Recent content by randommacuser

Beautiful. Thanks for your help.

Homework Statement Let E be the exchange matrix (ones on the anti-diagonal, zeroes elsewhere). Suppose A is symmetric and positive definite. Show that B = EAE is positive definite. Homework Equations The Attempt at a Solution I've tried showing directly that for any conformable vector h...

Perfect. Thanks for your help, Data.

moe_3_moe: Yes, I did this by hand to about f(5) and on Mathematica to f(10). But the goal is one expression of f(x), for all x, in terms of the four given parameters.

Well, that was the way the problem was stated. I've edited the OP to make it more standard. Now f is a function of the variable x, as usual.

I am trying to find a formula for f(x) = a*f(x-1) + b*f(x-2) for x≥1, assuming f(-1) and f(0) are known. In other words, I need some expression for f(x) just in terms of a, b, f(-1), and f(0). I tried plugging in stuff and factoring on paper for the first few iterations, but it quickly got out...

Got them all, I think. Thanks everyone!

I like this last suggestion for #1. I can do all that and I see where it is headed, but at the end I have expressions for phi(n) and phi(mn) that depend on different primes (or at least I can't prove they are the same). How do I use m|n to prove this? As I suspected, #2 is a lot easier than...

Hey all, I've got a few problems that are tripping me up tonight. 1. Let m,n be positive integers with m|n. Prove phi(mn)=m*phi(n). I know I can write n as a multiple of m, and m as a product of primes, and my best guess so far is that I can work with some basic properties of or formulas...

Well, it's crude, but I think I have #3 as well. Thanks for all the hints, guys. If anyone is interested in how I proved any of these questions, just ask and I will try to explain as best I can.

Yeah, I just had another look at #1 and it's not that difficult. Silly me... And once I saw the reasoning #2 followed fairly easily, though I haven't quite figured out the notation. So if anyone has suggestions for #3, I'd appreciate it!

Sure, that should be the easier case. How about the other way around?

I'm attempting a proof by contradiction on #1, along the lines of what Tide is hinting at. I just don't know how to show it formally. And I know sqrt(10) is not rational, I'm just not sure how to use that yet.

Hey all, I've got a few number theory exercises that are troubling me. 1. Prove a positive integer s is a square if and only if each of the exponents in its prime factorization is even. 2. Let c,d be positive, relatively prime integers. Prove that if cd is a square, c and d are squares. 3...

I have the second one now. In the first one, the operator is greatest common divisor.