All Questions
Tagged with elementary-proofs nt.number-theory
21 questions with no upvoted or accepted answers
17
votes
0
answers
891
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An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$
This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
- 10.5k
10
votes
0
answers
633
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 ...
- 2,661
7
votes
0
answers
275
views
Simple/Elementary derivation of Ramanujan's continued fraction for Hurwitz $\zeta(3,x+1)$
I came across this MSE post discussing a certain continued fraction for $\zeta(3)$ (more specifically, the Hurwitz zeta function $\zeta(s,z)$ at $s=3$) due to Ramanujan. I asked the original poster ...
- 831
7
votes
2
answers
451
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A set, product of any two elements minus one is a perfect square
The first problem of IMO 1986 asks the following:
Prove that, one can find two distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
Note that, this proves, for the ...
- 1,096
6
votes
0
answers
381
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
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 ...
- 2,661
4
votes
0
answers
235
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A combinatorial proof for equality of two $q$-series
Consider the following two $q$-series
\begin{align*}
f(q):&=\sum_{k=1}^{\infty} \frac{(-1)^{k-1}(1 + q^k)\,q^{\binom{k + 1}2}}{(1 - q^k)^2} \qquad \text{and} \\
g(q):&=\frac1{\prod_{j=1}^{\...
- 43.2k
4
votes
0
answers
140
views
Factorization in the ring of integers of a particular biquadratic number field, and questions about norms
Consider the number field $K={\mathbb Q}[\sqrt{2},\sqrt{3}]$ and its ring of integers ${\mathcal O}_K$. I have been doing some calculations with this number field as a toy example, to see what can be ...
- 25.8k
3
votes
0
answers
1k
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Formula for $\pi$ involving exponents of Mersenne primes
Can someone provide a proof for the following claim?
$$\pi=\dfrac{S_0S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
- 2,661
3
votes
0
answers
265
views
Conjectured primality test for numbers of the form $N=4 \cdot 3^n-1$
This is a repost of this question.
Can you provide proof or counterexample for the claim given below?
Inspired by Lucas-Lehmer primality test I have formulated the following claim:
Let $P_m(x)=2^{-m}\...
- 2,661
2
votes
0
answers
306
views
Conjectured initial values of Inkeri's primality test for Fermat numbers
This is a repost of this question .
Can you provide a proof or a counterexample to the claim given below ?
First , we shall give a definition of the Inkeri's primality test for Fermat numbers :
...
- 2,661
2
votes
0
answers
65
views
Can this function satisfy Song conditions?
Let $r_{s,k}(n)$ be the number of representation of a naturel number $l$ as a sum of $s$ positive $k$-th powers.
Joung Min Song introduced some conditions to study asymtotic behavior of some positive ...
- 1,257
1
vote
0
answers
127
views
Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
- 1,257
1
vote
0
answers
94
views
How to estimate the highest power of 2 in the partial sum of 2-adic $\log(-1)$ (i.e. $\sum_{i=1}^n\frac{2^i}{i}$)?
The estimate I wanna get is $$v_2(\sum_{i=1}^n\frac{2^i}{i})\geq\min_{t\geq n+1}\{t-v_2(t)\}\tag{*}$$
where $v_2$ is the 2-adic valuation, that is the highest power of 2 defined on $\mathbb{Q}$.
Set $$...
user178596
1
vote
0
answers
118
views
A primality criterion for specific class of $N=4kp^n+1$
Can you provide a proof for the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ .
Let $N= 4kp^n+1 $ such that $p$ is a prime number greater ...
- 2,661
1
vote
0
answers
93
views
Primality test for specific class of $N=12k \cdot 5^n-1$
Can you provide a proof for the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ .
Let $N= 12k \cdot 5^{n} - 1 $ where $n\ge3$ , $12k <5^n$ ...
- 2,661
1
vote
0
answers
112
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: ...
- 365
1
vote
0
answers
207
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Proofs for almost prime limits
A number $n$ with prime factorization $$n=\prod_{i=1}^rp_i^{a_i}$$
is a k-almost prime if it has a sum of exponents $$\sum_{i=1}^{r}a_i=k$$ i.e., when the prime factor (multiprimality) function $\...
- 1,903
0
votes
0
answers
183
views
A certain Pell Equation
Recently I came up with a positive solution $((x,y)\neq (\pm 1;0))$ to this diophantine equation
$$
x^2-\left(w^2(2^{n-2}p)^2+2^n(2^{n-2}p)\right)y^2=1,\qquad n\geq 2,
$$
where all variables are in $ ...
- 785
0
votes
0
answers
123
views
Testing the primality of Mersenne and Fermat numbers using third order recurrence relation
Can you prove or disprove the claims given below?
Inspired by generalization of Lucas-Lehmer test I have formulated the following claims:
Claim 1 Let $M_p=2^p-1$ where $p$ is an odd prime number , ...
- 2,661
0
votes
0
answers
165
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;
...
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