one set (3,5,7). How many factorials do not end in zero? Only five (0! through 4!) I have a pretty good feel for infinity and my intuition tells me that the perfect numbers are drying up as you go out to infinity, on orders of magnitude more than the way the primes thin out. And to the other poster, yes, we could show that the number of perfect numbers is finite by establishing a largest one. Not an easy task. It's been unsolved for centuries but new facts about primes are discovered now and then. One of those innovations will take down this problem in a few decades.
Finding successively larger perfect numbers may be a matter of computing power (for now at least). But determining whether or not there are infinitely many of them isn't, really. We'll either prove that there are or aren't. To prove that there aren't, we wouldn't necessarily need to find the largest one, just establish an upper bound for where they might be found.Establishing the largest perfect number is limited by our computing power, so that
number could keep going up as our computer power goes up. Say the "largest"
perfect number is a trillion X trillion X trillion digits long. 25 years later that gets blown
away by finding 3 more perfect numbers after that and the new "largest" perfect number
is a Quadrillion X Quadrillion X Quadrillion X Quadrillion digits long. And so on.
Not true. Computing power is practically irrelevant here. As in my example with factorial's, what's needed is an argument (proof) for why they run out. All factorials larger than 4 include factors of 2 and 5 which produce a zero at the end. Simple argument in this case. The proof that perfect numbers are finite will surely be intricate but will probably not need big computing power.Establishing the largest perfect number is limited by our computing power, so that
number could keep going up as our computer power goes up. Say the "largest"
perfect number is a trillion X trillion X trillion digits long. 25 years later that gets blown
away by finding 3 more perfect numbers after that and the new "largest" perfect number
is a Quadrillion X Quadrillion X Quadrillion X Quadrillion digits long. And so on.
I'll take your word for it. I'm not a math or numbers theory guy.Not true. Computing power is practically irrelevant here. As in my example with factorial's, what's needed is an argument (proof) for why they run out. All factorials larger than 4 include factors of 2 and 5 which produce a zero at the end. Simple argument in this case. The proof that perfect numbers are finite will surely be intricate but will probably not need big computing power.
Alrighty then. Thanks for coming out.I'll take your word for it. I'm not a math or numbers theory guy.
Ah, proof by obfuscation. My personal favorite. I will say that I'm probably in the minority among my peers. Next time I'm hanging out with my mathematician friends, I'll ask their opinion but I suppose a decent fraction of them disagree with me and predict that the perfect numbers are indeed infinite. That's what makes it a good question.My gut feeling was infinite, but all the big words @pecorino used has swayed my opinion.