Questions on finding integer/rational solutions of polynomial equations.

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16 views

quadratic diophantine's equation in form of $y=ax^2+bx+c$

I stumbled on this on Geogebra. Actually i would like to set integers pair $x$ $y$ that fits the general quadratic form. Given $(x_1,y_1)$ and $(x_2,y_2)$ are integers pairs, i am looking for set ...
0
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2answers
47 views

$x^2+y^2=N$, Diophantine equation

$$ x^2+y^2=N $$ $N$ integer, Find $x,y$ integer so that the Diophantine equation is fulfilled. If $N$ is a prime number, we can calculate all solutions very fast via Gauß reduction. Is ...
0
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0answers
32 views

Find all pairs of positive integers $(x,y)$ : $x(x+1) = y(y+1)(y+2)$

Find all pairs of positive integers $(x,y)$ : $$x(x+1) = y(y+1)(y+2)$$ I was able to find only two pairs: $(2,1)$ and $(14,5)$ and looks like no more exists. How to prove it?
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1answer
31 views

Prove that the solutions to the system of equations are integers

Let $a, b \in \mathbb{Z}$ and consider the system of equations below: $$\begin{cases} y -2x-a =0\\ y^2-xy+x^2-b=0\end{cases} $$ Prove that $x,y\in\mathbb{Q}$ implies $x,y\in\mathbb{Z}$. I ...
3
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2answers
51 views

Can $a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$ be an integer if $x$, $y$, and $z$ are not squares?

Let $\gcd(x,y,z)=1$.Can we find 3 non-perfect squares $x,y,z\in \mathbb{Z},$ such that $a \in \mathbb{Z} \geq 2$ $$a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$$ ...
7
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1answer
90 views

The Diophantine Equation $x^2+y^4=2z^4$

We know that the Diophantine equation $x^2+y^4=2z^4$ has infinitely many solutions . Some of them are shown below $$(y,z)=(1,1),(1,13),(1343,1525),(2372159,2165017).$$ I investigated the ratio of ...
4
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1answer
78 views

How many $s,t,u$ satisfy: $s +2t+3u +\ldots = n$?

Given $n\in \mathbb{N}^+$, what is the possible number of combinations $s,t,u,\ldots\in\mathbb{N}$, such that: $$s +2t+3u +\ldots = n\quad?$$ Additionally, is there an efficient way to find ...
5
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2answers
66 views

Integral solutions to $56u^2 + 12 u + 1 = w^3.$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$ This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 ...
4
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3answers
293 views

Just a 3rd grade math problem in my country. Please help.

Even though this is a 3rd-grade math problem, people found it extremely hard. Any people have a solution, or algorithm is welcome. I'll try make a program base on the algorithm and see if it's ...
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1answer
25 views

Solving the Diophantine equation $x^2-y^2=a$, $\{x,y,a \in \mathbb Z^+\}$

the diophantine equation is a simple case divider equation or Pell's equation?
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1answer
30 views

Diophantine equations in $\Bbb Z$ [duplicate]

$x + 2y + 3z = 4$ $w = x + 2\times y$, then the equation becomes $w + 3z = 4$. $\gcd(1, 3) = 1 | 4$, so this two variable equation is solvable. $w = -2, y = 2$ i can't seem to pass this point
3
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1answer
66 views

Solving a Diophantine equation3

The Diophantine equation that I have to solve is: $$343x^2-27y^2=1$$ This question has already been posted by other user but it has not received an answer. I proved to solve it. This is my attempt: ...
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3answers
53 views

Is the following inequality true $(a^3-b^6)^3+(3abc)^3 \leq (a^3-b^6+3cb^3)^3$?

Let $a,b,c$ be all positive integers greater than $1$. If $$a>b^2$$ and $$a^3-b^6> 3c$$ Is this the following inequality true?: $$(a^3-b^6)^3+(3abc)^3 \leq (a^3-b^6+3cb^3)^3$$ I tried to ...
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0answers
29 views

Find all solutions to the Diophantine equation or show that none exist [duplicate]

The equation is $17x^4 + 5y^4 = 35z^4$ I reduced $\pmod 5$ but that just told me $x$ has to be a multiple of $5$. Not sure where to go from here. Any help would be appreciated. I've only taken an ...
7
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2answers
157 views

Integer solutions of $x^3 = 7y^3 + 6 y^2+2 y$?

Does the equation $$x^3 = 7y^3 + 6 y^2+2 y\tag{1}$$ have any positive integer solutions? This is equivalent to a conjecture about OEIS sequence A245624. Maple tells me this is a curve of genus $1$, ...
0
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0answers
74 views

How to show that the equation $7^3d^2-3^3c^2=1$ has infinitely many integer solutions ?

How to show that the equation $7^3d^2-3^3c^2=1$ has infinitely many integer solutions ? Please help . Thanks in advance
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2answers
56 views

Does the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ?

How to show that the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ? ( If this can be shown then solutions of $12x^2-8y^2=4$ give infinitely many powerful numbers ...
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1answer
45 views

Cubic Congruence Solutions

While I was reading a paper on number theory, there was a claim which wasn't prove there and I couldn't find a way to justify it. The claim is as follows For a prime $p$, when $p\nmid a$, the number ...
4
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4answers
738 views

Correct statement of Fermat's Last Theorem

I'm looking at the wikipedia page on Fermat's Last Theorem In the statement it requires $a,b,c$ to be positive integers. Is that correct? I always took it to be no solutions in non-zero integers. ...
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1answer
71 views

Solving to find the general equation with a “mod” equation

They probably aren't called "mod" equation but i couldn't think how else to word them, so I have this equation $8x + 10y ≡ 8 \pmod 7$ And have been tasked with finding the general solution, I know ...
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1answer
21 views

Integer solutions of a degree 3 curve

Suppose you have a square pyramid made out of rigid balls and all these balls are equal. Suppose now that you want to fill a square with the same number of balls that the pyramid is made. If $x$ ...
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0answers
22 views

Would my paper be considered Diophantine?

From Wikipedia In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is ...
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2answers
46 views

Types of elliptic curves

I'm trying to research elliptic curves, and I always get the generic equation $$y^2 = a_0 x^3 + a_1 x^2 + a_2 x + a_3.$$ However, I'm looking for information on an equation like $$y^3 = a_0 x^3 + a_1 ...
3
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1answer
38 views

Solving Mordell Equations

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
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0answers
47 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
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3answers
62 views

Solve the equation $2xy+2x-5y=40$, if $x$and $y$ are whole numbers. [closed]

Solve the equation $2xy+2x-5y=40$, if $x$ and $y$ are whole numbers. Could anyone give me a step by step answer?
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1answer
74 views

equation $x^4 + y^4 = z^4$

Diophantine equations that are insoluble in $\mathbb{Z}$ may become soluble in finite integral domains. Show that \begin{equation*} x^4 + y^4 = z^4 \end{equation*} is soluble (as a congruence) in ...
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2answers
58 views

How to solve 3 variable in 2 equation?

This paper is abstracted from 2007 British Mathematics Olympiad Round 1 Question 2. I am currently practicing grade 8 (Singapore Secondary 2) for the upcoming Singapore Mathematics Olympiad(SMO). ...
4
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1answer
79 views

30th problem of the fifth book of Diophantus;

Is there a complete answer to this problem? I have found Saunderson's answer, but I believe it is missing a few answers. The problem states: $a^2+b^2=d^2 \\ a^2+c^2=e^2 \\ b^2+c^2=f^2$ Saunderson ...
2
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0answers
40 views

Diophantine eqution with odd prime

HOW to find all possible set of solutions of an equation type $y^p \pm 2 = x^2$, where $p$ is any odd prime High regards to one and all
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0answers
58 views

Determine all natural numbers n and m that satisfying in this equation. [closed]

I'm trying to solve the following question: Determine all natural numbers n and m such that: $$n ^ { n ^ n } = m^m.$$ I don't have any idea about this question. Can somebody help me or give ...
2
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1answer
51 views

Solvability of the equation $2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?

As a natural extension of the question titled Solvability of $a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?, I wonder if the equation $$2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + ...
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0answers
71 views

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

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 that $z^4-x_i^4$ is divisible by $d_4=5^4$. Additionally, Ward ...
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1answer
48 views

Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square.

I successfully thought of bounding our expression examining consecutive squares that attain values close to it, and this led to the solution I'll post as an answer, which was the one reported. ...
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2answers
46 views

Solving the equation in natural numbers

How can I find the solutions in natural numbers for the following equation? $$a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=b$$ Where $x_{1},...,x_{n}$ are unknown. I want to find the whole of solutions ...
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2answers
57 views

Solvability of $a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?

Aware of a Darmon-Merel theorem that asserts that if $n \geq 5$ is prime then the equation $a_{1}^{2} = a_{2}^{n} + a_{3}^{n}$ has no solution in relatively prime integers $a_{1}, a_{2}, a_{3},$ I ...
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0answers
44 views

prove that the number of solutions is finite

Prove that \begin{equation*} \frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\cdots +\frac{1}{x_n}=1,~∀i,x_i\in \mathbb{Z^+} \end{equation*} has a finite number of integer solutions. I tried to solve ...
4
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2answers
79 views

Find a non-trivial solution for the Diophantine Equation $17a^4 + 5b^4 = 35c^4$, or show that no non-trivial solutions exist

This is a problem on my practice exam for number theory, and we haven't had an example like this in class yet. The question is looking for a solution in $\mathbb{Z}$ for $a,b,c \in \mathbb{Z}$. I've ...
4
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1answer
63 views

Solve in positive integers: $5x^2+6x^3=z^3$

Solve in positive integers: $5x^2+6x^3=z^3$. $x^2(6x+5)=z^3$ If $(x,5)=5$, let $x=5k$. So $k^2(6k+1)=\left(\frac{z}{5}\right)^3$, we're left with solving $6n^3+1=m^3$. If $(x,5)=1$, ...
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0answers
41 views

A nice set of squares. [duplicate]

Are there integers $a, b, c, d$ such that $$a^2+b^2=c^2$$ $$a^2-b^2 = d^2?$$ I have tried by showing that $a^2 = b^2 + d^2$ and thus $a^2+ b^2 = 2b^2+d^2 = c^2$ But how do I show that there are no ...
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1answer
32 views

Exponential diophantine equation

Need some help regarding the equation $$2^a-3^b=(2^c-1)\cdot d >0$$ where $a,b,c,d$ are integers; $a,b$ are fixed; and $c>2$. Can we show that $c,d$ exist? Thank you!
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2answers
50 views

Show that there are exactly 16 pairs of integers $(x,y)$ such that $11x+8y+17=xy$.

Original problem Show that there are exactly 16 pairs of integers $(x,y)$ such that $11x+8y+17=xy$. My work From case by case analysis I come to know that the equation will hold if and only if $x$ ...
3
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3answers
37 views

Split 16 Consecutive Integers into Two Subsets of 8 Integers

Show that any given set of sixteen consecutive integers {$x+1,x+2,\ldots,x+16$} can be divided into two eight element subsets with the properties that they have the same sum, the sums of the squares ...
6
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2answers
94 views

Only finitely many $a, b$ such that $2+3^n+5^{n^2}=2^a7^b$ for some $n$?

Let $(a,b)$ be a pair of positive integers such that $$2+3^n+5^{n^2}=2^a7^b$$ for some positive integer $n$. Is it true that there are only finitely many such pairs? I don't know the answer to such ...
0
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0answers
21 views

Solving the inverse loop

I was making playing around with the solutions of the equation of a circle in the origin $\{ (x^2)+(y^2) = (r^2) \mid\text{ solutions }y = \pm((r^2)-(x^2))^{1/2} \}$, when I tried to find the inverse ...
4
votes
2answers
74 views

$2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$

How do I show that the diophantine equation $2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$
1
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1answer
74 views

Trouble forming general solution for linear congruence

I was given $$ 6x+14y=4 \space \mod 5 $$ I took this approach: $$ 6x+14y-5z=4, \space \text{ for some } z $$ Let $$ w=\frac{6}{(6,14)}x+\frac{14}{(6,14)}y $$ Then, $$ (6,14)w+5z=4 \quad , \quad ...
12
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2answers
151 views

Diophantine equation $x^2 + xy + y^2 = \left({{x+y}\over{3}} + 1\right)^3$.

Solve in integers the equation$$x^2 + xy + y^2 = \left({{x+y}\over3} + 1\right)^3.$$
1
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1answer
72 views

Solving a Diophantine equation: $y^x=x^{2007}$, $x$ and $y$ integers.

I found this Diophantine equation and to solve it I used the definition of logarithm but the solution doesn't require the use of logarithmic rules. I solved it in this way: $$y^x=x^{2007}$$ ...
1
vote
1answer
35 views

Splitting Field of a real cubic

Let $f(X)$ be a cubic with 3 real roots, integer coefficients irreducible over $\mathbb{Q}$. Let $\alpha$ be one of these roots, and consider the number field $\mathbb{Q}(\alpha)$. Dirichlet's unit ...