All Questions
8 questions
2
votes
1
answer
180
views
Only trivial solution to a pair of constrained linear diophantine equations
Given positive integer $n$, we are looking for a set
of $n$ positive integers $a_i$.
The following linear integer program must have only
the trivial integer solution of all ones.
$0 \le x_i \le \frac{...
- 25.4k
21
votes
1
answer
740
views
Does $A-A=\mathbb Q$ hold for $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$?
Let $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$. Then
$$A-A:=\{a-b:\ a,b\in A\}=\{u^4+v^4-x^4-y^4:\ u,v,x,y\in\mathbb Q\}.$$
Motivated by Question 415482, here I ask the following question.
Question. Is it true ...
- 15.6k
0
votes
0
answers
266
views
Is there a permutation $\tau\in S_n$ with $\tau(1)^{\tau(2)}+\cdots+\tau(n-1)^{\tau(n)}+\tau(n)^{\tau(1)}$ a square?
Let $n>1$ be an integer, and let $S_n$ be the symmetric group of all the permutatins of $\{1,\ldots,n\}$.
I'm curious whether there is a permutation $\tau\in S_n$ such that
$$\tau(1)^{\tau(2)}+\...
- 15.6k
6
votes
4
answers
628
views
Request for an exact formula related to a partition in number theory
The Frobenius equation is the Diophantine equation $$
a_1 x_1+\dots+a_n x_n=b,$$
where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$
must consist of non-...
- 10.5k
7
votes
2
answers
604
views
Density version of the Erdős-Graham conjecture
In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdős-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into finitely many subsets then one of the subsets ...
- 15.6k
5
votes
0
answers
196
views
Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?
Recall that the triangular numbers are those integers
$$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$
In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, ...
- 15.6k
15
votes
4
answers
575
views
Are all partial consecutive harmonic subsums distinct?
Let $b \gt a \geq 0$ be integers, and as elsewhere let $H_n$ be $\sum^n_{i=1} 1/i$. A partial consecutive harmonic subsum is a number $H(a,b)$ of the form $H_b - H_a$ (with $ H_0=0$). If $c=a$ and $...
- 13k
3
votes
0
answers
180
views
Additive combinatorics and a Diophantine equation
Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
$...
- 2,207