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1. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

But how should I verify that ## N\equiv 1\pmod {6} ##? And I think I made some mistakes in my previous proof attempts, because ## p\mid N ## and ## p\mid (2p_{1}\dotsb p_{n})^{2} ## implies that ## p\mid (N-(2p_{1}\dotsb p_{n})^{2}) ##, so ## p\mid 3 ##.
2. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

I apologize for the confusion. What I meant is the symbol for expressing Legendre symbol, not divisibility.
3. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

Because ## 3\nmid 2p_{1}p_{2}\dotsb p_{n} ##. And ## p\mid N, p\mid 3 ## implies ## p\mid (N-3) ##, so ## p\mid (2p_{1}p_{2}\dotsb p_{n}) ##. Also, how should I show that ## N ## has an odd prime divisor of the form ## 6k+1 ##? At first, I thought this is so because ## N ## itself is odd. But it...
4. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

Okay, so I revised this proof: Suppose for the sake of contradiction that the only primes of the form ## 6k+1 ## are ## p_{1}, p_{2}, ..., p_{n} ##. Consider the integer ## N=4p_{1}^{2}p_{2}^{2}\dotsb p_{n}^{2}+3=(2p_{1}p_{2}\dotsb p_{n})^{2}+3 ##. Since ## N ## is odd, it follows that ## N ##...
5. ### Prove that there are infinitely many primes of the form ## 6k+1 ##?

Proof: Suppose that the only prime numbers of the form ## 6k+1 ## are ## p_{1}, p_{2}, ..., p_{n} ##, and let ## N=4p_{1}^{2}p_{2}^{2}\dotsb p_{n}^{2}+3 ##. Since ## N ## is odd, ## N ## is divisible by some prime ## p ##, so ## 4p_{1}^{2}\dotsb p_{n}^{2}\equiv -3\pmod {p} ##. That is, ##...
6. ### I Is this the correct way to quantify these integers?

Thank you for the clarification, @andrewkirk @topsquark .
7. ### I Is this the correct way to quantify these integers?

Does the above quantifier represent/symbolize that all of the integers ## a, b, c, d ## cannot be ## 0 ##? Is this correct?
8. ### How to construct a table of all the real-valued Dirichlet characters?

So now we have that ## \chi(k)^{2}=\chi(k)\cdot \chi(k)=\chi(k\cdot k)=\chi(k^{2})=\chi(1)=1 ## for all ## k\{8, 13, 20\} ## and ## \chi(k)^{6}=\chi(k^{6})=\chi(1)=1 ## for all ## k\{2, 4, 5, 10, 11, 16, 17, 19\} ##. And this implies that ## \chi(n)=8, 13, 20 ## can either be ## -1, 1 ##. But...
9. ### How to construct a table of all the real-valued Dirichlet characters?

## 1, 8, 13, 20 ## ## 1^{2}\equiv 1\pmod {21}, 8^{2}\equiv 1\pmod {21}, 13^{2}\equiv 1\pmod {21}, 20^{2}\equiv 1\pmod {21} ##
10. ### How to construct a table of all the real-valued Dirichlet characters?

Since ## \varphi(21)=\varphi(3)\varphi(7)=2\cdot 6=12 ##, there are ## 12 ## elements such that ## G=\{1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20\} ##. So ## G ## can be generated by order ## 2 ## or ## 6 ##. And we have ## \chi(k)^{2}=\chi(k)\cdot \chi(k)=\chi(k\cdot k)=\chi(k^{2})=\chi(1)=1 ##...
11. ### How to construct a table of all the real-valued Dirichlet characters?

No. I do not know the structure of finite abelian groups. I do not know the characters of cyclic groups are. You said that my group of ## G ## is isomorphic to ## C_{6}\times C_{2} ##, which are the two cyclic groups of order ## 6 ## and ## 2 ##. But how did you get these?
12. ### How to construct a table of all the real-valued Dirichlet characters?

I already did. But I still don't understand. How should I find normal subgroups?
13. ### How to construct a table of all the real-valued Dirichlet characters?

This is my question, too. I do not know what they mean, I just posted them under the relevant equation(s) just because my book has these definitions. Since these definitions are preventing people to make sense of my question, then please ignore them. How should I find those values then, starting...
14. ### How to construct a table of all the real-valued Dirichlet characters?

\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 4 & 5 & 8 & 10 & 11 & 13 & 16 & 17 & 19 & 20 \\ \hline \chi_{1}(n) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline \chi_{2}(n) & 1 & -1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & -1 & 1 \\ \hline \chi_{3}(n) & 1 & 1 & 1 & -1 & 1...
15. ### A How should I write an account of prime numbers?

I wish there are more details in this question, but no. The question states: "Write an account of prime numbers in arithmetic progressions. Your account should be in the form of an essay of 500-1000 words."
16. ### A How should I write an account of prime numbers?

I am not completely sure of whether or not this will work for my purpose.
17. ### A How should I write an account of prime numbers?

How should I write an account of prime numbers in arithmetic progressions? Assuming this account should be in the form of an essay of at least ## 500 ## words. Should I apply the formula ## a_{n}=3+4n ## for ## 0\leq n\leq 2 ##? Can anyone please provide any idea(s)?
18. ### How to find the Gateaux differential of this functional?

After breaking down into smaller pieces, I got the following: \begin{align*} &(y'(x)+\tau\psi'(x))^{2}=y'^{2}(x)+2y'(x)\tau\psi'(x)+\tau^{2}\psi'^{2}(x)\\ &\omega^{2}(y(x)+\tau\psi(x))^{2}=\omega^{2}y^{2}(x)+2\tau\psi(x)\omega^{2}+\omega^{2}\tau^{2}\psi^{2}(x)\\...
19. ### How to find the Gateaux differential of this functional?

Do you mean ## S(y+\tau\psi)=\int_{a}^{b}\frac{d}{d\tau}(y'^{2}+\omega^{2}y^{2}+2yx^{4})dx ##? If so, then how to evaluate this?
20. ### How to find the Gateaux differential of this functional?

I am not sure if this is correct, but here is my work by using the definition of the Gateaux differential: \begin{align*} &dS(y; \psi)=\lim_{\tau\rightarrow 0}\frac{S(y+\tau\psi)-S(y)}{\tau}=\frac{d}{d\tau}S(y+\tau\psi)\biggr\rvert_{\tau=0}\\...
21. ### Prove that ## g(x)=f(x)\log {x}-\int_{2}^{x}t^{-1}f(t)dt ##.

So both ## \pi(x)\approx \frac{x}{\log {x}} ## and ## li(x)\approx \frac{x}{\log {x}} ##? How did you get ## (1+O(\log^{-1} (x))) ##?
22. ### Prove that ## g(x)=f(x)\log {x}-\int_{2}^{x}t^{-1}f(t)dt ##.

Thank you for pointing that out on part a). Also, another part of this question asks to prove that ## g(x)\sim\frac{1}{4}x ## by assuming that ## f(x)\sim\frac{1}{4}\pi(x) ##. By definitions, both ## \pi(x)=\sum_{\substack{prime p\leq x}}1 ## and ## v(x)=\sum_{\substack{prime p\leq x}}\log {p}...
23. ### Prove that ## g(x)=f(x)\log {x}-\int_{2}^{x}t^{-1}f(t)dt ##.

How did you get ## g(40)\approx 6.8 ##? I thought it's ## g(40)\approx 2.95 ##.
24. ### Prove that ## g(x)=f(x)\log {x}-\int_{2}^{x}t^{-1}f(t)dt ##.

a) ## f(40)=\sum_{\substack{prime p\leq x \\ p\equiv 3\pmod {10}}}1=3+13+23=39 ## ## g(40)=\sum_{\substack{prime p\leq x \\ p\equiv 3\pmod {10}}}\log {p}=\log {3}+\log {13}+\log {23}=\log {897} ## b) Proof: Let ## f(n)=\log {n} ## and ## a_{n}=1 ## if ## n\leq x ## is prime such that ##...
25. ### How to prove this using Abel's summation formula?

Just to confirm from where you left off, \begin{align*} &\frac{1}{\varphi(k)}+O(\frac{1}{\log {x}}+\int_{2}^{x}(\frac{1}{t\log {t}}\cdot \frac{1}{\varphi(k)}+\frac{R(t)}{t\log^2 {t}})dt\\ &=\frac{1}{\varphi(k)}+O(\frac{1}{\log {x}}+\frac{1}{\varphi(k)}\int_{2}^{x}\frac{dt}{t\log...
26. ### How to prove this using Abel's summation formula?

The following proof below is from the book: Let ## f(x)=\frac{1}{\log {x}} ## and ## a(n)=\frac{\log {n}}{n} ## if ## n\equiv h\pmod {k} ## is prime and ## 0 ## otherwise. By Dirichlet's Theorem, we have ## \sum_{n\leq x}a(n)=\frac{1}{\varphi(k)}\log {x}+R(x) ##, where ## R(x)=O(1) ##...
27. ### How to prove this using Abel's summation formula?

Before I try to prove the theorem, I want to know, how did you get those numbers from ## 3 ## to ## 1553 ## from the sequence?
28. ### How to prove this using Abel's summation formula?

From the textbook and notes. But if ##f(t)## can be any on ##[x, y]##, then what would you choose? How to choose this ##f(t)## function wisely?
29. ### How to prove this using Abel's summation formula?

Before I apply/use the Abel's summation formula, how should I find ## f(x) ##?
30. ### Proof about two disjoint non-empty sets ## S ## and ## T ##

Don't say sorry, it's okay. I am the one who should apologize, because many stuffs above are things I never learned, I have no knowledge in group theory. So these are all new to me. For example, ##A\trianglelefteq \mathbb{Z}_{23}^*## is a subgroup, I've never seen ## \trianglelefteq ## before...
31. ### Proof about two disjoint non-empty sets ## S ## and ## T ##

I still don't really understand. Can you please tell me why?
32. ### Proof about two disjoint non-empty sets ## S ## and ## T ##

So to rule out the following situation in which you listed above, one example would be ## S\cap B=S ## if ## B=S ##?
33. ### Proof about two disjoint non-empty sets ## S ## and ## T ##

Aren't the two statements of ## A=S ## and ## B=T ## ("automatically") the same as proving what's already proven from the first decomposition: ##S\cdot S\subseteq S\, , \,T\cdot T \subseteq S\, , \,T\cdot S\subseteq T.##? What's different in the second decomposition from the first...
34. ### Proof about two disjoint non-empty sets ## S ## and ## T ##

Proof: Let ## p ## be an odd prime and ## G=\left \{ 1, 2, ..., p-1 \right \} ## be the set which can be expressed as the union of two nonempty subsets ## S ## and ## T ## such that ## S\neq T ##. Observe that ## p-1=22\implies p=23 ##. Let ## g\in G ##. Since ## g ## is either an element of ##...
35. ### Finding square root of number i.e. ##\sqrt{\dfrac{16}{64}}##

The answer is ## 0.5 ## if you enter ## \sqrt{\dfrac{16}{64}} ## into any online calculator. So I wouldn't say ## \frac{1}{2} ## is wrong. And ## \frac{4}{8}=\frac{1}{2} ##. It's the same thing.
36. ### Without evaluating the Legendre symbols, prove the following....

What a long proof. But what does ## aG^{2} \mapsto ({\frac{\cdot }{p}}) ## mean/indicate? I've never seen this before.
37. ### Without evaluating the Legendre symbols, prove the following....

Let ## a ## be a primitive root of ## p ##. Then the integers ## a^{1}, a^{2}, ..., a^{p-1} ## form a reduced residue system modulo ## p ## such that ## \varphi(p)=p-1 ##, where ## r\in\left \{ 1, 2, ..., p-1 \right \} ##. This implies ## r\equiv a^{k}\pmod {p} ## for ## 1\leq k\leq p-1 ##. By...
38. ### Without evaluating the Legendre symbols, prove the following....

I was thoughtless.
39. ### Without evaluating the Legendre symbols, prove the following....

From ## \frac{p-1}{2} ##, to find the number of quadratic residues and non-quadratic residues.
40. ### Without evaluating the Legendre symbols, prove the following....

That seems to be right. Because ## 73 ## has 36 quadratic residues and 36 non-quadratic residues. So they should cancel each other out leaving the answer to ## 0 ##. Also, earlier you mentioned that there's another theorem which claims ## \sum_{r=1}^{p-1}(r|p)=0 ## for any prime, where both the...
41. ### Without evaluating the Legendre symbols, prove the following....

So ## \sum_{k=1}^{p-1}k\cdot (\frac{k}{p})=[1+72\cdot (\frac{72}{73})]+[2+71\cdot (\frac{71}{73})]+\dotsb +[36\cdot (\frac{36}{73})+37\cdot (\frac{37}{73})]=73+73+\dotsb +73 ##?
42. ### Without evaluating the Legendre symbols, prove the following....

Maybe by substituting ## (r|p)=(-1)^{(p-1)/2}(p-r|p) ## into ## \sum_{r=1}^{p-1}r(r|p)=0 ##? But then we have ## \sum_{r=1}^{p-1}r(r|p)=\sum_{r=1}^{p-1}r(-1)^{(p-1)/2}(p-r|p) ##. How should I proceed from here and simplify to ## \sum_{r=1}^{p-1}r(r|p)=0 ##?
43. ### Without evaluating the Legendre symbols, prove the following....

Since ## p=73 ## in this problem, how should I prove that ## \sum_{r=1}^{73-1}r(r|73)=0 ##? Given that ## 73=1\pmod {4} ##.
44. ### Prove that there are infinitely many primes of the form ## 8k-1 ##

Proof: Suppose for the sake of contradiction that the only primes of the form ## 8k-1 ## are ## p_{1}, p_{2}, ..., p_{n} ## where ## N=16p_{1}^2p_{2}^2\dotsb p_{n}^2-2 ##. Then ## N=(4p_{1}p_{2}\dotsb p_{n})^2-2 ##. Note that there exists at least one odd prime divisor ## p ## of ## N ## such...
45. ### How many birds of each kind did I buy?

I found my mistakes.
46. ### How many birds of each kind did I buy?

Let ## x ## denote the number of sparrows, ## y ## denote the number of turtle doves and ## z ## denote the number of doves. Then we have ## \frac{1}{3}x+\frac{1}{2}y+2z=30 ## such that ## x+y+z=30 ##. Observe that \begin{align*} &\frac{1}{3}x+\frac{1}{2}y+2(30-x-y)=30\\...
47. ### Determine the set of odd primes ## p ##?

Thank you for this!
48. ### Determine the set of odd primes ## p ##?

And I have another question, since you said that ## 92 ## is therefore not an upper bound for possible values ## p ##, shouldn't the answer be infinitely many primes ## p ##? Don't we have infinitely many prime numbers?
49. ### Determine the set of odd primes ## p ##?

That's what I was thinking earlier too. Sorry for the late response. I see it now.
50. ### Determine the set of odd primes ## p ##?

From the second case, I found out that ## (7|23)=(11|23)=(19|23)=-1 ## because ## 23\equiv 3^{2}\pmod {7}, 23\equiv 1^{2}\pmod {11}, 23\equiv 2^{2}\pmod {19} ## from solving ## 23\equiv x^2\pmod {p} ## for all ## p\in\left \{ 7, 11, 19 \right \} ##. But I do not see where you lost the solutions...