User BalsamicVinegar

5 votes

1 answer

310 views

Solve $x^{2} = -1$ in $\mathbb{Q}_{5}$.

elementary-number-theory p-adic-number-theory

asked Feb 27, 2020 at 9:22

3 votes

2 answers

82 views

Show : $(m,n)=1\implies(mx+ny,mn)=(m,y)(n,x)\;\forall x,y\in\mathbb Z$.

asked Feb 5, 2020 at 11:01

3 votes

2 answers

166 views

Show that $\sum_{d \mid n} (-1)^{\frac{n}{d}} \varphi(d) = 0$ for $n$ even.

elementary-number-theory gcd-and-lcm totient-function

asked Feb 11, 2020 at 23:52

2 votes

1 answer

185 views

Let $f(n) = \sum_{(k,n)=1} e^\frac{2\pi ik}{n}$. Show that $f(n) = \sum_{k=1}^{n}\left(\sum_{d\mid (k,n)} \mu(d)e^{\frac{2\pi ik}{n}} \right) $

elementary-number-theory proof-explanation

asked Apr 2, 2020 at 9:43

2 votes

2 answers

327 views

Finding the lim inf and lim sup of a sequence

real-analysis sequences-and-series limsup-and-liminf

asked Mar 7, 2019 at 12:03

2 votes

3 answers

86 views

Proving that a sequence converges if $|a_n - a_{n+1}| < Mr^n$ for some $M > 0$ and $r \in (0,1).$

real-analysis sequences-and-series cauchy-sequences

asked Mar 7, 2019 at 13:14

1 vote

1 answer

64 views

Find a function $f(x)$ which is defined at every real number but is continuous at $0$ and is not continuous at every other number [duplicate]

calculus real-analysis continuity

asked Oct 5, 2016 at 4:14

1 vote

3 answers

29 views

Finding a Set Which Spans a Given Set

linear-algebra

asked Apr 7, 2018 at 12:45

1 vote

2 answers

67 views

Proving the supremum of a set

real-analysis supremum-and-infimum

asked Feb 12, 2019 at 9:59

1 vote

1 answer

100 views

For which $t \in \mathbb{N}$ does $\varphi(t) \mid t$? [duplicate]

elementary-number-theory

asked May 12, 2020 at 15:54

1 vote

2 answers

59 views

Convergence of $\frac{z^{n}}{n}$ in $\mathbb{C}$

complex-analysis limits

asked Sep 7, 2020 at 8:22

1 vote

1 answer

40 views

Prove that $a^{b} \equiv 3 \,( \text{mod}\, 4)$ implies $a,b$ odd.

elementary-number-theory modular-arithmetic

asked Feb 12, 2020 at 7:46

1 vote

1 answer

58 views

Verification of proof of convergence in $p$-adic fields

limits elementary-number-theory solution-verification p-adic-number-theory

asked Mar 2, 2020 at 0:53

1 vote

0 answers

49 views

Can you apply the Residue Theorem to real domains?

complex-analysis

asked Dec 4, 2020 at 20:44

0 votes

0 answers

18 views

Multivariable Optimization of Microbes

multivariable-calculus optimization

asked Feb 11, 2021 at 3:49

0 votes

1 answer

342 views

Show that $\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}$.

number-theory elementary-number-theory analytic-number-theory riemann-zeta mobius-function

asked Apr 2, 2020 at 5:09

0 votes

1 answer

41 views

Prove that $\frac{\pi(2x)\text{log}\, x}{x}$ has a constant upper bound.

elementary-number-theory prime-numbers upper-lower-bounds

asked Apr 20, 2020 at 6:53

0 votes

2 answers

47 views

Sum over all characters $\chi$ mod $m$

elementary-number-theory analytic-number-theory

asked Apr 30, 2020 at 1:37

0 votes

1 answer

234 views

Let there be a sequence such that the distance between two consecutive terms converges to 0. Must this sequence converge? [duplicate]

real-analysis convergence-divergence cauchy-sequences

asked Mar 7, 2019 at 7:36

0 votes

2 answers

43 views

Verification of proof of nonconvergence

real-analysis proof-verification convergence-divergence cauchy-sequences

asked Mar 7, 2019 at 10:28

0 votes

1 answer

74 views

Proving a Linear Transformation

linear-transformations

asked Apr 29, 2018 at 21:04

0 votes

1 answer

32 views

Proving an Inequality with the Mean Value Theorem

analysis

asked Apr 11, 2019 at 13:24

0 votes

2 answers

39 views

Proving $f(x) \ge 1+x$ if $f'$ increasing and $f(0)=f'(0)=1$

real-analysis derivatives

asked Apr 11, 2019 at 13:46

0 votes

2 answers

41 views

Let $\{a_n\}_n$ be a sequence and suppose $\{a_n\}_n$ isn't bounded above. Prove that there's a subsequence $\{a_{n_k} \}_k$ such that $a_{n_k} → ∞.$

sequences-and-series proof-verification

asked Apr 18, 2019 at 8:48

-1 votes

2 answers

44 views

If $f$ is entire and $f(3/n) = 6- \frac{10}{n} - \frac{20}{n^2}$, then find $f$ in explicit form.

complex-analysis

asked Oct 27, 2020 at 23:48

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25 Questions

Solve $x^{2} = -1$ in $\mathbb{Q}_{5}$.

Show : $(m,n)=1\implies(mx+ny,mn)=(m,y)(n,x)\;\forall x,y\in\mathbb Z$.

Show that $\sum_{d \mid n} (-1)^{\frac{n}{d}} \varphi(d) = 0$ for $n$ even.

Let $f(n) = \sum_{(k,n)=1} e^\frac{2\pi ik}{n}$. Show that $f(n) = \sum_{k=1}^{n}\left(\sum_{d\mid (k,n)} \mu(d)e^{\frac{2\pi ik}{n}} \right) $

Finding the lim inf and lim sup of a sequence

Proving that a sequence converges if $|a_n - a_{n+1}| < Mr^n$ for some $M > 0$ and $r \in (0,1).$

Find a function $f(x)$ which is defined at every real number but is continuous at $0$ and is not continuous at every other number [duplicate]

Finding a Set Which Spans a Given Set

Proving the supremum of a set

For which $t \in \mathbb{N}$ does $\varphi(t) \mid t$? [duplicate]

Convergence of $\frac{z^{n}}{n}$ in $\mathbb{C}$

Prove that $a^{b} \equiv 3 \,( \text{mod}\, 4)$ implies $a,b$ odd.

Verification of proof of convergence in $p$-adic fields

Can you apply the Residue Theorem to real domains?

Multivariable Optimization of Microbes

Show that $\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}$.

Prove that $\frac{\pi(2x)\text{log}\, x}{x}$ has a constant upper bound.

Sum over all characters $\chi$ mod $m$

Let there be a sequence such that the distance between two consecutive terms converges to 0. Must this sequence converge? [duplicate]

Verification of proof of nonconvergence

Proving a Linear Transformation

Proving an Inequality with the Mean Value Theorem

Proving $f(x) \ge 1+x$ if $f'$ increasing and $f(0)=f'(0)=1$

Let $\{a_n\}_n$ be a sequence and suppose $\{a_n\}_n$ isn't bounded above. Prove that there's a subsequence $\{a_{n_k} \}_k$ such that $a_{n_k} → ∞.$

If $f$ is entire and $f(3/n) = 6- \frac{10}{n} - \frac{20}{n^2}$, then find $f$ in explicit form.