Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

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Diophantine Simultaneous Equations

Solve the set of Diophantine equations: $\frac{x}{y} - \frac{1}{y^2} = a^2 - \frac{64}{x}$ $a^{2x} + 9 = y - \frac {1} {x^2}$ Edit: I have got as far as this (by multiplying, simplifying and solving ...
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2 votes
1 answer
118 views

What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$?

What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$, where $p$ is a prime number and $p>3$?. Calculations suggest that the number of solutions to this equation is $p$ if $p\...
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4 answers
82 views

Show that there are infinitely many solutions $x, y, z \in\mathbb N$ of the following equation $x^2 + y^3 = z^7.$ [closed]

Show that there are infinitely many solutions $x, y, z \in\mathbb N$ of the following equation $x^2 + y^{3} = z^7$. I am thinking about using proof by infinite descent, but I am not too sure how to ...
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In search for methods for finding a closed form for the roots of a non-homogenous diophantine equation.

I'm trying to solve a special case of a number theoretical problem, and it relies on finding a closed form for the roots of a non-homogenous diophantine equation with four variables, but I could only ...
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0 votes
0 answers
16 views

Creating non-intersecting linear Diophantine equations

I have a problem involving linear Diophantine equations which I know should be fairly simple, but I've forgotten the math. Let $r(t)$ and $w(t)$ be two linear, partial Diophantine equations: $$ r(t) = ...
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1 answer
43 views

$\mathbb{Z}$ solutions for a system of equation

An example: Let $x_1,x_2,x_3$ be variables satisfying the equations given by \begin{align*} \begin{bmatrix} x_1&x_3&x_2\\ x_3&x_2&x_1\\ x_2&x_1&x_3 \end{bmatrix}\begin{bmatrix}...
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  • 1,587
1 vote
1 answer
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Find solution using infinite descent.

Can someone help with a task? Need to find a solution other than $(0,0,0) $ with infinite descent. $x,y,z\in\mathbb{Z}$. Any help would be appreciated. The equation is $x^2-3y^2=2z^2$. I tried to ...
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  • 23
8 votes
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123 views

Positive integers satisfying $a^b = cd$ and $b^a = c+d$

Yesterday, at 23:18, I thought it was a remarkable moment of the day. The digits on the watch were providing a quadruplet of positive integers that satisfy the following system of equations: $$\begin{...
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1 vote
1 answer
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Find the smallest value of the product $ab$

From the 2018 Moroccan Mathematics Olympiad: Let $(a,b) \in \mathbb{Z^2}$ such that $a+b$ is a solution of the equation $x^2+ax+b=0$. Find the smallest value of the product $ab$. ($\mathbb Z$ ...
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Showing that the only integer solutions to $y^3=4x^2+47$ are $x=250,-250$

We’ve computed that the class number for $\mathbf Q(\sqrt{-47})$ is $5$. From my attempt at this question, we have the equation of ideals $$(y)^3=(2x+\sqrt{-47})(2x-\sqrt{-47}).$$ Next we show that ...
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0 answers
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A system of cubic diophantine equations over positive integers

I was trying to solve an exercise and it led me to this system of equations: $$a^3 + b^3 = c^3 + x^3\\ c^3 + e^3 = a^3 + y^3\\ c^3 + d^3 = b^3 + z^3\\ c^3 + d^3 + e^3 = a^3 + b^3 + t^3$$ I need to ...
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3 votes
2 answers
146 views

How to solve this eliptic equation? [duplicate]

I want to find a solution to $$ \frac{x_1}{x_2 + x_3} +\frac{x_2}{x_1 + x_3} + \frac{x_3}{x_1+x_2} = 4 $$ for $x_1,x_2,x_3>0$, and $x_1+x_2+x_3=1$. We have $$\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+\...
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  • 451
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Number of values of $n\in \mathbb{Z}$ such that $n^2+n+2$ is a perfect square [duplicate]

My approach: Case $1$: $n$ is positive: Let $x\in\mathbb{Z}$, so $$n^2+n+2=x^2$$ $$\implies n+2=x^2-n^2=(x+n)(x-n)$$ $$\therefore x-n=\frac{n+2}{x+n}$$ $\because x\in \mathbb{Z}$ and $n\in \mathbb{Z}$,...
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2 votes
2 answers
221 views

The number of integer points on the curve $(7x-1)^2+(7y-1)^2=n$

The number of integral solutions to the equation $$x^2+y^2=n$$ is defined to be $r_2(n)$ and if $n=2^ap_1^{a_1}\dots p_k^{a_k}q^2$ where $p_i\equiv 1\mod 4$ and $q$ is the product of primes which are $...
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0 votes
1 answer
57 views

Rational solutions of the equation $2xy^2=kx^2+1$

I am looking for the rational solutions of the equation: $$2xy^2=kx^2+1$$ where $k$ is a fixed positional natural number. For $k=6$, Maple shows it is irreducible and doesn't produces the rational ...
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2 votes
1 answer
80 views

Find all $x,y\in\mathbb{Z},$ such that $x^{2008}+{2008}!=21^{y}$

Find all $x,y\in\mathbb{Z},$ such that $x^{2008}+{2008}!=21^{y}$ My try: So,I want to solve this problem without 'Lifting the exponent lemma'.Here is what I have tried: $x^{2008}+{2008}!=21^{y}\...
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  • 126
1 vote
1 answer
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Proof why solution of linear diophantine equation $ax-by=1$ is found by sign change of solution for $b$ in solution of $ax+by=1$

Say, given LDE $113 x +42y=1$ have solution given by $$113 = 2.42 +29\implies 29= 113 - 2.42$$ $$42= 1.29+13 \implies 13=42 -1.29$$ $$29= 2.13 +3 \implies 3=29 -2.13$$ $$13= 4.3 +1\implies 1=13 -4.3$$ ...
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Diophantine equations and equations over free groups

I'm interested in quadratic diophantine equations such as the ones addressed [here] 1. I have come through a series of papers that solve equations over groups. For instance, here. Since $(\mathbb{Z},+)...
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1 vote
1 answer
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If $p$ is prime and $x \in \mathbb{N}$ then $a^2+a = p^{2x}(b^2+b)$ has no solutions for $a,b \in \mathbb{N}$

I am in a 3rd year university course in elementary number theory, and one of the problems I am given to solve is as follows. If $p$ is prime and $x \in \mathbb{N}$ then $a^2+a = p^{2x}(b^2+b)$ has no ...
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0 votes
0 answers
34 views

Existence and uniqueness of solution of Diophantine equations

Let $a_1,a_2,b_1,b_2,c_1,c_2,d_1$ be known positive integers. Let $x$ be an unknown satisfying $$ 4(a_1d_1+1) \mid (a_1x+d_1a_2)^2$$ $$ 4(b_1d_1+1) \mid (b_1x+d_1b_2)^2$$ $$ 4(c_1d_1+1) \mid (c_1x+...
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1 vote
1 answer
62 views

Diophantine Equation solved with elliptic curves

I want to know how to find all solutions in $\mathbb{Z}$ for $$ 2a^2 -3ab +5c^2 =0. $$ I already solved it and I will post my solution soon. One solution for example is $(15,11,3).$
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A help with a number theory proof. [duplicate]

I've been doing some proofs about the number theory section in The art and Craft of problem solving by Paul Zeitz. I've already done (a) and I think I've been trying to solve (b) but I don't really ...
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2 answers
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Method to solving $a^n + b^n = c$

I have been wondering for a while if there is a method to solving the following form of equation $a^n + b^n = c$ Where a,b,c are all integers. For example, $2^n + 5^n = 133$. One can quickly see that ...
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  • 21
0 votes
1 answer
51 views

Solving the equation $xy=z^n$ in a Dedekind domain

Let $xy=z^n$ where $x$, $y$ and $z$ belong to a Dedekind domain $R$, with $n>1$, and $(x,y)=1$. We can also assume that the ideal class group of $R$ is torsion-free. Then I’d like to show that $x=...
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0 votes
0 answers
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Given $n\geq 0, j\geq 1.$ How many solutions to: $\sum_{i=1}^j x_i=n,$ where $x_i\geq 0$ and $x_i$ are all different?

Given $n\in\mathbb{N}\cup\{0\},\ j\in\mathbb{N}.$ How many solutions are there to: $\sum_{i=1}^j x_i=n,$ where $x_i\in\mathbb{N}\cup\{0\}$ for all $i\in\{1,\ldots,j\}$ and $x_k\neq x_l$ if $k\neq l\ ?$...
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1 vote
2 answers
60 views

Find all pairs $(a,b)$ of positive integers satisfying $6a^2+a=b^2$.

I have already tried treating the equation as a quadratic on a and b, but it doesn't work. I also have plugged in some values. $(6,10)$ is a solution, but I didn't manage to find any other. Are there ...
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0 votes
0 answers
48 views

Find all $(m,x)\in\mathbb{N},$ such that $m^x +x=m^2 +x^2$

Find all $(m,x)\in\mathbb{N},$ such that $m^x +x=m^2 +x^2$ My try: The equation $m^x +x=m^2 +x^2$ can be further written as $m^2 (m^{x-2}-1)=x(x-1).$ If $x=3\implies m^2 (m-1)=6\implies\text{No ...
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  • 126
0 votes
1 answer
71 views

The diophantine equation $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{n}{a+b}$

How do I solve This diophantine equation $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{n}{a+b}$ of unknown $(a,b,n)\in \mathbb{(N^*)^3}$.
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1 vote
1 answer
46 views

Solving an exponential diophantine equation

I came across the following problem: Find all positive integers $a,b$ such that The following equation is satisfied by the two integers: $5^b+2=7^c$ My Attempt: I tried equating both sides modulo 6. ...
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2 votes
1 answer
70 views

How many ordered pairs of integers, $(m, n)$, satisfy \[ 5m^2 + 9n^2 = 1234567? \]

How many ordered pairs of integers, $(m, n)$, satisfy $ 5m^2 + 9n^2 = 1234567? $ I started by trying to just solve the equation but ended up with either $m$ in terms of $n$ or $n$ in terms of $m$. I ...
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0 votes
0 answers
33 views

How to solve a quadratic Diophantine equation without trial and error by just inputting integer values for one of the variables?

I’m looking for a way to find only integer solution pairs to a dual-variable quadratic equation without trial and error. For example: $$(a+3\sqrt 5)^2+a-b\sqrt 5=51$$ Valid solution pairs are any ...
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3 votes
1 answer
70 views

Find all integer solutions of $3^a +7 = 2^b$ [duplicate]

I want to find all integer solutions of $3^a + 7 = 2^b$ I have found (by brute force) the two solutions $3^0 + 7 = 2^3$ and $3^2 + 7 = 2^4$ but I want to see if there are more solutions. I have ...
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0 votes
0 answers
40 views

solve the positive integers $(m,n)$ such $(F_{m})^m=((F_{n})^n+1)^2$

Suppose $F_n$ to be the nth term of the Fibonacci sequence.($F_1=F_2=1,F_{n+2}=F_{n+1}+F_n$) (1)Find all duals $(m,n)$, so that $$F_m^m=(F_n^n+1)^2$$. (2)Find all triples $(m,n,t)$, so that $$F_m^m=...
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0 votes
1 answer
65 views

Find solution set of Diophantine equation: $\frac{a^b-1}{a-1}=\frac{c^d-1}{c-1}$

After reading up on Catalan's Conjecture, a related equation piqued my interest: Let $a>c>1$ and $b,d>2$, where $a,b,c,d \in \mathbb{Z}$. I am looking for integer solutions to $$ \frac{a^b-1}{...
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0 votes
2 answers
103 views

Solve the Diophantine equation $2xy + 3y^{2} = 24$

Solve the equation $$2xy + 3y^{2} = 24, \qquad x, y \in \mathbb{Z}$$ I tried by stating that $x$ and $y$ cannot both be odd and that y cannot be odd. So either $x$ and $y$ are both even or $x$ is ...
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  • 86
2 votes
2 answers
124 views

Solve this equation $12^x-5^y=19$ positive integers

Find all $x,y$ be positive integers,such $$12^x-5^y=19$$ I found $(x,y)=(2,3)$ is solution,maybe have other,so I consider case $x,y>3$ and $\pmod 9$,since $$12^x\equiv 0\pmod 9,x\ge 2$$ then $5^y\...
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0 votes
1 answer
42 views

A Diophantine equation along with an inequality

Find all positive integers $a, b, c$ and integers $x, y, z$ such that $$\begin{align*} ax^2+by^2+cz^2=abc+2xyz-1\\ ab+bc+ca\geq x^2+y^2+z^2 \end{align*}$$ My Progress - By $\pmod 4$. I could conclude ...
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  • 139
0 votes
1 answer
86 views

Rational points on $y^2=5(x^4+1)$

How would one find $\mathbb{Q}$ -rational points on $y^2=5(x^4+1)$. I do not think there is one but I would also love to see a proof of it. I thought about writing them in terms of fractions and got ...
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-7 votes
2 answers
73 views

Solve $4(x+y+z+4) = xyz$ [closed]

How do I solve the following equation: $xyz = 4(x+y+z+4)$, subject to the conditions that $x,y,z$ are positive integers?
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  • 516
1 vote
1 answer
64 views

Prove that a Diophantine equation $ax^3 + by^2 = c$ has or doesn't have integer solutions

I got a diophantine equation, specifically $5x^3 + 4y^2 = 535$ where I have to prove that there are or are not integer solutions. I tried using modular artimetic, this is my process: $$4y^2 = 535 - 5x^...
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1 vote
1 answer
21 views

About a quartic diophantine equation in several variables

I have a nonlinear diophantine equation of the form $$g(x,y)t^4+h(x,y)t^3+w(x,y)t^2+f(x,y)t+d(x,y)=0$$ such that $t$ is a positive integer variable and $g(x,y),h(x,y),w(x,y),f(x,y),d(x,y)$ are ...
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  • 3,715
1 vote
0 answers
29 views

How to recover a matrix with "linear" entries from its homogenous characteristic polynomial?

Suppose I have a given monic homogeneous polynomial $p$ of total degree $n$ in $a_0,...,a_k$ and $x$ with integer coefficients, and suppose I know or "reasonably conjecture" that there ...
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1 vote
0 answers
129 views

Finding all integers $m$ and $n$ satisfying $\frac{1}{m}-\frac{1}{n}=\frac{1}{100}$

So my older brother posed this question to me and I've been stuck on it for a long time. Find all integers $m$ and $n$ that satisfy $$\frac{1}{m}-\frac{1}{n}=\frac{1}{100}$$ (He actually said a $2$ ...
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7 votes
2 answers
1k views

Is there any variation known to the sum of two squares theorem?

Originally posed by Fermat and subsequently generalized as sum of two squares theorem, we can see the following statement. An integer greater than one can be written as a sum of two squares if and ...
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  • 331
0 votes
1 answer
35 views

Solving the exponential Diophantine equation $2^{m+1} = zk + 1$ where $m,k,z \in Z$ for a given very large $z$

I am working on a problem where I have ended with an exponential Diophantine equation of the form $$2^{m+1} = zk + 1$$ where $m,k,z \in Z$ for a given very large $z$ (i.e., factoring $z$ is ...
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  • 1,075
1 vote
1 answer
156 views

Find all integers $x,y$ such that $x^5-x^3-x^2+1=y^2$

Find all integers $x,y$ such that $x^5-x^3-x^2+1=y^2$ I think we need to factor this out and I've managed to factor it to $(x-1)(x+1)^2(x^2-x+1)=y^2$, but I'm not sure what to do here. Have I done ...
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1 vote
1 answer
115 views

On relation between prime numbers and exponential Diophantine equation $c\cdot a^x\pm b=z$, feat. $71999999\cdots$

While dealing with some integers which are the elements of the following set $$\{p\mid p\in\mathbb{P}, p=72\times(10^n)-1\}$$ I've could observed that when $n\in\{6,7,8,9\}$, they are all primes.(Such ...
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  • 331
0 votes
4 answers
127 views

Is there any solutions to the diophantine equation $x^2 + y^2 = 2z^3, x \neq y$? [closed]

So far, I have been unable to find any solutions to the diophantine $ x^2 + y^2 = 2z^3, x \neq y $, and I believe there to be none for natural x,y,z, but I cannot find any ways to prove this.
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  • 13
1 vote
1 answer
90 views

Writing an integer $m$ as the sum of $2m$ integers contained in $[-2,2] \cap \mathbb Z$

Suppose that $m$ is a positive integer. I want all the possible ways to write $$ m=k_1+\ldots+k_{2m} $$ where $k_i \in [-2,2] \cap \mathbb Z$. For example, if $m=1$, I can write $$ 1=1+0 \quad \text{...
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  • 402
0 votes
1 answer
74 views

Solution to Pell's equation $a^2 - Nb^2 = 1$ when $a$ is an even integer

Given an integer $a$, show that there is a square-free integer $ N > 1$ and an integer $b$ such that $a^2 - Nb^2 = 1$. I'm not too sure how to go about proving this. I believe I'm supposed to use ...
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