Is infinity even or odd? If it's even (or both), then it would mean there's a finitely largest prime. By coarsely applying limit concepts, and lim(x->inf)x/2 does not yield a remainder.
You shouldn't assume anything in math. I seem to remember seeing a proof of that somewhere, but I can't remember any of the details. My memory could be wrong. Surely somebody will know though...
Suppose there are finitely many primes. Can you find a number divisible by all of them? Can you find a number not divisible by any of them?
Wouldn't the question be: suppose there are finitely many primes, can a combination of their products be used to construct any integer?
It's even odder since infinity=2*infinity+1, hence it's odd as well as even. (joking :tongue2:) Seriously though, infinity is not a number that you can perform arithmetic with. You can add it to the reals and make what's often called the extended reals, but don't expect it to play nice with the rest of the numbers. Certainly don't expect it to have any nice properties like even or odd.
Not every number can be formed from a product if primes if there were only a finite number of them: multipply them all together and add 1.
The answer is yes. If you can find a number not divisible by all the primes, then you would have found a number that is not a combination of their products. Suppose their are finitely many primes, [itex]p_1,p_2,\ldots,p_n[/itex]. Then consider their product [itex]M=p_{1}p_{2}\ldots p_{n}[/itex]. Then M is divisible by all the primes. This answers Hurkyl's first question (from post#6). Can you use it to answer the second? edit:I type slower than matt!
In another thread on this forum I've seen posts claiming that an integer cannot be infinite (ie only finite integers are allowed). See https://www.physicsforums.com/showthread.php?p=478006#post478006 Any thoughts on this subject anyone? If an infinite integer is allowed, then it must also have an infinite number of digits, right?
They aren't elements of [tex]\mathbb{Z}[/tex] or [tex]\mathbb{N}[/tex] by definition so your argument is baseless. You are saying: if we assume that when cantor said the natural numbers he actually meant something entirely different then his argument is wrong. As he didn't mean something entirely different his argument is correct and you are wrong.
infinity is a description not a numerical value and since only numerical values can be odd or even, infinity is neither.
Is green even or odd? Is salt even or odd? Is Fred even or odd? (Oops, bad example, Fred is definitely odd!) The point is that "infinity" is not a member of the set of integers and that is all the terms "even" or "odd" (in the mathematical sense) apply to.
If infinity could be reguarded as a number, I think it would e both. It would also be the only number bigger, smaller and the same size as itself at the same time. This is because infinity is not a number. If we were to fill infinity with the highest known number, it would still fall short. Therefore infinity would have to e flled with every impossible number (including decimals). That is the only way it would come close to becoming a "proper" number.
Hydrogen is kind of like "Pat What's That". Potassium, on the other hand, is male, while Chlorine is female.