# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### System of diophantine equations $x^2+3y=u^2$, $y^2+3x=v^2$

Solve the following system of Diophantine equations(the unknowns are positive integers): $$\left\{ \begin{array}{c} x^2+3y=u^2 \\ y^2+3x=v^2 \end{array} \right.$$ I worked as follows: ...
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### How many generators needed for Pell-equation-related group

Let $d$ be a positive integer which is not a perfect square. We have the norm multiplicative group homomorphism, $N:{\mathbb Q}[\sqrt{d}] \to {\mathbb Q}$ defined by $N(x+y\sqrt{d})=x^2-dy^2$. It ...
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### Diophantine equations using Euclidean algorithm

I solved two systems of Diophantine equations using the Euclidean algorithm and I can't figure out where I went wrong because the solutions I test aren't working but I have rechecked my work several ...
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### Showing that there are infinitely many integer solutions for the hyperbolic formula $|a^2 - 26 b^2| = 1$

I want to show that the formula $$| a^2 - 26\cdot b^2| = 1$$ has infinitely many solutions $(a, b) \in \mathbb{Z}^2$. First I tried to solve the formula for one of the two variables, to get ...
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### Diophantine equation with binomial coefficient

Suppose that $p$ is a prime number and $p \le q \le p^2$ is an integer. How many solutions are there to the following equation? $$\binom{p^2}{q}-\binom{q}{p}=1$$ This question was proposed ...
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### Solve the following diophantic equations

I can't seem to find the solution to two problems in my textbook. They ask us to solve the diophantic equations: 1) $xy²-2y²-x-6=0$ $4x²-4xy+y²-9=0$ I tried several things but these two just ...
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### Solutions to $\lfloor x\rfloor\lfloor y\rfloor=x+y$

Find all solutions to $$\lfloor x\rfloor\lfloor y\rfloor=x+y$$ and show that the non-Integral solutions lie on two unique lines. Also determine the equations of these 2 lines. I divided the problem ...
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### Nature and number of solutions to $xy=x+y$

Find all solutions to $$xy=x+y$$ Initially the given condition was $x,y\in \Bbb{Z}$. In this case, I just guessed that the solutions were $(0,0)$ and $(2,2)$. As far as I can see, these are the ...
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### Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
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### Three Colour Analogue of Boolean Pythagorean Triples Problem

Having read about the Boolean Pythagorean Triples Problem (see here and this question), it occurred to me that a related problem would require the integers to be coloured in three rather than two ...
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### Positive integer solutions to $1\cdot m!\cdot(m^2)!\cdots (m^p)!=2(n^p)!$

Let $m$ and $n$ be natural numbers and $p$ a positive integer such that $$1\cdot m!\cdot(m^2)!\cdot\ldots\cdot(m^p)!=2(n^p)!$$ One solution is $(m,n,p)=(2,1,1)$. Are there any others?
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### Sum of the cubes of a Pythagorean triple equal a cube.

Apart from (3, 4, 5, 6) are there any more primitive solutions to $x^3+y^3+z^3=w^3$ where $x^2+y^2=z^2$ ? I’ve noted that if gcd(x ,y ,z) = k, then k divides w, so non-primitive Pythagorean triples ...
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### Finding all pairs of integers that satisfy a bilinear Diophantine equation

The problem asks to "find all pairs of integers $(x,y)$ that satisfy the equation $xy - 2x + 7y = 49$. So far, I've got \begin{align} xy - 2x + 7y &= 49 \\ x\left(y - 2\right) + 7 &= 49 \\ ...
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### Minimum of $|ax-by+c|$

Find the minimum of the function $$f(x,y)=|ax-by+c|$$ where $a,b,c \in \mathbb N$ and $x,y \in \mathbb Z$. The questions here and here are similar but they are in cases where $x, y$ are ...
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### Finding smallest positive value of a function.

Given four positive integers $A,B,C$ and $D$, we have to find the minimum absolute difference between $A+qC$ and $B+wD$ where $q$ and $w$ are non-negative integers. I know it has something to do with ...
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### Solution of diophantine equation with lowest c

Lets say I have a diophantine equation , aX - bY = c Now, for some (a,b,c) I may not have any integer solution at all. But lets say , I write the equation in this way , aX - bY = c + p p is an ...
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### The numbers that can be written as the sum of squares of two **natural** numbers [closed]

It's easy to solve for sum of two squares.but it becomes hard when we want numbers that can written as sum of squares of two natural number.For example given number $n$ can be written as the sum of ...
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### A bilinear diophantine problem

Suppose we know $a,b,c,d,e,f,m\in\Bbb Z$ in $$(a^2c+b^2d)y+ab(vy)+(a^2e+b^2f)v=m$$ how do we find $v,y\in\Bbb Z$?
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### Show that the equation $x^2+y^2+z^2=x^2y^2$ has no integer solution,except $x=y=z=0$

Show that the equation $x^2+y^2+z^2=x^2y^2$ has no integer solution,except $x=y=z=0.$ Let one of the $x,y,z$ be even number.Let $x=2p$ $x^2+y^2+z^2=x^2y^2$ This gives $y^2+z^2$ is also even,which ...
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### Find the value of $a$ if $x^2+y^2=axy$ has positive integer solution.

Find the value of $a$ if $x^2+y^2=axy$ has positive integer solution. My try: Let g.c.d of $x$ and $y$ is $d$ i.e.$(x,y)=d$ and let $x=dx',y=dy'.$ Then $x'^2+y'^2=ax'y'$ I am stuck here.The answer ...
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### If $1/x-1/y=1/z$, where $x,y,z$ are positive integers, prove that $xyz\gcd(x,y,z)$ is a perfect square

I found that $(z+y)(z-x)=z^2$ but I don't know how to continue
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### General Conic and its Rational Solutions

Suppose you have a rational conic $ax^2+bxy+cy^2+dx+ey+f=0$. There is a theorem that states if a conic has 1 rational solution it has infinitely many rational solutions. How can you prove this ...
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### Integers of the form $m^k-n^k$ [closed]

We know that an integer number is the difference of two squares if and only if it is not congruent to 2 mod 4. As a generalization, do we have a similar statement for integers of the form $m^k-n^k$, ...
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### Find all $(a,b) \in \Bbb Z^2$ such that $b \equiv 2a \pmod 5$ and $28a+10b=26$

I'm stuck with this exercise: Find all $(a,b) \in \Bbb Z^2$ such that $b \equiv 2a \pmod 5$ and $28a+10b=26$ It's from my algebra class, we are looking into diophantic and congruence equations. ...
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### Find all positive integers that solve Mordell's equation $y^2=x^3+37$

Find all Mordell's equation: $$y^2=x^3+k$$ where $k=37$ positive integer numbers,I can't find the when $k=37$ the mordell equation solution with some result,and we can known this equation have ...
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### Prove this diophantine equation $2^a-3^b=5~，a,b\in N^{+}$ has no postive integers solution

show that the diophantine equation $$2^a-3^b=5~~~~,a>5,b>3，a,b\in N^{+}$$ has no postive integers solution maybe is old problem,But I try somedays,can't solve it by now
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### Power Diophantine equation involving primes: $(p+q)^q-p^q-q^q+1=n^{p-q}$

Suppose $p$ and $q$ are prime numbers, and $n>1$ is a positive integer. Find all solutions to the following Diophantine equation:$$(p+q)^q-p^q-q^q+1=n^{p-q}$$ What I have tried: Obviously $p>q$...
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### Proving that an equation doesn't have integer solutions

I need to prove that there are no integer solutions for a bunch of equations like the following: $$15x^2 - 7y^2 = 9$$ I was able to solve some simpler ones by picking a dividend and looking into it's ...
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### Normalizing an elliptic curve to find integer solutions

I have an elliptic curve $$c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6$$ with integers $a_1,a_2,a_3,a_4,a_6,c_1,c_2$ and I would like to find all integer solutions of this elliptic curve. I ...
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### Find all integers $a,b,c$ that satisfy: $a^3 - 3a^2b - 3c+2b^2 = c^3 -3ab^2 + 3c^2 +1$

(From a math competition) Question: Find all integers $a,b,c$ that satisfy: $$a^3 - 3a^2b - 3c+2b^2 = c^3 -3ab^2 + 3c^2 +1$$ What I have tried/attempted basically I've been looking for ...
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### A quick method to solve $89y-273x=40$

how to solve this equation $$89y-273x=40$$ I saw this question somewhere and this obviously can be solved by hit and trial but is there an easier method to solve it, something more definite? I need ...
### Positive integers $a,b$ satisfying $a^3+a+1=3^b$
How to prove that $a=b=1$ is the only positive integer solution to the following Diophantine equation?$$a^3+a+1=3^b$$