I need your help for these 3 exercises. Ex.1 Considering that x+y=a+b and X2+y2=a2+b2 demonstrate that xn+yn=an+bn which n is a natural number. (n is the exponent.) Ex.2 Demonstrate that the quadratic sum of 2 odd numbers is not a full quadrate number. (It means (a+b)2 Ex.3 Considering that a x b> a+b a>0 and b>0 demonstrate that a+b>4. I hope you understand me and I am sorry for my english.
I think I understand the second problem . . .
2) Demonstrate that the sum of the squares of two odd integers is not a square.
Consider the square of an even integer: .
Consider the square of an odd integer: .
Now consider the sum of the squares of two odd integers:
This is two more than a multiple of 4 . . . It cannot be a square.
IF then a + b > 4.
In other words, show that .
Your counter-example doesn't satisfy the necessary condition. (And yes, I too fiddled around for a bit thinking a counter-example might exist).
Perhaps the fact that is only true when
leads somewhere ....
For b > 1, this gives . The graph of has a minimum turning point at b = 2 (which satisfies b > 1), corresponding to y = 4 ......