Intereting Posts

Let $a,b,c$ positive integers such that $\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right) = 3$. Find those triples.
Find the sum of digits in decimal form of the given number
Intuitive explanation of the tower property of conditional expectation
Proving that $\lim\limits_{x\to 0}\frac{f(x)}{g(x)}=L$ implies $\lim\limits_{r\to 0}\frac {\int_R f(ry)h(y)\,dy}{\int_R g(ry)h(y)dy}=L$
Question on using sandwich rule with trig and abs function to show that a limit exists.
Find the infinite sum $\sum_{n=1}^{\infty}\frac{1}{2^n-1}$
Twin primes of form $2^n+3$ and $2^n+5$
Prove $2(7^n) + 3(5^n)-5$ is always divisible by $24$
subgroup generated by two subgroups
limit $\lim\limits_{n\to\infty}\left(\sum\limits_{i=1}^{n}\frac{1}{\sqrt{i}} – 2\sqrt{n}\right)$
When does a group of dilations/scalings exist in a metric space?
Is total boundedness a topological property?
Joint distribution of $n$ Bernoulli variables equal to binomial distribution, how?
$19$ divides $ax+by+cz$
Are there any good algebraic geometry books to recommend?

I know to use Wilson’s Theorem and that each element in the second half is congruent to the negative of the first half, but I’m not sure how to construct a proof for it.

- Limit inferior of the quotient of two consecutive primes
- How to prove these two ways give the same numbers?
- Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.
- number theory $\gcd(a,bc)=\gcd(a,c)$
- Structure of $\mathbb{Z}]/(x-n)$
- Solving the equation $ x^2-7y^2=-3 $ over integers
- Representation of integers by Fibonacci numbers
- A number-theory question on the deficiency function $2x - \sigma(x)$
- Applications of additive version of Hilbert's theorem 90
- A subset whose sum of elements is divisible by $n$

$p-r\equiv -r\pmod p\implies r\equiv-(p-r)$

For uniqueness, $r\le p-r$ or $2r\le p\implies r\le\frac p 2$

So, $1\le r\le \frac{p-1}2$ as $p$ is odd

Putting $r=1,2,3,\cdots,\frac{p-3}2,\frac{p-1}2$ we get,

$1\equiv-(p-1)$

$2\equiv-(p-2)$

…

$\frac{p-3}2\equiv-(p-\frac{p-3}2)=\frac{p+3}2$

$\frac{p-1}2\equiv-(p-\frac{p-1}2)=\frac{p+1}2$

So, there are $\frac{p-1}2$ pairs so,

$(p-1)!=(-1)^{\frac{p-1}2}\left((\frac{p-1}2)!\right)^2$

Using Wilson’s theorem, $(-1)^{\frac{p-1}2}\left((\frac{p-1}2)!\right)^2\equiv-1\pmod p$

If $p\equiv3\pmod 4,p=4t+3$ for some integer $t$,

So, $\frac{p-1}2=2t+1$ which is odd, so $(-1)^{\frac{p-1}2}=-1$

$\implies \left((\frac{p-1}2)!\right)^2\equiv1\pmod p$

$\implies \left(\frac{p-1}2 \right)!\equiv\pm1\pmod p$

- Why do we find Gödel's Incompleteness Theorem surprising?
- Why do we use “if” in the definitions instead of “if and only if”?
- Every preorder is a topological space
- What's the quickest way to see that the subset of a set of measure zero has measure zero?
- Every absolute retract (AR) is contractible
- problems with singularity $0$ of $\int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz$.
- Given matrix A with eigenvalue $\lambda$ and corresponding eigenvector x, prove $A^k$ has eigenvalue $\lambda^k$
- Motivating implications of the axiom of choice?
- Abel/Cesaro summable implies Borel summable?
- Why does the sign $\times$ vanish in mathematical expressions?
- Example of a very simple math statement in old literature which is (verbatim) a pain to understand
- Find the possible values of $a$, $b$ and $c$?
- $C^\infty$ approximations of $f(r) = |r|^{m-1}r$
- Proof on Riemann's Theorem that any conditionally convergent series can be rearranged to yield a series which converges to any given sum.
- Cardinality of the intersection of 2 power sets