Can there be a largest prime within the concept of infinity?

  • Context: Graduate 
  • Thread starter Thread starter Icebreaker
  • Start date Start date
  • Tags Tags
    even Infinity
Click For Summary
SUMMARY

The discussion centers on the concept of infinity and its relationship to prime numbers, specifically questioning whether there can be a largest prime. Participants assert that infinity is not a number but a concept, emphasizing that it cannot be classified as even or odd. The conversation references the proof of the infinitude of primes, highlighting that if there were finitely many primes, one could construct a number not divisible by any of them. Ultimately, the consensus is that infinity does not possess numerical properties and cannot be treated as a finite integer.

PREREQUISITES
  • Understanding of basic number theory concepts, particularly prime numbers.
  • Familiarity with the concept of infinity in mathematics.
  • Knowledge of Euclid's proof of the infinitude of primes.
  • Basic comprehension of mathematical terminology such as even, odd, and divisibility.
NEXT STEPS
  • Study Euclid's proof of the infinitude of primes in detail.
  • Explore the concept of infinity in set theory and its implications.
  • Research the properties of prime numbers and their distributions.
  • Investigate the philosophical implications of infinity in mathematics.
USEFUL FOR

Mathematicians, educators, students of mathematics, and anyone interested in the foundational concepts of number theory and the philosophical aspects of infinity.

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.
 
Physics news on Phys.org
I'm no math Phd, but I do know this: infinity is not a number!
 
Infinity isn't a number; it's a concept used in mathematics.
 
So, what, we can assume that there will ALWAYS be larger primes?
 
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?
 
Interesting. Is hydrogen male or female? The answer is similar.
 
Hurkyl said:
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?
 
Can you find a number not divisible by any of them?

Would the answer to this answer your version?
 
  • #10
infinity = 2 times infinity, hence infinity is even, which may seem odd. :-p
 
  • #11
It's even odder since infinity=2*infinity+1, hence it's odd as well as even. (joking :-p)

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.
 
  • #12
Hurkyl said:
Would the answer to this answer your version?

What is the answer?
 
  • #13
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.
 
  • #14
Icebreaker said:
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!
 
  • #15
Icebreaker said:
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 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?
 
  • #16
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.
 
Last edited:
  • #17
infinity is a description not a numerical value and since only numerical values can be odd or even, infinity is neither.
 
  • #18
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.
 
  • #19
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.
 
  • #20
Chronos said:
Interesting. Is hydrogen male or female? The answer is similar.

Hydrogen is kind of like "Pat What's That".

Potassium, on the other hand, is male, while Chlorine is female.
 

Attachments

  • Positive Alkali Metals.JPG
    Positive Alkali Metals.JPG
    34.9 KB · Views: 826
  • #21
Mmmm so... electronegative...

- Warren
 
  • #22
Actually my original question was badly formulated. My question was about whether there can be a largest prime. I approached the problem by asking can any arbitrarily large number can be divided by 2, and somehow got infinity entangled. It was indeed the wrong approach.
 
  • #23
great idea infinity is a prime number
 
  • #24

Similar threads

Insights Groups, The Path from a Simple Concept to Mysterious Results
  • · Replies 17 ·
Replies
17
Views
9K
MHB Prove that there are infinitely many irreducible polynomials in Z5.
  • · Replies 3 ·
Replies
3
Views
3K
Python Who can find the largest prime number with their own programmed code?
  • · Replies 28 ·
Replies
28
Views
4K
High School Is Infinity a Prime Number: The Confusing Concept of Infinity Explained
  • · Replies 19 ·
Replies
19
Views
5K
Undergrad Question about two interference patterns viewed at infinity
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Prime Number Powers of Integers and Fermat's Last Theorem
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Infinity: The Limit Concept and Cantor Transfinites
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Can This Formula Identify All Prime Numbers?
  • · Replies 2 ·
Replies
2
Views
1K
Graduate Extended Goldbach: every odd number the sum of 5 primes
  • · Replies 1 ·
Replies
1
Views
4K
Can the limit of a quotient of trig functions approach a specific value?
  • · Replies 2 ·
Replies
2
Views
1K