All Questions
14 questions with no upvoted or accepted answers
12
votes
0
answers
704
views
Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?
Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...
- 12.6k
10
votes
0
answers
205
views
Does the diophantine equation $\,\prod_{k=1}^n(p_k^{x_k}-1)=y^2\,$ have always at least a solution for $\,n\gt2\,$?
P.G.Walsh proved in this paper that the diophantine equation $\,(2^{x_1}-1)(3^{x_2}-1)=y^2\,$ has no solution in positive integers $\,x_1$, $\,x_2\,$ and $\,y$.
If we generalize the previous equation ...
- 1,008
10
votes
0
answers
223
views
Product of four consecutive primes plus $1$ equals square
Some days ago, I noticed that $3\cdot 5\cdot 7\cdot 11 +1=34^2$.
I am almost sure that if we denote four consecutive primes by $p, q, r, s$ then the equation
$$p\cdot q\cdot r\cdot s+1=x^2 \quad ...
- 3,866
4
votes
0
answers
238
views
On the values of $\prod_{k=1}^{(p-1)/2}(e^{2\pi i/12}-e^{2\pi i k^2/p})$ for primes $p>3$
In a recent preprint, I investigated
$$S_p(x):=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p}),$$
where $p$ is an odd prime and $x$ is a root of unity.
Motivated by Question 337879 and Question 338325, ...
- 15.6k
4
votes
0
answers
408
views
Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$
Suppose we're given a particular number $n \in \mathbb{N}$.
We're also given that $n=pq$ where $p,q$ are unknown primes satisfying
$$
p=a^2+b^2
$$ and
$$
q=2ab+1
$$
for some $a,b$.
Is there an ...
- 141
2
votes
0
answers
207
views
n-ary quadratic forms with $S$-integer values
Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
- 1,639
1
vote
0
answers
243
views
Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?
I posted this question on SE, and was told I should repost it here.
The Goormaghtigh conjecture explores the Diophantine equation of the form
$$
\frac{a^b-1}{a-1}=\frac{c^d-1}{c-1},
$$
where $a>c&...
- 119
1
vote
0
answers
104
views
Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers
In April 2018, I noted that the first integer $n>1$ with $n^2\not\in\{x^2+2y^2+3\times 2^z:\ x,y,z=0,1,2,\ldots\}$ is $$5884015571=7\times17\times49445509.$$
Question. Is it true that for each ...
- 15.6k
1
vote
0
answers
156
views
On segments of the series $\sum_p\frac1{p-1}$
Here I ask a question concerning segments of the divergent series
$$\sum_p\frac1{p-1}=\sum_{k=1}^\infty\frac1{p_k-1},\tag{$*$}$$
where $p$ runs over all the primes, and $p_k$ denotes the $k$-th prime.
...
- 15.6k
1
vote
0
answers
159
views
The existence of solution for special equation on integer ring
I have a question which belongs to the field of number theory. Can we prove or disprove the following claim:
For all prime number $p=24t+1$ and the natural number $n=6t+1$, there is at least, one ...
0
votes
0
answers
241
views
Conjecture about some recurrent primes
I want to know if there are conjectures similar to this one, I know there is the Bell primes conjecture or Gardner conjecture (mentioned in this page https://en.wikipedia.org/wiki/Bell_number), but ...
0
votes
0
answers
112
views
The number of solutions of $2^xpx+k=y^2$
Let's consider the family of diophantine equations
$$2^xpx+k=y^2$$
being $p\gt2$ a prime and $k$ a positive integer.
An example is given by the equation
$$2^x\cdot3x+97=y^2$$
that presents, at least, ...
- 1,008
0
votes
0
answers
72
views
Superfluousness of ET-type $I$ for ES-equation (?)
You may consult the following paper by Christian Elsholtz & Terence Tao: https://terrytao.wordpress.com/tag/erdos-straus-conjecture/
A natural solution $\ (p\ x\ y\ z)\ $ of Erdös-Straus equation
...
- 7,500
0
votes
0
answers
322
views
When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?
It is easy to show that the following problems are equivalent.
a. When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square?
and
b. When is $X^2-PY^2=k$ ...
