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Suppose G is a 2x2 matrix with real entries, such that det(G)=+/- 1 and G^2 = - I (I = id 2x2 id matrix) What can G be? I came with matrices of the following form. Zeros on the main diagonal and b and -1/b on the off diagonal where b is any non-zero real number. I was wondering if you can show...

EDIT: I figured it out! I had the wrong roots for x^3-2. They are 21/3, w21/3 and w221/3, where w= e2pi*i/3 I think that I am doing something wrong and am hoping that someone can point it out to me. Find the Galois group for x^3-2 Step 1: the roots are 21/3, w and w2, where w= e2pi*i/3 Step...

Does any one understand a reason why W/A shows the graph of tan θ = 5 as a circle of radius 5 with center at (0,0)? I would think that it is just the line y=5x.

I read a proof from Copilot that basically used the following to show that a mapping was 1-1. Suppose there are two sets A and B with an equivalence relation, ~, on A. There is a homomorphic mapping T from A to B. In order to show that T is one-to-one the proof basically said suppose that T(x)...

I was given a matrix A. I found the Jordan matrix for A Let x = lamda. The characters equation for |A-xI| = (x-1)3 So I tried to find the null space for (A-xI)3. However, that matrix was the zero matrix. So I am at a loss as how to find P as I only have two Linear Independent vectors for the...

I am not sure what it means by ZxZxZ are represented by the column vectors. I know which column vectors that is be referred to. I also have no idea what they mean by Z^3/MZ^3. I can't show any work since I don't know the meaning of what is given. As always, any help would be appreciated. Steven

Suppose n=72*3 = 147 Clearly 72|n 73\nmidn n\neq73m n\neq727m n\neq72c 72\nmidn But 72|n I see where the error is. The problem is that this video I was watching basically says just that (in my opinion) The link to the video is here . Please watch from 39:30 for just a few seconds.

Happy New Year!

Show that 1! + 2! + 3! +...+ n! is a perfect square only for n=1 and n=3.

Below is a proof in my Number Theory book. I have a problem accepting that the summation in the statement of the theorem equals f(n) as sometimes that summation equals 0. Shouldn't it be necessary that f(n) = 0 as well when the summation is 0? There is no reason (that I see) why f(n)=0 when the...

What is the connection between alpha and theta? I tried drawing all kinds of lines but need a little help here. I refuse to show my work so please don't ask!!

Show that if p and q are distinct primes, then p^(q-1) + q^(p-1) = 1 (mod pq)

Prove that 1^5+2^5+...+100^5=0 (mod 4)

Show that x^2+1=0 (mod p), where p is prime, has a solution iff p=1 (mod 4)

Prove that if p>q>5, where p and q are both prime, then 24| p2-q2

Show that there are no arithmetic progressions, a, a+1b, a+2b, ...,that contain solely prime numbers.

Show that given any n>0 we can find n consecutive composite integers.

As a student I chose not to study number theory as I though that it would be boring compared to say algebra classes. Having been studying number theory for a while now I can say that this subject is not exciting at all! However, I am still enjoying studying this subject as I am learning about...

Please show that p, p+2 and p+4 can't all be primes. It's trivial, if you have the correct ammunition. I'd like to see how one would prove this without this special ammunition. Remember, have fun with this.

Claim: sqrt(2) is irrational Proof: Suppose otherwise. That is assume sqrt(2) = a/b where gcd(a,b)=1 Then a^2=2b^2=(2b)b =>b|a^2 If b>1, then the Fundamental Theorem of Arithmetic guarantees that there exists a prime p such that p|b. Since p|b and b|a^2 => p|a^2. That is p|a*a. Well p has to...