All Questions
39 questions
18
votes
2
answers
1k
views
Lower bounds on the easier Waring problem
The easier Waring problem asks for the least number $v=v(k)$ such that every every integer is a sum of $v$ $k$'th powers with signs, i.e. every $n\in \mathbb{N}$ is of the form $$n=x_1^k\pm x_2^k\pm\...
- 7,836
13
votes
1
answer
524
views
The number of representations of an integer as the inner product of integral lattice points
I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a question I asked about four years ago. In particular I wrote that for an ...
- 2,506
12
votes
1
answer
499
views
A diophantine equation in $\mathbb{N}$
While I was working on a paper on graph theory, I encountered a problem which I think is a number-theory-problem. I don't know if there are any tools to answer the question.
Find all natural numbers $...
- 351
11
votes
1
answer
699
views
"strange" diophantine and parity of the partition function
Let $\{x_i\}:=\{x_1=5, x_2=13, x_3=29, x_4=37, x_5=45, \dots \}$
be the sequence of those positive integers of the form
$$
p^{4\alpha+1}n^2$$
in increasing order where $p\equiv 5\pmod 8$ is prime ...
- 43.2k
11
votes
0
answers
431
views
Growth of $n=n(k)$ for which there's a non-trivial solution to $x_1^k+\cdots+x_n^k=y^k$?
Walter Hayman just asked me the following question. What, if anything, is known about the growth of the function $n(k)$, where $k\geq1$ is an integer, and $n=n(k)\geq2$ is the smallest integer for ...
- 41.4k
10
votes
1
answer
2k
views
Is new $n$-conjecture as follows correct?
Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...
- 1,776
8
votes
0
answers
271
views
Restricted divisor summatory function
I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where
$$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$
and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
- 361
6
votes
1
answer
462
views
The number of integral solutions to $x^2+y^2-az^2=0$
I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation
$$ x^2+y^...
- 6,224
5
votes
1
answer
442
views
polynomials in many variables and Hasse principle
I was wondering whether there exists any result of the form
"if $f \in \mathbb{Z}[x_1, ..., x_k]$ is a polynomial (not form! I don't require homogeneity) of total degree $n$, with $k \geq \delta n$...
- 51
5
votes
1
answer
430
views
How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?
Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...
- 13.9k
5
votes
6
answers
826
views
Representations with Triangular Numbers
A well known theorem of Gauss says that any natural number $n$ may
be written as the sum of three triangular numbers -
$$
n={a_{1} \choose 2}+{a_{2} \choose 2}+{a_{3} \choose 2}
$$
The following ...
5
votes
1
answer
383
views
What analytic tools can provide a lower bound for this Diophantine equation?
The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken.
I felt that the following ...
- 53
5
votes
0
answers
606
views
Necessary and Sufficient condition for Sharpness of Bombieri and Vaaler's result on Siegel's lemma?
This Wikipedia page currently quotes Bombieri and Vaaler's result on Siegel's lemma:
Suppose we are given a system of $m$ linear equations in $n$ unknowns such that $n>m$, say
$a_{11}x_1+\dots+...
- 13.9k
4
votes
1
answer
306
views
Waring's problem over $\mathbb Q_{\ge0}$
Let $k$ be a positive integer. Note that $a/b=ab^{k-1}/b^k$ for any integers $a$ and $b>0$.
If every $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x_1^k+\cdots+x_{s}^k$ with $x_1,\ldots,x_s\...
- 15.6k
4
votes
1
answer
193
views
Small linear relations between primitive Pythagorean triples $\mathsf{II}$
WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$.
Now let $a^2+b^2=c^2$ be a primitive Pythagorean ...
- 1,826
4
votes
1
answer
332
views
Counting integral points on a surface
Let $f$ be a homogeneous polynomial with integral coefficients of 4 variables $a$, $b$, $c$ and $d$. Suppose $f$ is invariant under the rotation that rotates $(a,b)\in\mathbb{R}^2$ and $(c,d)\in\...
- 5,169
4
votes
0
answers
78
views
Repeated values of a monomial
Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer ...
- 1,990
4
votes
0
answers
211
views
Rational solutions to Catalan's equation
Famous Catalan's conjecture, now a theorem proved by Mihăilescu, states that the only solution in the natural numbers of the equation
$$
x^{a}-y^{b}=1.
$$
for $a, b > 1$ and $x, y > 0$ is $x = 3,...
- 7,182
3
votes
1
answer
370
views
Close integer solutions to $ab-cd=1$?
I am looking for infinite set of Diophantine solutions.
Suppose we require
$$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$
$$a,b,c,d\in\mathbb Z$$
then can we still find ...
- 1,826
3
votes
1
answer
164
views
Existence of Pillai equations with Catalan type solutions?
In Catalan's conjecture we have $$x^m-y^n=1$$ having solution $(3,2,1,1)$ and $(3,2,2,3)$.
Call $$ax^m-by^n=k$$ to be Pillai Diophantine equation.
Is it true no Pillai Diophantine equation exists ...
- 13.9k
2
votes
3
answers
416
views
Density of $d$ for which a generalized Pell equation has a solution
For how many $0 < d \leq D$ is there an integer solution to
$$x^2-dy^2 = -n$$
for $n > 1$? I have circumstantial reason to believe it might be $\sim D^{\frac{1}{2}}$ but I'd be interested in any ...
- 479
2
votes
0
answers
356
views
Classifying solutions of a certain Diophantine Equation
The following question arose from a problem I am working on.
Let $N, k$ be positive integers. Consider the Diophantine equation in $a, b, c$:
$$
\frac{1}{a} + \frac{N - 1}{b} = \frac{N^k}{c}
$$
with ...
- 791
1
vote
2
answers
127
views
Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known ...
- 3,625
1
vote
1
answer
209
views
Representing integers efficiently with quadratic polynomials
For every large enough $T$ given four integers $a,b,c,d$ with absolute value less than $T^2$ are there integers $w_1,x_1,y_1,z_1,w_2,x_2,y_2,z_2$ with absolute value less than $T$ such that
$$w_1x_1+...
- 13.9k
1
vote
2
answers
289
views
Solutions of a linear diophantine equation
Let $N(h)$ be the number of solutions of the following linear diophantine equation:
\begin{equation}
x_1 + 2x_2 + 3x_3 + \dots + (h-1)x_{h-1} = 6h-6;
\end{equation}
where $h\geq 2$ and solution means ...
- 8,998
1
vote
1
answer
193
views
Fundamental solutions to linear Diophantine equations and their existence and computation
$T>0$ is a parameter.
Consider the linear Diophantine equation $ax+by=c$ where $a,b$ are coprime.
Suppose $a,b$ are of magnitude $T^{1+\epsilon}$ and $c$ is of magnitude $T^2$.
For how many such ...
- 13.9k
1
vote
0
answers
98
views
Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
- 25.4k
1
vote
0
answers
115
views
Integral points in smooth cubic curves
Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
- 403
1
vote
0
answers
102
views
Finding number fields over which Diophantine equations are solvable
Given a Diophantine equation $f(x_1, \dots, x_n) \in \mathbb{Z}[x_1, \dots, x_n]$ and a family of number fields $K$ (say, the number fields of a specified degree and signature), are there techniques ...
- 479
1
vote
0
answers
88
views
Distribution of number of integer solutions in box to bivariate polynomials?
Take a bivariate polynomial of degree $d_x+d_y>\max(d_x,d_y)>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
What is the ...
- 13.9k
1
vote
1
answer
189
views
Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?
http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf gives a general method to solve quadratic bivariate diophantine equation while Coppersmith introduced a method to solve bivariate ...
- 13.9k
1
vote
0
answers
132
views
Probability of small solutions to an uniform random linear diophantine equation?
Take the set $$T(c_1,\dots,c_t)=\{(x_1,\dots,x_t)\in\mathbb Z^t\backslash\{(0,\dots,0)\}:\sum_{i=1}^tc_ix_i\equiv0\bmod q\}$$ where $c_1,\dots,c_t\in(-q/2,q/2)\cap\mathbb Z$.
What is probability ...
- 13.9k
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
0
votes
1
answer
394
views
Diophantine equations and modular forms
Let $D$ be a square-free positive integer which is the fundamental discriminant of a real quadratic field. Consider the following quadratic form $$Q_{D}(x,y)=x^2+Dy^2.$$
My questions are :
What is ...
- 347
0
votes
1
answer
152
views
Counting solutions of a certain diophantine equation
For some $s, k$, let $J_{s, k}(X; \mathbf{n})$ be the number of solutions to the system $$\sum_{i\le s} (x_i^j - y_i^j) = n_j$$ for $j\le k$ with $x_1, \dots, x_s, y_1, \dots, y_s\in [1, X]\cap\mathbb{...
- 1,890
0
votes
0
answers
115
views
Maximum number of integer solutions with some size constraints to bivariate polynomials?
Take a bivariate polynomial of total degree $d$ satisfying $d=d_x=d_y>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
Given a ...
- 13.9k
0
votes
1
answer
74
views
Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples
$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$.
Is it true that there are ...
- 1,826
0
votes
0
answers
133
views
What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?
It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...
-2
votes
1
answer
201
views
Solutions to a diophantine system
What is the smallest $\gamma_1,\gamma_2,\gamma_3>0$ such that given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ there are coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ ...
- 13.9k