Questions on finding integer/rational solutions of polynomial equations.

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Generalization of a Diophantine Equation Problem

I've been working a lot with Pythagorean triples and sums of squares that are themselves squares, specifically interlocking sums (where one square is part of two or more sums). As part of my work I ...
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24 views

What are the general solutions of the Diophantine equation $ ax+by+cxy+d=0 $

Does the diophantine equation $$ ax+by+cxy+d=0 $$ always have solutions ?
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30 views

Integer solution for $Rx^2+Sy^2=1$ .

Is there any integer solution in-terms of $R,S$ for the equation $Rx^2+Sy^2=1$ , . For example $(\frac{1}{\sqrt {2R}},\frac{1}{\sqrt {2S}})$ is a solution but not integer solution . Is there any ...
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3answers
85 views

For what $a,b$ such that $ax^2+by^2 = z^2$?

This post made me think about this question. What is the criterion on positive integer $a,b$ such that, $$ax^2+by^2 = z^2$$ can be solved in positive integers $x,y,z$? (Three broad classes are: 1) ...
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1answer
90 views

If $a,b > 1$ and $r>2$ does $ax^2+by^2=z^r$ have any rational solutions?

I have been trying to solve the following equation for months without much success. It has been so far a very frustrating endeavor.Please help. Consider the diophantine equation: $x^2+y^2=z^r$ where ...
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80 views

Equation $a^5+15ab+b^5=1$

What are the integer solutions of $a^5+15ab+b^5=1$? The equation is symmetric in $a$ and $b$, so let's assume $a\geq b$. When $a=b$, we have $2a^5+15a^2=1$, which has no solution by the Rational Root ...
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36 views

$8x +9y = 5$ where $x,y \in \mathbb{Z}$

Solve the following Diophantine equation algebaically: $$8x+9y=5$$ Give 3 possible solutions for the equation I have the following: The Diophantine equation has solutions $x,y \iff ...
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57 views

What is stopping every Mordell equation from having a [truly] elementary proof?

The Mordell equation is the Diophantine equation $$Y^2 = X^3-k \tag{1}$$ where $k$ is a given integer. There is no known single method — elementary or otherwise — to solve equation $(1)$ for all $k$, ...
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71 views

Positive integer solutions to $a^{a^a}=b^b$

What are all positive integer solutions to $a^{a^a}=b^b$? $(a,b)=(1,1)$ works. If we take log on both sides, we get $a^a\log a=b\log b$, which is still hard to analyze. (It helps in equations like ...
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26 views

Is there an $n$ such that $p|n^2+1$ with $2n<p<2n+\sqrt n$?

Is there an integer $n$ such that $n^2+1$ is divisible by a prime $p$ with $2n<p<2n+\sqrt n$? It's complicated to describe my interest, but these are near-missed for arc-cotangent reducible ...
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216 views

What are the necessary and sufficient conditions for a cubic equation to have integers roots

Let's start with Fermat equation with the lowest power, $x^3 + y^3 = z^3$. Now let's set $y = x + a, z = x + b$ with $b > a$ and $a,b$ integers. then the equation becomes $$x^3 + (3a-3b)x^2 + ...
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76 views

Prove there are no non-trivial solution to $3x^2 - 5y^2 + 7z^2 = 0$

I've tried using modulo $3$, and I get it down to $y^2 + z^2 = 0 \pmod 3$ ; I don't know where to go from here though. I justified my answer by stating that, because we're in $\pmod 3$ and we ...
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5answers
180 views

When does $x^3+y^3=kz^2$?

For which integers $k$ does $$ x^3+y^3=kz^2 $$ have a solution with $z\ne0$ and $\gcd(x,y)=1$? Is there a technique for counting the number of solutions for a given $k$?
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130 views

how do you solve $a^2+b^2+c^2=d^3$

let $ a,b,c,d$ be 4 integers such that $\gcd(a,b,c,d)=1$. How do you find the integral solutions of the equation: $$a^2+b^2+c^2=d^3$$
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28 views

Does this “distribution of factors” cover all possibilities?

I have the Diophantine equation $$3a^2(4a^2+1)=b(b+1). \tag{$\star$}$$ Each side can evidently be “separated” into two [integer] factors as $$3a^2 \cdot (4a^2+1) = b \cdot (b+1).$$ Now I believe I ...
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89 views

An annoying Pell-like equation related to a binary quadratic form problem

Let $A,B,C,D$ be integers such that $AD-BC= 1 $ and $ A+D = -1 $. Show by elementary means that the Diophantine equation $$\bigl[2Bx + (D-A) y\bigr] ^ 2 + 3y^2 = 4|B|$$ has an integer ...
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56 views

Diophantine $7^a+2=3^b$

I want to find the solutions $(a,b)\in\mathbb{Z}^+\times\mathbb{Z}^+$ of $7^a+2=3^b$. One such solution is $(a,b)=(1,2)$. Looking modulo $4$, we have $(-1)^a+2\equiv(-1)^b$, so $a$ and $b$ are of ...
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226 views

At which p-adic fields does the equation have no solution?

I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't have, I have to find at which p-adic fields it has ...
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34 views

Diophantine Equation 1

I want to solve for positive integral values of $x$ and $y$: $$1216562x=87654321y+a$$ Here $a$ is a positive integer. For example if $a=40642509$ then one solution is : $x=37716$ and $y=523$ How do I ...
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81 views

Why doesn't the equation have a solution in $\mathbb{Q}_2$?

I have to find for which primes $p$, the equation $x^2+y^2=3z^2$ has a rational point in $\mathbb{Q}_p$. According to my notes: Obviously, $\forall p \in \mathbb{P}, p \nmid 2 \cdot 3$, there is a ...
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Undecidebility in Number Theory [duplicate]

Recently one of my teachers says that it is not impossible that we find a problem in number theory that is undecidable in usual system of set theory. This was so wonderful for me. When I say this ...
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85 views

What are the integer solutions of the system $a^2+b^2=c^2$, $a^3+b^3+c^3=d^3$?

How to solve these equations to find the integer numbers (a, b, c, and d)? $$a^2+b^2=c^2\tag{1}$$ $$a^3+b^3+c^3=d^3\tag{2}$$ I know one of solutions which is $a=3, b=4, c=5, ...
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Why do we conclude that $(a,b)=1$, having found that $(a',b'=1)$?

Suppose that we have the equation $ax^2+by^2+cz^2=0, a,b,c \in \mathbb{Q}$. Without loss of generality, we suppose that $gcd(a,b,c)=1$. Also, we can consider that $a,b,c$ are square-free. We can ...
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176 views

Does the equation has a non-trivial solution?

Could you give me some hints how I can solve the following exercise? Check if the equation $3x^2+7y^2-5z^2=0$ has a non-trivial solution in $\mathbb{Q}$ . If it has a solution, find at least one. If ...
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1answer
21 views

Infinite geometric series whose coefficients correspond to the number of solutions of a Diophantine equation.

This is problem 2 from Polya's Problems and Theorems in Analysis I. The question is as follows, Let $A_n$ denote the number of solutions to the Diophantine equation $x+5y+10z+25u+50v=n$. What is the ...
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When $Ax^2+By^2=z^2$ has a solution in integers?

Consider the Diophantine equation $Ax^2+By^2=z^2$, with positive integer parameters $A$ and $B$ (not necessarily square-free or co-prime). When does this equation have a non-trivial solution? Can one ...
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2answers
71 views

Existence of integer solution to 63x+70y+15z=2010

I have an equation $63x+70y+15z=2010$. The question asks me to conclude whether it has an integral solution or not? Any help on how to proceed?
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1answer
34 views

Diophantine solution set for $\frac{n(n-1)}2 = b(b-1)$

By Diophantine solution set I mean solutions where n and b are integers. I have one solution I found by trial and error but ...
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16 views

Interval in future

I have two intervals (times). For example, t1= 0:17, in this interval, we are now and interval t2=0:12, and the time, when was the time "time2". time2WasBefore = 0:04( which means that t2 was from now ...
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Prove that $ x^xy^y=z^z $ has infinite integral solutions [duplicate]

Show that there exist an infinite number of solutions for $$ x^xy^y=z^z $$ where $x,y,z \gt 1$ & $x,y,z\in \mathbb Z$ I don't know how to even start,infact I am not able to find a particular ...
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25 views

Solve this Simple quadratic equation $cU^2-2(a+ b)U+2(a-b)V- cV^2=0$

I need help solving this symmetrical quadratic equation where $\gcd(a,b,c)=1$$cU^2-2(a+ b)U+2(a-b)V- cV^2=0$ Is there an easier method than the quadratic formula? Any hint?
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92 views

Solve $3x^2-y^2=2$ for Integers

If $x$ and $y$ are integers, then solve (using elementary methods) $$3x^2-y^2=2$$ I tried the following If $y$ is even, then $4|y^2$ and hence $2|y^2+2$ (and $4$ doesn't divide it), but ...
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9 views

Parametrization of quadratic (diophantine) equations with a nth power

Is it always the case that the general solutions can be readily found if the primitive ones are known? If so, can this be applied to $ax^2+by^2=cz^n$ if the primitive solutions of $x^2+y^2=c^r$ are ...
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56 views

solve this simple equation:$ax^2+byx+c=0$

I need help solving the diophantine equation:$$ax^2+bxy+c=0$$ The quadratic formula does not seem to help much. Please help.
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how do you find the parametrization of $ax^2+by^2=z^r$?

how do you find the parametrization of $ax^2+by^2=z^r$ if a non-trivial solution $(x_0, y_0, z_0)$ is known? Any hint?
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84 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
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1answer
40 views

Coprime numbers and equations

Suppose $~m~$ and $~n~$ are coprime and both of them are greater than one. Is it right that equation $~mx + ny = (m-1)(n-1)~$ has solutions over non-negative integers? For example $~ (x,y) = (6,0) ...
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Parametrization of $ax^2+bxy+c=0$

Can I just fix $y=t$ and use quadratic formula to get the rational points of the diophantine $$ax^2+bxy+c=0?$$ or is there another method? I feel like I am turning in circles with the quadratic ...
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105 views

Rational points of $ax^2+by^2=z^r$, $r $ odd integer.

I am trying to find the rational points of:$$ax^2+by^2=z^r$$ I am aware that:$$(u^r-2^{r-2}v^r)^2+(2uv)^r=(u^r+2^{r-2}v^r)^2$$ How can I deduct the results?
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185 views

$2 \times 3 = 5+1$ and $2+3 = 5 \times 1$. When else can we switch the operators like this? [duplicate]

I noticed the following: $$2 \times 3 = 5+1$$ If you switch the operators, it is still true: $$2+3 = 5 \times 1$$ There is another obvious/trivial example where you can swap the operators: $$2\times 2 ...
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70 views

integral solutions of $ ax^2+by^2=c$ [closed]

Let $a,b,c,x,y$ be all non-zero positive integers, $\gcd(a,b,c)=1$, find the integral solutions of:$$ ax^2+by^2=c$$ Any hint? Can I use Euler's identity to get the solutions?
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61 views

If $(a^2+b^2) \mid (c^2+d^2)$ and $\gcd(a,b)=\gcd(c,d)=1$ and $\gcd(a,c)>1$, what can be said about the components?

While working on a divisibility problem in integers $a,b,c,d$, with $\gcd(a,b)=\gcd(c,d)=1$, I've come up against the hypothetical condition $$ (a^2+b^2) \mid (c^2+d^2), \tag{$\star$} $$ where, also ...
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275 views

Positive integer solutions to $x^4+y^7=z^9$

A while ago, a maths teacher gave me this problem: find solutions to $x^4+y^7=z^9$ with $x,y,z>0$. I found $(2^{56})^4+(2^{32})^7=(2^{25})^9$. In general, if $k=8+9l$ then ...
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72 views

Can Legendre's theorem really help solve this equation: $ax^2+by^2=cz^2$?

let $a,b,c,x,y$ be non-zero positive integers such that $$\gcd(x,y,z)=1$$ find all the non-trivial integral solutions of the diophantine equation:$$ax^2+by^2=cz^2$$ I know that the Legendre's theorem ...
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How many natural solutions does the equation $x^2 - c y^2 = 1$ have?

How many natural solutions has equation $x^2-cy^2=1$ depends on value of $c$ . I think I've seen this problem somewhere as a theorem but I can't remember where .
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51 views

List one of the ways in which Mario could buy the stars and comets. Note: Mario needs to spend all of his gold coins

Mario has 773500 gold coins to purchase a number of stars and comets. Each star costs 299 gold coins, and each comet costs 208 gold coins. If the number of stars that Mario buys is at least twice the ...
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1answer
39 views

Linear Diophantine Equation Question

Mario has $773500$ gold coins to purchase a number of stars and comets. Each star costs $299$ gold coins, and each comet costs $208$ gold coins. If the number of stars that Mario buys is at least ...
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1answer
87 views

Find all positive integers s.t. $10^m-8^n=2m^2$

Find all pairs of positive integers $(m,n)$ such that $10^m-8^n=2m^2$
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2answers
101 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
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29 views

Can you write a variable as the sum of two variables?

I was seeing this question and, in the develop of an answer, a question arised: I have a variable $k\in [1,40]_{\Bbb N}$ and I want write it as the sum of one variable and something more with the ...