Is this proof acceptable ? Lemma Let $M_p=2^p-1$ with $p$ prime and $p>2$ , thus If $M_p$ is prime then $3^{\frac{M_p-1}{2}} \equiv -1 \pmod {M_p}$ Proof Let $M_p$ be a prime , then according to Euler’s criterion : $3^{\frac{M_p-1}{2}} \equiv \left(\frac{3}{M_p}\right) \pmod {M_p}$ , where $\left(\frac{3}{M_p}\right)$ denotes Legendre symbol . If $M_p$ is prime then […]

Let $\mathbb{N}$ denote the set of natural numbers (i.e., positive integers). A number $N \in \mathbb{N}$ is said to be perfect if $\sigma(N)=2N$, where $\sigma=\sigma_{1}$ is the classical sum of divisors. For example, $\sigma(6)=1+2+3+6=2\cdot{6}$, so that $6$ is perfect. (Note that $6$ is even.) Denote the abundancy index of $x \in \mathbb{N}$ as $I(x)=\sigma(x)/x$. Euler […]

I’ll be grateful for any help with the foollowing question. I think the solution must be easy enough but i haven’t figured it out yet. Let a and b be positive integers such that 1) $\exists c \in \mathbb{Z}: ~~ a^2 + b^2 = c^3$ 2) $\exists d \in \mathbb{Z}: ~~ a^3 + b^3 = […]

I am working on proving that $\sqrt{5}$ is irrational. I think I have the proof down, there is just one part I am stuck on. How do I prove that $x^2$ is divisible by 5 then $x$ is also divisible by $5$? Right now I have $5y^2 = x^2$ I am doing a proof by […]

I’m studying for a number theory exam. Our review sheets offers the question: Under what conditions will $n$ divide $n \choose k$ for all 1 $ \leq k \leq n-1$? I can see that this will be true for any prime $n$, and don’t think that it would be true for any composite $n$, but […]

I am looking for integer pairs $(x,y)$ that respect $$2x^2 = y^2 + y$$ For example $(6,8)$ is a solution for that. Simple solution is to enumerate on $x$ or $y$ and test if the corresponding variable is an integer. However, I am searching for number too big to enumerate and test (just takes too […]

Main Question What is wrong with this proof that there are no odd perfect numbers? The “Proof” Euler proved that an odd perfect number $N$, if any exists, must take the form $N = q^k n^2$ where $q$ is the Euler prime satisfying $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$. Denote the sum […]

For any real number $a$ and a positive integer $n$, there is a concise formula to calculate $$a + 2a + 3a + \cdots + na = \frac{n(n+1)}{2} a.$$ The proof for the same is given in Mathematical literature. Is there any such formula to calculate: $$\lfloor a\rfloor + \lfloor 2a\rfloor + \lfloor 3a\rfloor + […]

This question already has an answer here: Why $a^n – b^n$ is divisible by $a-b$? 9 answers

Let $N(m)$ be the number of primitive Dirichlet characters modulo $m$. Could someone please explain me why it satisfies the following relation?: $$ \phi(m) = \sum_{d|m} N(m). $$ Thank you very much! PS here $\phi$ is the Euler totient function.

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