Difference between any two odd numbers is even

In summary: Yes, I agree that the difference between any two odd numbers is even, as well as the difference between two even numbers being even. This may seem trivially obvious, but it has been formally proven using algebraic equations and the concept of even and odd numbers. This proof is also related to the famous Goldbach Conjecture, which states that every even number greater than 4 can be expressed as the sum of two odd prime numbers. While the sum of any two odd numbers is always even, the proof for this conjecture has not yet been formally proven. However, it is considered a trivially obvious statement by many mathematicians and can be easily seen through various mathematical proofs and examples.
  • #1
Brad_Ad23
502
1
Is there a law or theorem somewhere that states the difference between any two odd numbers is even? Or the difference between 2 even numbers is even?
 
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  • #2
It's trivially obvious. Why do you ask?
 
  • #3
Well its also trivially obvious that the sum of any two odd primes yields an even integer over 4, yet that has not been proven yet. I just wanted to make sure that these were because someone keeps telling me they aren't. Luckily in the meanwhile I came up with my own proof for them, so it is all good.
 
  • #4
Let m= 2n+1
Let r= 2s+1

r and m are arbitrary odd numbers(1 greated then an even number)

so
r+m = (2n +1)+ (2s+1)= 2n+2s+2 = 2(n+s+1)

r+m is even.
QED
You can do something similar for 2 even numbers.
 
  • #5
Yes that is the proofs I came up with...different notation but same message.
 
  • #6
Well its also trivially obvious that the sum of any two odd primes yields an even integer over 4, yet that has not been proven yet.

What's not trivially obvious is what you MEAN by "an even integer over 4". If you mean, literally, "an even integer divided by 4" then it doesn't make sense because the sum of two odd primes certainly has to be an INTEGER, not a half. If you mean "divisible by 4" then it's not true: 3+ 7= 10.
The fact that the sum of two odd PRIMES is even is "trivially obvious" because the sum of two odd numbers is always even. That certainly has been proven, in fact, I've seen it given as an exercise in high-school algebra texts.
 
  • #7
I forgot that mathematicians need to be exact here.

Every even integer greater than 4 is the sum of two odd primes.

It is indeed obvious the sum of any two odd primes is even (after all, they are just special odd numbers), but that seems to be obvious as well at first glance, but not proven. It is, after all, the Goldbach Conjecture.
 
  • #8
It is indeed obvious the sum of any two odd primes is even
Actually, the sum of ANY two odd numbers is an even number (wether they are primes or not).
Look at integral's proof.
Is there a law or theorem somewhere that states the difference between any two odd numbers is even?
You can adjust Integral's proof to proove this.
Let A be an odd number, B another odd number
A = 2k + 1
B = 2n + 1
(where both k and n are integers)
A - B = (2k + 1) - (2n + 1) = 2k-2n + (1-1) = 2(k+n)
Since k and n are integers, k+n is an integer too, and A-B is even (since it can be expressed as 2*integer).
 
  • #9
Merci
 
  • #10
Originally posted by Brad_Ad23
Merci

LOL

"It is, after all, the Goldbach Conjecture," is a great line.
I suspect that you set up this thread with that line
in mind from the start
in order to have an opportunity to deliver it.

Perhaps I'm easily amused today but find it difficult
to stop chuckling at this thread.
 
  • #11
Hehe, I wish I was ingenious enough to have that planned from the start. The actual purpose was to make sure my arguments were correct. Of course the sum of any two odd primes will be even since the sum of any two odds is even. As I then explained to my friend, the other way...all even numbers are the sum of two primes, is much less obvious and so far not proven.
 
  • #12


Originally posted by Brad_Ad23
Is there a law or theorem somewhere that states the difference between any two odd numbers is even? Or the difference between 2 even numbers is even?

The difference between the number êÄ and the number Ç{ is DEFINITELY ODD (don't you think?)
 

1. What does it mean when the difference between two odd numbers is even?

When the difference between two odd numbers is even, it means that the result of subtracting one odd number from the other is an even number. For example, the difference between 7 and 3 is 4, which is an even number.

2. How can you prove that the difference between two odd numbers is always even?

To prove that the difference between two odd numbers is always even, we can use the fact that an odd number can be represented as 2n+1, where n is any integer. So, if we subtract one odd number (2n+1) from another (2m+1), we get (2m+1) - (2n+1) = 2m-2n = 2(m-n), which is always an even number.

3. Are there any exceptions to the rule that the difference between two odd numbers is even?

No, there are no exceptions to this rule. As mentioned earlier, the difference between two odd numbers will always result in an even number. This is because odd numbers are always separated by an even number.

4. Can the difference between two odd numbers ever be odd?

No, the difference between two odd numbers can never be odd. This is because two odd numbers will always be separated by an even number, which results in an even difference. If the difference were to be odd, it would mean that the two odd numbers are not actually odd.

5. How is the concept of the difference between two odd numbers being even useful in mathematics or science?

The concept of the difference between two odd numbers being even is useful in many areas of mathematics and science. For example, it can be used to simplify and solve equations, as well as in number theory and probability. It also helps in understanding patterns and relationships between numbers.

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