Proving a theory

[squarebear]

I had some spare time yesterday and was thinking about maths and I realised that any odd number can be made from two consecutive numbers. eg:

3+4=7
7+8=15
50+51=101

and so on. I'm sure this has already been discovered before but my question is, how would you prove my "Squarebear theorem" for any odd number, as it was a long time since I did maths at school.

[jake-the-peg]

Your theory needs an exception

0+1=1 odd

[jake-the-peg]

Sorry scratch that not had enough coffee yet

[squarebear]

But 1 can be made from zero and one, which are consecutive.

[squarebear]

No problem Jake. It is still a bit early :-)

[ChuckFickens]

as any two sequential numbers will always be one odd and one even number Surely it's just the old rules.

Even + Even = Even

Even + Odd = Odd

Odd + odd + even

[jake-the-peg]

If a number x is even x+x+1 must be odd as 2x is even

If a number x is odd x+x+1 must be odd
Consider y where y=x-1 y must be even id x is odd

so we have y+1+y+1+1 or 2y+2+1

As x must be odd or even the sum of two consecetive numbers is odd

[squarebear]

True Chuck but an odd and even number are not necessarily consecutive.

Jake's reply is the sort of thing I was looking for. I just need to get my head round it now :-)

Thanks for your answers guys. It's amazing what the mind thinks up when bored.

[razza]

Zzzzzzzzzzzzzzzzzzzzzzz

[Prudie]

A slightly simpler answer would be that two consecutive whole numbers can be expressed as x and x+1
Adding together gives x + x + 1 = 2x +1
As 2x must be even (integer divisible by 2) the 1 left over makes it odd.

[David90]

What you're really saying is 3+3+1 = 7, 7+7+1 = 15 etc or if you like 2n+1 must be an odd number because 1 is an odd number. In fact 2n+m must be odd if m is odd and even if m is even.

[squarebear]

Pefect. I understand it now. Thanks for your help.

[bibblebub]

your theory is more like an axiom

[squarebear]

I don't know the correct terminology. As I say, it's a long while since I was at school.

[bibblebub]

Frankly, it's fairly self evident which is why it's strikes me as being axiomatic in all of maths other than number theory (sorry to be brutal).

[squarebear]

It may be self evident but I doubt say Pythagoras would have got away with his idea by saying, "It's obvious. Everyone knows that". I was just wondering how you would prove something like that and got my answer.

This is where the "close question" button would be handy.

[bibblebub]

Sorry to have upset you.

Euclid's Elements book 7 contains the definitive answer to your original question.

[squarebear]

No problem bibblebub :-)

[David90]

Yesterday I said 2n+m must be even if m is even and odd if m is odd. Obviously this is only true when you are dealing with whole numbers and does not apply if (say) n = 1.5

[chakka35]

If you want something to prove that needs proving how about tackling the proposition that every even number can be expressed as the sum of two primes - and sometimes more.

E.g. 8=5+3 40=37+3 124=113+11 36=23+13 or 36=23+11+2 or 36=19+11+3+3

No-one has done it yet.

[factor-fiction]

Well clearly every even number can be written as the sum of primes since 2 is a prime number. So 12= 2+2+2+2+2+2, etc, 8 = 2+2+2+2 etc.
Or did you mean the sum of two or more prime numbers, 2 or more of which are different?