# Is infinity even or odd?

by Icebreaker
Tags: infinity
 P: n/a 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.
 Mentor P: 5,190 I'm no math Phd, but I do know this: infinity is not a number!
 P: 483 Infinity isn't a number; it's a concept used in mathematics.
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## Is infinity even or odd?

So, what, we can assume that there will ALWAYS be larger primes?
 Mentor P: 5,190 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...
 PF Patron Sci Advisor Emeritus P: 16,094 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?
 PF Patron Sci Advisor P: 8,889 Interesting. Is hydrogen male or female? The answer is similar.
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 Quote by Hurkyl 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?
PF Patron
Emeritus
P: 16,094
 Can you find a number not divisible by any of them?
 HW Helper Sci Advisor P: 9,371 infinity = 2 times infinity, hence infinity is even, which may seem odd.
 HW Helper Sci Advisor P: 1,996 It's even odder since infinity=2*infinity+1, hence it's odd as well as even. (joking ) 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.
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 HW Helper Sci Advisor P: 9,395 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.
HW Helper
P: 1,996
 Quote by Icebreaker What is the answer?
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, $p_1,p_2,\ldots,p_n$. Then consider their product $M=p_{1}p_{2}\ldots p_{n}$. 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!
P: 1,603
 Quote by Icebreaker 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.
In another thread on this forum I've seen posts claiming that an integer cannot be infinite (ie only finite integers are allowed).

See http://www.physicsforums.com/showthr...006#post478006

Any thoughts on this subject anyone?

If an infinite integer is allowed, then it must also have an infinite number of digits, right?
 HW Helper Sci Advisor P: 9,395 They aren't elements of $$\mathbb{Z}$$ or $$\mathbb{N}$$ 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.
 P: n/a infinity is a description not a numerical value and since only numerical values can be odd or even, infinity is neither.
 PF Patron Sci Advisor Thanks Emeritus P: 38,416 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.

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