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.
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.
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.
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.
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
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
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?