# 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.'

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### Two conjectural infinite series for $\pi$

I am looking for a proofs of the following two claims: Claim 1. $$\frac{2\pi}{\sqrt{3}}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\Omega_1(n)}}{n}$$ where $\Omega_1(n)$ is the number of prime ...
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### Pseudo-Droz-Farny circles

I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
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### An infinite series involving harmonic numbers

I am looking for a proof of the following claim: Let $H_n$ be the nth harmonic number. Then, $$\frac{\pi^2}{12}=\ln^22+\displaystyle\sum_{n=1}^{\infty}\frac{H_n}{n(n+1) \cdot 2^n}$$ The SageMath ...
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I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...
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### Four infinite series involving Riemann zeta function

Can you provide a proof for at least one of the claims given below? It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta ...
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### The 4th vertex of a triangle?

I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...
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### The 4th Lemoine circle

The first and second Lemoine circles are well-known to geometers. According to this article the third Lemoine circle has been first discovered by Jean-Pierre Ehrmann in 2002. It is worth noting that ...
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### An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$

Can you prove or disprove the following claim: Claim: $$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$ The SageMath cell that demonstrates this claim can be found ...
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### Nonstandard proofs of the fundamental theorem of arithmetic

Thirty or so years ago, someone showed me a clever proof of the Fundamental Theorem of Arithmetic that did not make use of the lemma "If $p\mid ab$ then $p\mid a$ or $p\mid b$". I'm unable ...
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### Collinearity in tangential pentagon

I am looking for a proof of the following claim: Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...
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### The product of the lengths of two line segments that belong to Newton line

I am looking for the proof of the following claim: Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...
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### A tricky integral to evaluate

I came across this integral in some work. So, I would like to ask: QUESTION. Can you evaluate this integral with proofs? $$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
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### A binomial convolution of Catalan numbers vs "utterly odd numbers"

An integer is called utterly odd if the terminal string of $1$’s in its binary representation has odd length. A number $2^{k+1}m+(2^k-1)$ where $m\geq0$ (every non-negative integer has this form) is ...
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### Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments

It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
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