Homework Statement
Prove that there are infinitely many primes using Mersenne Primes, or show that it cannot be proven with Mersenne Primes.
Homework Equations
A Mersenne prime has the form: M = 2k - 1
The Attempt at a Solution
Lemma: If k is a prime, then M = 2k - 1 is a prime.
Proof of...
Homework Statement
So basically for n ∈ {1, ... , 16}
Find the lowest t to satisfy nt ≡ 1 (mod 17)
Homework Equations
Euler's Theorem tells us that the order, t, must be a divisor of φ(17), which is Euler's Phi Function.
φ(17) = 16
t ∈ {1, 2, 4, 8, 16}
The Attempt at a Solution
n = 1
11 ≡ 1...
Not quite exactly. Here's a picture of my notes since I don't know how to type it here with proper notation.
Link: https://dl.pushbulletusercontent.com/wwTIGu5Cjgt9OlhFWP6geOc1nuuUgJJX/20151024_164749.jpg
Homework Statement
f(x,y) =
(xy) / (x2 + y4), when (x, y) ≠ (0,0)
0, when (x,y) = (0,0)
Homework Equations
Explicitly show that f(x,y) does not satisfy
lim h -> 0 [ E(v,h) / ||h|| ] = 0 when v = 0
(h, v, and 0 are all vectors; I'm not sure how to put a hat on them)
The Attempt at a Solution
I...
Oops, I haven't been on for a while and am readjusting to the forums new style, my bad! I'll make sure to post in the correct sections next time! Thanks.
The problem:
Relevant Equations:
A = l * w
So I drew a rectangle like this:
And I'm not really sure how to proceed from this point.. I always have a hard time setting up the problems for optimization :/
Any tips to the right direction would be greatly appreciated! Just to clarify, this is...
Wow, thankyou again HakImPhilo, that diagram really helped!
Am I still doing problem 1 incorrectly? I seem to get different points of intersection when I graph
sin^2x + sinx = 0
compared to sinx + 1 = cosx
sinx + 1 = cosx
(sinx + 1)^2 = 1 - sin^2x
sin^2x + 2sinx + 1 = 1 - sin^2x...
Mmm, so just to double check my answers...
Question 1:
sinx + 1 = cosx
(sinx + 1)^2 = 1 - sin^2x
sin^2x + 2sinx + 1 = 1 - sin^2x
2sin^2x + 2sinx = 0
sin^2x + sinx = 0
x = 0, pi, 2pi, 3pi, 4pi
(And for the record, how would I input sin^2x into my calculator anyways? I think it's...
Thank you Sammy for clearing that up! I thought it had to satisfy both equations, since I was finding the point of intersection, but I guess it only needs to satisfy one or the other.
And thanks again HakImPhilo for the in depth explanation, it really helped clear my understanding.
Mmm, I think I'm still a little slow on the uptake. I still don't really understand why [π, 3π] work, because the cos values of [π, 3π] are both -1, and if cosx = 1, then that doesn't make the equation true, since -1 doesn't equal 1. I think I'm really missing some fundamental basic idea here...
Thanks a lot HakimPhilo, you really cleared it up for me, but I still don't understand question 2 though.
I thought that the only time cosx = 1 was when x = 1 (Likewise for sinx = 0 being y = 1).
So I have several problems that ask me to find all points of intersection algebraically, but I haven't been able to make much headway on most of them.
The first problem
Homework Statement
Find all the points of intersection algebraically of the graphs of ... on the interval [0, 4π]...