Discrete Mathematics questions - need help now?

Question

I'm having horrible trouble with these questions.

1. Prove that if n is a perfect square, then n+2 is not a perfect square.
2. Prove that if n is a positive integer, then n is even if and only if 7n+4 is even.
3. Prove that either 2x10^500 + 15 or 2x10^500 + 16 is not a perfect square.
4. Prove or disprove that if a and b are rational numbers, then a^b is also a rational number.
5. Find a counterexample to the statement that every positive integer can be written as the sum of squares of three integers.

Really, it's been YEARS since I've had a math course before, and I really haven't the slightest idea of even where to start any of these... I'm not asking for a full proof for everyone, but just where I should start and the direction I should go from there...

Answer 1

1) a^2 - b^2 = 2, so (a+b)(a-b) = 2, so a+b = 2 and a-b=1, but then a and b are not both integers.
2) If n is even 7n is even and so is 7n+4. If n is odd, 7n is odd and so is 7n+4.
3) 2x10^500 + 15 is not a perfect square since it has a remainder of 3 when divided by 4
4) Disprove. Let a be 2 and b be 1/2. then a^b = sqrt(2)
5) 7

Answer 2

4 is false: 2^½ is irrational but 2 and 1/2 are rational.
5. 7 cannot be written as the sum of 3 squares.
In fact, no number of the form 4^k(8k+7) can be written
as the sum of 3 squares.
Interestingly enough, all other positive integers can be so written!
This is a deep theorem of Gauss.
1. Let's assume n is an integer
Let n = y². Then y²+2 = x².
Let's also assume x and y are positive integers.
So
x²-y² = 2.
Factor
(x+y)(x-y) = 2.
Since x+y and x-y are integers and their
product is 2, x+y must be 2 and x-y must be 1
since x+y > x -y.
So
x + y = 2
x - y = 1
Which gives x = 3/2, which is not an integer.
BTW, this equation does have a solution in
rational numbers: 1/4 + 2 = 9/4.

2. If n is even, 7n is even and so is 7n+4.
If 7n+4 is even, say it equals 2m
then 7n = 2m-4
So 2 divides 7n. Since 7 is odd, 2 divides n,
so n is even.

3. I'll prove that n = 2*10^500 + 15 is not a square.
Note that 2^10^500 is divisible by 4.
So n can be written 4k + 12 + 3 = 4(k+3) + 3 = 4m + 3.
Claim: Any odd square is always of the form 4m+1,
Once this is proved, it follows that n is not a square.
Proof of claim:
Any odd number is one of the 2 forms
4n + 1 or 4n + 3.
Now square each of these:
(4n+1)²= 16n² + 8n + 1 = 4(4n²+2n) + 1 = 4m + 1.
and
(4n+3)² = 16n² + 24n + 8 + 1 = 4m + 1,
which proves the claim.

Answer 3

each of these diserves its own question on this website. none of them have easy solutions or proofs (except # 2 and 3). this represents an entire week's problem ~2 hours of work. post them individually and I'll answer them all eventually.

or take this hint:
when adding a small n (like 2 in problem 1) to x
if f>n divides evenly into x, it will not divide evenly into (x + n)

Answer 4

See the site http://zimmer.csufresno.edu/~larryc/proofs/proofs.contrapositive.html