Questions on finding integer/rational solutions of polynomial equations.

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0
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4answers
53 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 ...
4
votes
3answers
308 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 ...
-6
votes
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?
7
votes
1answer
91 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 ...
0
votes
0answers
49 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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?
4
votes
1answer
79 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 ...
3
votes
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 $a \in \mathbb{Z} \geq 2$ $$a=\left(\sqrt{2(\sqrt{y}+\sqrt{z})(\sqrt{x}+\sqrt{z})}-\sqrt{y}-\sqrt{z}\right)^2$$ ...
5
votes
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 ...
7
votes
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 $1$, ...
3
votes
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: ...
-1
votes
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?
0
votes
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
0
votes
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
1
vote
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 ...
15
votes
5answers
1k views

Linear diophantine equation $100x - 23y = -19$

I need help with this equation: $$100x - 23y = -19.$$ When I plug this into Wolfram|Alpha, one of the integer solutions is $x = 23n + 12$ where $n$ is a subset of all the integers, but I can't seem ...
0
votes
1answer
42 views

Linear diophantine equation $97y-299x=10$

Here is my equation: $$97y-299x=10$$ I tried to solve like this: $$-299 =-3\cdot97-8$$ $$97=-12\cdot-8+1$$ $$-8=-8\cdot1+0$$ I'm not sure if I am correct or can I ignore the negative signs?
0
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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 ...
1
vote
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 ...
2
votes
1answer
376 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 + ...
17
votes
2answers
369 views

Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?

Are there solutions in integers $a,b>1$ to the following simultaneous congruences? $$ a^4\equiv 1 \pmod{b^2} \quad \mathrm{and} \quad b^4\equiv1 \pmod{a^2} $$ A brute-force search didn't turn up ...
4
votes
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. ...
3
votes
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 ...
1
vote
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 ...
2
votes
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 ...
2
votes
7answers
291 views

Find all integral solutions to $a+b+c=abc$.

Find all integral solutions of the equation $a+b+c=abc$. Is $\{a,b,c\}=\{1,2,3\}$ the only solution? I've tried by taking $a,b,c=1,2,3$.
0
votes
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 $x$ ...
0
votes
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 ...
3
votes
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 ...
1
vote
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 ...
11
votes
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 ...
8
votes
1answer
382 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
4
votes
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 ...
0
votes
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 ...
1
vote
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). ...
2
votes
2answers
311 views

Prove that there are infinitely many integer solutions to a diophantine equation

Prove that there are infinitely many integer solutions to the diophantine equation: $(x-y)^7 = x^3y^3$
9
votes
2answers
119 views

How to prove the cubic formula without root extraction

I'm trying to prove the cubic formula, in the following form: Given a field $F$ and $x,p,q\in F$, define $m=\frac p3$ and $n=\frac q2$, and suppose also that $\gamma,\tau$ are given such that ...
2
votes
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
3
votes
0answers
48 views

A question on the Pell equation $x^2-pqy^2 = -1$, with prime $p,q$.

We know that a necessary but not sufficient condition such that, $$x^2-dy^2 = -1\tag1$$ is solvable is that $d$ is not divisible by a prime of form $4m+3$. It is not sufficient because the prime ...
2
votes
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} + ...
45
votes
3answers
709 views

Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$

When I was playing with numbers, I found that there are many triplets of three positive integers $(a,b,c)$ such that $\color{red}{2\le} a\le b\le c$ $\sqrt{abc}\in\mathbb N$ $\sqrt{abc}$ divides ...
0
votes
1answer
49 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. ...
0
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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 ...
6
votes
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 ...
4
votes
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 $(x,5)=1$, ...