Articles of elementary number theory

Proof concerning Mersenne primes

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 […]

If $N=q^k n^2$ is an odd perfect number and $q = k$, why does this bound not imply $q > 5$?

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 […]

Maximal sum of positive numbers

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 = […]

Proving the irrationality of $\sqrt{5}$: if $5$ divides $x^2$, then $5$ divides $x$

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 […]

Conditions for which $n | {n \choose k}$ for all $k$

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 […]

Distribution of integer solution pairs (x,y) for $2x^2 = y^2 + y$

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 […]

What is wrong with this proof that there are no odd perfect numbers?

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 […]

Formula and proof for the sum of floor and ceiling numbers

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 + […]

Prove by induction that $a-b|a^n-b^n$

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

Number of primitive characters modulo $m$.

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.