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0
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0answers
40 views

solve nonlinear congruence modulus prime [on hold]

I would like to solve the following congruence equation in positive integers $a$ and $b$. I would be grateful if anyone can give some hints and references. $$ 4\equiv (a+b)/(ab) (\mod p) $$ where ...
2
votes
0answers
162 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
668 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
849 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
226 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
776 views
+50

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. ...
5
votes
2answers
591 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 ...
1
vote
1answer
117 views

How to solve a quadratic diophantine equation [closed]

I'm trying to solve $y^2=3x^2+3x+1$ for integers, which transforms into $(2y)^2-3(2x+1)^2=1$. I know how to solve pell's equation, but how can we extract only (odd,even) pair from the solutions of the ...
0
votes
2answers
131 views

Special type Diophantine equations with integer solutions

The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method. Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not? ...
10
votes
3answers
722 views

Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$

Two years ago, I made a conjecture on stackexchange: Today, I tried to find all solutions in integers $a,b,c$ to $$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$ I have found some ...
2
votes
1answer
71 views

Sufficient condition for solvability of linear diophantine system

I would like to know under what conditions does an integer solution exist to the under-determined linear system: Ax = b. (without constraints) Where A is m x n matrix with positive integers entries ...
14
votes
2answers
734 views

Origin of the term “Diophantine equation”

It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus ...
13
votes
1answer
538 views

Algorithmic (un-)solvability of diophantine equations of given degree with given number of variables

Question: For which $d, k \in \mathbb{N}$ is there an algorithm to determine whether a polynomial diophantine equation $$ P(x_1, \dots, x_k) = 0, \ \ \ P \in \mathbb{Z}[x_1, \dots, x_k] $$ ...
4
votes
1answer
176 views

Numbers represented by inhomogeneous forms

I have a family of Diophantine equations that I am trying to solve, and I am trying to figure out what methods could be used to prove existence of solutions. Unfortunately, the equations are ...
2
votes
3answers
598 views

A Diophantine equation with prime powers

Let $p$ and $q$ be prime numbers such that $p^2+p+1=3q^a$: is it true that $a=1$? This specific equation appears when computing order components of finite groups.
5
votes
1answer
210 views

Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?

(This was posted previously in MSE without getting any answers.) It is known that given primitive (co-prime) integer solutions to, $$x_1^4+x_2^4+x_3^4+x_4^4 = z^4$$ then there is one $x_i$ such ...
9
votes
1answer
419 views

On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation $$\frac{x^{n}-1}{x-1} = y^{2}$$ doesn't admit solutions in ...
3
votes
0answers
72 views

Low height integer points on a rank variety

Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition $$ \mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1. ...
13
votes
4answers
760 views

Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random ...
11
votes
0answers
427 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 ...
7
votes
0answers
195 views

Integer solutions of $x^2=4+8y^2+13z^2$

I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being $x^2=4+8y^2+13z^2$. The ideal answer would be a way to parametrize all the integer ...
8
votes
1answer
219 views

Integers $d$ for which the Negative Pell equation is soluble for both $d$ and $2d$?

Let $\text{NPE}_d$ denote the negative Pell equation: $$ x^2-dy^2=-1$$ Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y. we know that (in this paper): ...
7
votes
3answers
282 views

Deciding a quadratic diophantine equation

Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$ I am more interested in seeing if there is a quick way to test for case when ...
2
votes
0answers
148 views

Diophantine equations and the numbers $4,7,8$

Consider the diophantine equation $$ x^n+y^n+z^n=k\cdot xyz, $$ where $n,x,y,z$ are positive integers. Now consider $k\in\left\{4,7,8\right\}$. It is well-known or easily provable that for $n=1$ and ...
1
vote
3answers
141 views

Decidability of sum of powers exponential diophantine equation

I want to ask about decidability of exponential Diophantine equation: $z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables. Can we find ...
8
votes
2answers
589 views

Sum of consecutive cubes

I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does $$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$ have nontrivial solutions ...
24
votes
5answers
1k views

Parametric solutions of Pell's equation

Given a positive integer $n$ which is not a perfect square, it is well-known that Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$. Question: Let $n$ be a ...
-8
votes
1answer
209 views

Proof of a cubic equation problem [closed]

Well I was doing some questions and i found something. This equation $x^3+y^3+z^3=w^3$ has only one solution which is $x=3,y=4,z=5,w=6$. And what I have have proposed is that there is not other ...
7
votes
2answers
265 views

Decidability of diophantine equation in a theory

Given a theory $T \subseteq \operatorname{Th}(\mathbb{N})$, define the decision problem $D_T$ as follows: Given a polynomial $p$ with integer coefficients and variables $\bar{x}$, decide whether ...
1
vote
0answers
450 views

Is $ a^2b^2-ab-ap$ a perfect square for suitable $a,b\in \mathbb{Z}^+$?

Consider the expression $$ a^2b^2-ab-ap\qquad (a,b\in \mathbb{Z}^+), $$ where $p\equiv1\pmod{4}$. Question. For every prime $p\equiv1\pmod{4}$ do there exist $a,b\in\mathbb{Z}^+$ such that ...
5
votes
1answer
538 views

The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$

I. If there are $a,b,c,d,e,f$ such that, $$a+b+c = d+e+f\tag1$$ $$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$ $$3u^3-3uv+w=-def\tag3$$ where $u=a+b+c,\; v = ab+ac+bc,\;w = abc$, then, $$(a + u)^k + (b + ...
1
vote
1answer
185 views

Diophantine equation with factorials

Find all natural solutions $m!=a^2n!$ . It's clear that $m>=n$. When $m=n$ we have solutions $(1,m,m)$. When $m=n+1$ we have solutions $(a,a^2,a^2-1)$. I think that when $m>n+1$ we have no ...
1
vote
1answer
128 views

basis of the lattice generated by the integer points inside a subspace of R^L

Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., ...
5
votes
2answers
234 views

Density of multi-grade solutions to $x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k$ for $k = 5$ or $6$?

Given the Diophantine equation, $$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$ there is the rather curious observation that the smallest positive solutions for $k=5$ or $6$ is multi-grade. ...
3
votes
0answers
241 views

On 7th and 8th powers for $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$

The Diophantine equation, $$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$ for either $k=5$ or $6$ is quite well explored, and it has long been known that it has an infinite number of primitive ...
24
votes
1answer
1k views

$x^4+y^4$ powerful for relatively prime $x,y$

I asked this question on the NMBRTHRY mailing list on 17 February 2014, but it remains unsolved as far as I know. Recall that a "powerful number" is a positive integer whose prime factorizations $m = ...
5
votes
1answer
294 views

On the elliptic curve $x(x+a^2)(x+b^2) = y^2$

Ajai Choudhry showed that special cases of the elliptic curve, $$x(x+a^2)(x+b^2)=y^2\tag1$$ can be used to prove that, $$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$ has an infinite number of primitive ...
11
votes
1answer
799 views

More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.) The equation discussed in a paper by Jacobi and Madden, $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$ or equivalently, $$(p-2q + ...
3
votes
2answers
214 views

Sets of squares representing all squares up to $n^2$

Let $S_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$. Say that a subset $S \subseteq S_n$ square-represents $S_n^2$ if every square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting ...
29
votes
3answers
1k views

Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit, ...
7
votes
1answer
393 views

rational points of a hyperelliptic curve

I have the following hyperelliptic curve of genus $2$: $$ y^2 = 561 x^6 - 41904 x^5 + 627264 x^4 + 11860992 x^3 - 197074944 x^2 + 124416^2 $$ I need to find all the rational points on this curve. ...
33
votes
4answers
3k views

Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite. Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite? Here ...
0
votes
1answer
210 views

The number of solutions of a Diophantine equation [closed]

Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity. That is, I am asking whether the number ...
2
votes
0answers
199 views

Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
11
votes
2answers
2k views

How many Pythagorean triples are there in which every member is triangular?

How many Pythagorean triples $(a,b,c)$ are there such that $a, b$ and $c$ are triangular? Any two solutions with only $a$ and $b$ interchanged are considered equivalent. The question of existence ...
0
votes
0answers
146 views

The Diophantine equation $x^2 + bxy + cy^2 = p^z_1 \cdots p^{z_k}$

Let $b,c \in \mathbb{Z}$ and let $p_1,\ldots,p_k$ be given primes. Is there an effective algorithm to find all the solutions of the Diophantine equation $$x^2 + bxy + cy^2 = p_1^{z_1} \cdots ...
3
votes
3answers
411 views

Pairs of quadratic polynomials taking values pairs of consecutive squares

Let $f,g \in \mathbb{Z}[x]$ be quadratic and neither square. For $x,y,z \in \mathbb{Z}$ what is the maximal number of solutions to $f(x)=z^2,g(y)=(z+1)^2$? Solutions are integral points on the genus ...
4
votes
0answers
173 views

Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either returns true or false, we say that $\mathcal{A}$ works for ...
10
votes
1answer
226 views

Schoenberg's Rational Polygon Problem

"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by ...
3
votes
4answers
488 views

solutions to special diophantine equations [closed]

Let $0\le x,y,z,u,v,w\le n$ be integer numbers obeying \begin{align*} x^2+y^2+z^2=&u^2+v^2+w^2\\ x+y+v=&u+w+z\\ x\neq& w \end{align*} (Please note that the second equality is ...