# Questions tagged [elementary-proofs]

For questions related to 'elementary' proofs in a technical sense, which has nothing to do with the difficulty of the argument or result. A typical example would be 'elementary' proofs of the Prime Number Theorem, which avoid complex analysis. The tag is however not limited to this particular notion of 'elementary.'

112 questions
Filter by
Sorted by
Tagged with
74 views

216 views

### How to prove there is infinite prime numbers of form $5n+3$ without Dirichlet theorem? [closed]

Is there a nice elementary way to prove there is infinite prime numbers of form $5n+3$ (also for $5n+2$) with $n\in \mathbb{N}$? I know how to do it for primes of form $pn+1$ for any prime $p\geq 3$ ...
167 views

### Perfect squares between certain divisors of a number

Let $n$ be a positive integer. We will call a divisor $d(<\sqrt{n})$ of $n$ special if there exists no perfect squares between $d$ and $\frac{n}{d}$. Prove that $n$ can have at-most one special ...
138 views

### Diophantine equation $10^n-a^3-b^3=c^2$

Consider the Diophantine equation: $10^n-a^3-b^3=c^2$, for $a$, $b$, $c$, and $n$ positive. Has this equation infinitely many solutions?
252 views

### “Strengthening” the mean value theorem for the sine function

The present discussion arises from this MO question. Below, $e$ stands for Euler's number and let $$\tau:=\arccos\left(\frac{\sin e-\sin 1}{e-1}\right)\approx 1.82\cdots.$$ An application of the Mean ...
469 views

47 views

### Exponential sums involving completely multiplicative functions

It is well-known that exponential sums is used as a tool from the analytic number theory to optimize or to compute asymptotic formulas. My question is the following: Given a completely multiplicative ...
217 views

### Can I get away without using Arzela-Ascoli?

I am currently thinking of function-valued random variables. In order to prove a result, I need to approximate by (function-valued) step functions. This naturally leads to the idea of chopping up the ...
211 views

### Number of distinct terms in the dRRS for n

For OEIS sequence A061498, the "Number of distinct terms in the first difference sequence of reduced residue system[=dRRS] for n", I am wondering if there is a simple proof that this sequence is ...
142 views

### Some questions in a paper by E. H. Neville (1949) about Farey series?

I am reading the paper MR0029924: Neville, E. H. The structure of Farey series. Proc. London Math. Soc. (2) 51, (1949). 132–144. (Reviewer: W. H. Simons) and by now two questions raised for me; ...
229 views

### Area method in Lobachevskian geometry

There are many proofs in Euclidean geometry using the area method; for example, Ceva's theorem or the proof of Pythagorean theorem shown below. Do you know such proofs in hyperbolic geometry? I ...
86 views

### The $p$-adic valuation of powers of consecutive integers

Let $n > 0, K > 0$ integers and, for $i \in \{1,...,n\}$, let $k_i$ and $l_i$ be integers such that $k_i + l_i = K$. Assume that for some $i,j \in \{1,...,n\}$ we have $k_i \neq k_j$. Claim: ...
146 views

### Squares in Lucas sequences

Good night, everyone! According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
85 views

366 views

66 views

### Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector

Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector. Question $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$ Observation This paper allows us to ...
458 views

640 views

### Primality test for generalized Fermat numbers

This question is successor of Primality test for specific class of generalized Fermat numbers . Can you provide a proof or a counterexample for the claim given below? Inspired by Lucas–Lehmer–Riesel ...
### $2$-adic valuations and sum of divisor function
Consider the sum of $k^{th}$-power of divisors of $n$, denoted $$\sigma_k(n)=\sum_{d\vert n}d^k.$$ Let $\nu_p(x)$ stand for the $p$-adic valuation of the integer $x$. The following appears to be ...
This is probably easy but it might be interesting. Here goes $\dots$ Let $P\in\mathbb{R}[x]$ be a polynomial of degree $n>2$ and $P'=\frac{dP}{dx}$. If $x_1, x_2, \dots, x_n$ are the roots of \$P(x)...