Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. ...

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4
votes
0answers
44 views

Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...
1
vote
0answers
71 views

Simple Diophantine equations for Cartan matrices of Kac-Moody algebras

Let us consider the matrix $A$ defined as follows: $A_{i i} =2$, $A_{i j} = - \frac{2 d_i}{d - 1 - d_{j}}, \qquad i \neq j$; $i, j = 1, \dots, n$. Here $d_1,...,d_n$ are natural numbers, $n > 1$ ...
11
votes
3answers
429 views

How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?

It is well known that $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ and this is the only solution to $\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}=1$ with $2\leq x_1<x_2<x_3$. My question is: Let ...
-2
votes
0answers
304 views

Complete Local Ring and Fermat's Last Theorem

Let us consider the infinitely many variables formal power series ring over a finite field, viz. $R:= F_p[[S_1,...,S_∞]].$ Question: Choose an arbitrary finitely generated ideal $I$ of $R$. Is ...
0
votes
1answer
115 views

Find the rational cases where ${t}^{2} - 4$ is a perfect square with height bound $|t| \le N$ for positive integer $N \ge 1$

Find the unique cases when ${t}^{2} - 4$ is a perfect square say, ${n}^{2}$, with height bound $|t| \le N$ for positive integer $N \ge 1$, when $t$ is a rational where $t = p/q$ and integers $p$ an ...
2
votes
0answers
84 views

Logarithmic bound for Diophantine equation

Let $a_1 \geq a_2 \geq a_3$ be given positive integers and let $N(a_1,a_2,a_3)$ be the number of solutions $(x_1,x_2,x_3)$ of the equation $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = ...
0
votes
0answers
107 views

A trivial application of Wilson's theorem to Brocard's Problem

Proposition: Let $W(1)$ be the set of all Wilson primes of order $1$ and suppose $n=p-1,$ where $p$ is a prime such that $p\notin W(1)$, then there are no integer solutions to the equation ...
-1
votes
4answers
178 views

Finding integer zeroes for a particular family of equations [closed]

Given $p,q\in\mathbb Z^+$, and a vector $v=(x_1,\dots,x_{p+q})$ we consider the function $\chi(v)$: $$\chi(v)=x_1^2+\dots+x_p^2-x_{p+1}^2-\dots-x_{p+q}^2$$ We wish to find solutions to $\chi(v)=0$ ...
2
votes
0answers
141 views

Number of solutions to pentagonal-pentagonal numbers

Continuing the investigation from this question on CGSE about pentagonal-pentagonal numbers: Defining $p(n)$ as the $n$th pentagonal number (a positive integer of the form $n(3n−1)/2,\ n\geq 1$), and ...
30
votes
2answers
1k views

Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ? (I do not know any such numbers). B. Suppose that $\frac{a}{b+c} + ...
0
votes
0answers
134 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 ...
2
votes
1answer
288 views

On the Diophantine equation $x^2 = y^p + 2^{r}z^p$ where $p\geq 7$ is an odd prime and $r \geq 2$

It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: ...
12
votes
1answer
727 views

On cubic reciprocity for $x^3+y^3+z^3 = 996$?

I. The Diophantine equation, $$x^3+y^3+z^3 = 3w^3\tag1$$ with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...
6
votes
1answer
232 views

Why are some solutions of these diophantine equations off the usual patterns?

This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...
3
votes
1answer
250 views

Is this problem of Schinzel and Tijdeman misquoted? It appears easy with Pell equations

In Diophantine equations over the twentieth century: a (very) brief overview , p. 5 Problem Let $f(x) \in \mathbf{Z}[x]$ be an irreducible polynomial of degree at least 2. Do the Diophantine ...
1
vote
1answer
258 views

How can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution for every prime $p>3$?

Without applying Fermat's Last Theorem, how can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution $(x,y) = (0, \frac{1}{2})$ for ever prime $p ...
6
votes
2answers
725 views

A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$ [closed]

The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$. I am a prospective undergraduate mathematics student in Zimbabwe ...
2
votes
1answer
237 views

rational numbers and triangular numbers

This question is an offshoot of Ratio of triangular numbers. Suppose $ka(a+1)=nb(b+1)$, where $k,n >1$ are relative prime integers, and $a,b \geq 0$ are integers. Which $k,n$ pairs have no solution ...
18
votes
2answers
677 views

Rational points on the “quintic circle” $x^5 + y^5 = 7$

I suspect that the curve $x^5 + y^5=7$ has no $\mathbb Q$ points, and a brief computer search verifies this hypothesis for denominators up to $10^4$. What techniques can be used to show that there are ...
0
votes
2answers
197 views

Minimal solution of simultaneous congruences

I would to determine the set of values $\lbrace a_1,a_2,a_3,\ldots,a_n \rbrace$ that minimizes the value of $x$ such that: $$x\equiv a_1\mod p_1$$ $$\vdots$$ $$x\equiv a_n\mod p_n$$ where every ...
5
votes
1answer
208 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 $$ ...
10
votes
2answers
489 views

Equation $x^2=y^p + 1$

can you help me please for solving this diophantine equation : $x^2=y^p+1$ and if you can give me a general method to studying such equation : $x^2=y^p+t$ Thanks
4
votes
1answer
347 views

On the mixed sum of three k-th powers

Let the set $S_k=\{\pm x^k \pm y^k \pm z^k \ \vert \ x,y,z \in \mathbb{Z} \}$. Note that the signs are independently positive or negative. First of all $S_2 = \mathbb{Z}$ because (see the answers ...
0
votes
2answers
332 views

For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$?

(Much revised for clarity.) I was considering the system of equations, $$-a+nb+c = -d+ne+f\tag1$$ $$a+b+c = d+e+f\tag2$$ $$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$ $$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$ ...
0
votes
2answers
165 views

How to solve the following system of diophantine equations? [closed]

We have the equations $$a_1x+b_1y+c_1z=d_1,$$ $$a_2x^2+b_2y^2+c_2z^2=d_2,$$ $$a_3x^3+b_3y^3+c_3z^3=d_3,$$ where $a_i,b_i,c_i \in\Bbb N$ at $i \in \{1,2,3\}$ are known. Is there an efficient ...
11
votes
1answer
387 views

Why does genus control the number of points

Often number theorists can bound the number of solutions to a diophantine equation based on the size of the points and the size of the coefficients. But this, as I understand it, can be a bit of a red ...
3
votes
1answer
151 views

infinite solution of a diophantine quadratic equations

Let $a,b,c,d$ be integers such that $GCD(a,b,c,d)=1$. Assume that the diophantine equation $ax^2+bxy+cxz+dyz-x=0$ has a non-zero solution.Can we assert that it admits infinitely many solutions? ...
2
votes
2answers
201 views

Parametrizing the solutions to a diophantine equation of degree four [closed]

Good evening, Consider $x^4+y^4+z^4=2t^4$ where x,y,z,t integer. Is it known how to find all parametrisation of this equation ? If you have any parametrisation or reference of this equation, please ...
2
votes
0answers
110 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\}. ...
0
votes
0answers
75 views

Systems of linear modular equations with unknowns in the moduli

I am interested in systems of linear modular equations, where the unknowns also appear in the moduli. The general form would be: $A \vec{x}= \vec{b} \;\textrm{mod} \; (C \vec{x}+\vec{d})$ where A ...
-1
votes
1answer
168 views

Diophantine equations over natural numbers [closed]

Are there versatile techniques that are applicable to deciding if a system of multivariate quadratic diophantine equations with one determinantal restriction has solutions over natural numbers? In ...
18
votes
2answers
3k views

Does the equation $241+2^{2s+1}=m^2$ have a solution?

Let $p$ be a prime congruent to $1$ mod. 8. If $p= 17$ one has : $p+ 8 = 5 ^2$. If $p= 41$ one has : $p+ 8 = 7 ^2$. If $p= 73$ one has : $p+ 8 = 9 ^2$. If $p= 89$ one has : $p+ 32 = 11 ^2$. If ...
1
vote
2answers
174 views

Sharply Estimating Pythagorean Triples [closed]

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$ Is there a way to give a sharp estimate?
1
vote
0answers
151 views

System of congruences

I have a system of $n$ congruences. the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form: $(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq ...
0
votes
1answer
191 views

A three variable linear diophantine promise problem

Given $a,b,c,s\in\Bbb N$ such that $(a,b,c)=1$ with promise that we have at most one triple $x,y,z\in\Bbb N$ such that $ax+by+cz=s$, what is a good algorithm that runs in $O(\log(abcs))$ time to find ...
2
votes
0answers
92 views

Possible argument against Height bound hypothesis

From this paper. $f(x,y)$ is polynomial with integer coefficients. $s(f)$ is its size, the sum of the logarithms of the absolute values of the nonzero coefficients, defined on p. 6. From p. 7. ...
2
votes
1answer
349 views

Quadratic Diophantine equation in $\mathbb Z[T]$

I am trying to solve the following quadratic diophantine equation in $\mathbb Z[T]$: $$((T+1)X+TY-1-Z)((T+1)X+TY-1+Z)=24XY$$ One has the following trivial solutions: $(X,Y,Z)=(0,Y,\pm(1-TY))$, ...
2
votes
1answer
188 views

Finite sequences which happen to be the sequence of orders of elements of a simple group

Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ ...
6
votes
4answers
257 views

Application and usage of representation of integers as sum of powers?

We know that there are many articles and manuscripts from the ancient to date talking about representation of integers as sum of squares, cubes etc. I would like to know what is it the usage and ...
2
votes
0answers
117 views

Egyptian fractions similar to Erdos-Straus conjecture

It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions. My question is whether it is known that if $a>4$ $$ \frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k} ...
1
vote
1answer
311 views

Particular case of Beal's Conjecture

Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of Beal's Conjecture.
1
vote
0answers
156 views

Probability of correlated residues

Given $N,c\in\Bbb N$, where $c\ll(\log N)^{1/b}$ for any $b>1$ is fixed, what is the probability that given $A_1,A_2,A_3\in\Bbb N$ with ...
3
votes
4answers
491 views

Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that $$ ...
0
votes
1answer
135 views

Quadratic diophantine equation in $\mathbb C[T]$

I am trying to solve the following quadratic diophantine equation in $\mathbb C[T]$, but I did not manage. I hope someone could give some hints or solutions to my problem. Here is the equation ...
2
votes
0answers
184 views

Can estimate upper bound of $|p_{i}|$ or $|q_{i}|?$

when I Find the diophantine-equation rational points $$2y^2=x^6-x^2+2$$ I using Faltings's theorem showed that there are only finitely many solutions,if we assmue that ...
14
votes
4answers
826 views

Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...
7
votes
2answers
1k views

How to prove that this equation has only one solution?

I can't find a way to prove that the following equation has only one solution : $$ X = \frac{2^Q - 1}{2^{P+Q} - 3^P} $$ with $X,P,Q$ integers $> 0$. One trivial solution is $X = 1, P = 1, Q = ...
2
votes
2answers
275 views

Is it possible that $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are Pythagorean triples simultaneously? [closed]

Do there exist postive integers $a,b,c,x,y,p,q$ such $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are all Pythagorean triples? That is, does the system $$\begin{cases} a^2+b^2=c^2\\ x^2+y^2=a^2\\ p^2+q^2=b^2 ...
4
votes
2answers
944 views

Find all rational solutions of this diophantine-equation?

Now, today, my friend tell me this problem was posted by American Mathematical Monthly (Vol. 111, No. 2 Feb., 2004), p. 165 by Wu wei Chao ,and It is said that this problem is unsolved, until now. ...
6
votes
2answers
641 views

When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$?

When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$ ($n$ being an integer) , i .e., when does $(-1+\sqrt[3]{2})^n$ not have a non-zero term in $\sqrt[3]{4}$. As you might have noticed, I'm ...