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Results tagged with elementary-proofs
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user 88804
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.'
66
votes
3
answers
6k
views
Chebyshev polynomials of the first kind and primality testing
Can you provide a proof or a counterexample for the claim given below ?
Inspired by Agrawal's conjecture in this paper and by Theorem 4 in this paper I have formulated the following claim :
Let …
- 2,661
14
votes
4
answers
1k
views
Six points on an ellipse
Can you prove the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with centroid $G$. Let $D,E,F$ be the points on the sides $AC$,$AB$ and $BC$ respectively , such tha …
- 2,661
12
votes
2
answers
958
views
Intersection point of three circles
Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with orthocenter $H$. Let $D,E,F$ be a midpoints of the $AB$,$BC$ and $AC$ , respectiv …
- 2,661
12
votes
2
answers
656
views
A conjectural infinite series for $\frac{\pi^2}{5\sqrt{5}}$
I am looking for a proof of the following claim:
First define the function $\chi(n)$ as follows:
$$\chi(n)=\begin{cases}1, & \text{if }n \equiv \pm 1 \pmod{10} \\
-1, & \text{if }n \equiv \pm 3 \pmod{ …
- 2,661
11
votes
2
answers
910
views
Primality test for specific class of Proth numbers
Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
Let $N=k\cdot 2^n+1$ such …
- 2,661
10
votes
0
answers
631
views
Primality testing using Chebyshev polynomials
Can you provide a proof or a counterexample for the claim given below?
Inspired by an alternative definition of the Frobenius primality test which is given in this paper I have formulated the followin …
- 2,661
9
votes
1
answer
686
views
An infinite series involving harmonic numbers
I am looking for a proof of the following claim:
Let $H_n$ be the nth harmonic number. Then,
$$\frac{\pi^2}{12}=\ln^22+\displaystyle\sum_{n=1}^{\infty}\frac{H_n}{n(n+1) \cdot 2^n}$$
The SageMath cel …
- 2,661
8
votes
3
answers
1k
views
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Can you prove or disprove the following claim:
Claim:
$$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
The SageMath cell that demonstrates this claim can be found h …
- 2,661
8
votes
4
answers
2k
views
Three circles intersecting at one point
Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with nine-point center $N$ and circumcenter $O$. Let $A',B',C'$ be a reflection points …
- 2,661
7
votes
1
answer
460
views
Primality test for $N=2^a3^b+1$
Can you prove or disprove the following claim:
Let $N=2^a3^b+1$ , $a>0 , b>0$ . If there exists an integer $c$ such that $$c^{(N-1)/3}-c^{(N-1)/6} \equiv -1 \pmod{N}$$ then $N$ is a prime.
You can r …
- 2,661
7
votes
2
answers
453
views
An infinite series involving the mod-parity of Euler's totient function
Can you prove or disprove the following claim:
First, define the function $\xi(n)$ as follows: $$\xi(n)=\begin{cases}-1, & \text{if }\varphi(n) \equiv 0 \pmod{4} \\
1, & \text{if }\varphi(n) \equiv 2 …
- 2,661
6
votes
1
answer
363
views
The square root of natural number expressed by an infinite series
Can you prove or disprove the following claim:
Let $U(n,P,Q)$ be the nth generalized Lucas number of the first kind and let $m$ be a natural number. Then,
$$\sqrt{m}=1+\displaystyle\sum_{n=1}^{\infty …
- 2,661
6
votes
1
answer
224
views
Necessary and sufficient condition for tangential polygon to be cyclic
Can you prove or disprove the following claim?
Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the inscrib …
- 2,661
6
votes
4
answers
570
views
Necessary and sufficient condition for quadrilateral to be cyclic
Can you provide a proof for the following proposition:
Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ an …
- 2,661
5
votes
0
answers
586
views
Primality test for specific class of generalized Fermat numbers
Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
Let $F_{p,n}= (2p)^{2^n}+1 $ where $p$ is a prime number greate …
- 2,661