Recent content by Heute

Ah! Figured it out! The breakthrough was realizing z = -(p-z) (mod p) for all z and the negatives canceling. Whew! Crazy stuff. Thanks all for contributing your time to help random people like me out. I'm sure it's a thankless job a lot of the time, but you all have helped me a lot in the...

In class we haven't touched on how modulus works with rational numbers. Although I know (p-1)/2 is an integer, I want to use properties of modulus to say that -1 \equiv (p-1)/2 (mod p) - but I don't think I can do that "legally" unless I can say 1/2 \equiv 1/2 (mod p). I wrote out a few...

Homework Statement Prove that if p is prime and p \equiv 1 (mod 4), then x^{2} \equiv -1 (mod p) has a solution (x). Homework Equations We already have proved (p-1)! \equiv -1 (mod p) Hint: Use the properties of Z_{p} - a field that partitions the integers into p equivalence classes...

Apparently, I can't edit the original post, so I'll re-write it here: The Attempt at a Solution let X = (x_{1},x_{2},x_{3}) We need X*Y = cos(\pi/4) |x| |y| and X*Z = cos(\pi/3)|x||z| We want the length of X, that is |x|, to be 1 so I'll assume that it is 1 for now (this could be...

Apologies! That sqrt(2) should be a 1 (the length of Z). I copied that sqrt(2) from the memory of another similar problem I was working on. I'll edit the original post! This doesn't, however fundamentally change things. The question still remains.

Homework Statement Find all the unit vectors X element of R^3 that make an angle of pi/4 radians with vector Y = (1,0,1) and an angle of pi/3 radians with vector Z = (0,1,0) Homework Equations For any two vectors X and Y element of R^n, the dot-prodict of X and Y is equals to the...

I thought of that - but I felt like there was still a logical leap from let N = max(N1, N2) to n > N implies (what we're looking for) Maybe I'm trying to be too pedantic.

Homework Statement Given that limit of s_{2n} is L and limit of s_{2n+1} is L, prove that lim s_{n} is also L. Homework EquationsThe Attempt at a Solution This seems very obvious: If the even terms of a sequence approach a number and the odd terms of that sequence approach the same number...

Ah! It's so obvious now! Thanks.

Homework Statement find the limit of \sqrt{x^2+x}-\sqrt{x^2-x} as x approaches infinity Homework Equations The Attempt at a Solution Multiplying the original expression by \frac{sqrt(x^2+x)+sqrt(x^2-x)}{sqrt(x^2+x)+sqrt(x^2-x)} I get the following...

Ah! I think I've got it. Proof: Assume a_{n} is convergent and b_{n} is divergent. Now suppose that a_{n}+b_{n} is convergent. Then [for every ε > 0 there exists a natural number N_{1} so that n > N_{1} implies |a_{n}+b_{n} - L|< ε/2 We know by our assumption that there also...

Homework Statement Prove if sequence a_{n} converges and sequence b_{n} diverges, then the sequence a_{n}+b_{n} also diverges. Homework Equations The Attempt at a Solution My professor recommended a proof by contradiction. That is, suppose a_{n}+b_{n} does converge. Then, for...

I think I follow you. Suppose 3 does not divide a. Then a can be written as 3n-1 or 3n-2 for some natural number n. Case 1: a = 3n-1. Then a^2 = 9n^2-6n+1 which is not divisible by 3 since (3n^2-2n+1/3) is the sum of two natural numbers and a fraction, which is not a natural number. Case...

Homework Statement Assume a is a natural number and that a^2 is divisible by 3 (that is, there exists natural number n so that 3n = a^2) Homework Equations The Attempt at a Solution I thought about doing this one by contradiction. Suppose a is not divisible by 3. Then a/3 can be...

Ok, that helps a lot. Let me see if I can pull all of this together into a proof now. Given some \epsilon > 0, let N be any natural number greater than |bc-ad|/c^(2)\epsilon - (d/c) so that N is also greater than d/c Now assume that n > N. Then, n > |bc-ad|/c^(2)\epsilon-(d/c) n +...