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Difference between any two odd numbers is even 
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#1
Jun503, 05:54 PM

P: 499

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?



#2
Jun503, 06:32 PM

Sci Advisor
P: 6,080

It's trivially obvious. Why do you ask?



#3
Jun503, 06:49 PM

P: 499

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
Jun503, 09:42 PM

Mentor
P: 7,321

Difference between any two odd numbers is even
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
Jun503, 09:50 PM

P: 499

Yes that is the proofs I came up with...different notation but same message.



#6
Jun703, 07:36 AM

Math
Emeritus
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PF Gold
P: 39,568

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 highschool algebra texts. 


#7
Jun703, 08:10 AM

P: 499

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
Jun1403, 02:46 PM

P: 353

Look at integral's proof. 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) = 2k2n + (11) = 2(k+n) Since k and n are integers, k+n is an integer too, and AB is even (since it can be expressed as 2*integer). 


#9
Jun1403, 03:33 PM

P: 499

Merci



#10
Jun1403, 04:30 PM

Astronomy
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PF Gold
P: 23,235

"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
Jun1403, 06:01 PM

P: 499

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
Jun2103, 11:02 PM

P: 155




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