Questions on finding integer/rational solutions of polynomial equations.

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5
votes
1answer
48 views

Pythagorean Triples : Is every positive integer $\gt$ $2$ part of at least one Pythagorean triple?

I was doing some basic number theory problems from Rosen and came across this problem: Show that every positive integer $\gt$ $2$ is part of at least one ...
5
votes
1answer
84 views

Pythagorean Triples : Show that exactly one of $x$, $y$, and $z$ is divisible by $5$

I was doing some basic number theory problems from Rosen and came across this problem: Show that if $(x, y,z)$ is a primitive Pythagorean triple, then exactly one of $x$, $y$, and $z$ is divisible ...
6
votes
3answers
99 views

Finding integer solutions of $m^2-n^5 = m - n$

How to list all integer solutions of $m^2-n^5 = m - n$ Here $m$ and $n$ are some positive integers. Also, I want to know the name of this type equations (if name exist). Regards Rosy
1
vote
1answer
26 views

If equation has integer solution it has solution for every prime p.

How to prove that if the equation in the form: $a_0 x_0^2 + a_1 x_1^2 + \dots + a_nx_n^2 = 0$ where $a_0, a_1, \dots , a_n \in \mathbb{Z}$, has an integer solution, then it has solution in ...
2
votes
0answers
42 views

How to find whole number answers in systems of square root equations

Given the following 4 equations, can you find 4 whole number answers using whole number variable inputs? $x,y,z$ where $x>y>z$ $Eq 1 = (x^2-2xy+y^2-2xz+z^2)^{\frac{1}{2}} $ $Eq 2 = ...
1
vote
1answer
69 views

Diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{3}{5}$

I'm trying to solve the diophantine equation $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{3}{5}$ for all $x,y,z \in \mathbb{Z}$. I did this. Since that the diophantine equation is symmetric, we can ...
1
vote
0answers
36 views

The Nagell-Ljunngren Equation

I have been trying to find the papers of Nagell and Ljunngren, which deal with the equation $$\frac{x^n - 1}{x - 1} = y^2$$ and solve it completely. Many papers cite these papers, but I haven't found ...
1
vote
3answers
72 views

Solve in non-negative integers: $m^2+n^2=1997 (m-n)$

Solve in $\mathbb{N}$:$$m^2+n^2=1997(m-n)$$ I try with quadratic equation or with factorising, but I have no idea what to do after that.
0
votes
0answers
28 views

Argument for finite solution of power Diophantione Equation.

Assume the equation $4x^3=y^2+3$ has infinite positive integer solution. If $x,y$ has general solution then it is clear that for any $x$(rational, integer), there is a $y$. It can be said there is a ...
0
votes
1answer
76 views

Integer solution to the equation

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation? $$x_1x_2...x_n+(2y+1)z+y=4m+3$$ where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...
0
votes
0answers
13 views

minimising multivariate quadratic function over integer variables

I have a quadratic function $x_1^2+x_2^2-(u_1x_1+u_2x_2)^2$ which I need to minimise over integer $x_1$ and $x_2$; also, the coefficients $u_1,u_2<1$. In other word, assuming coefficients ...
1
vote
1answer
41 views

How do I solve these questions using Diophantine equations?

I have been told that it is easier to solve the below 2 questions using Diophantine equations instead of simply trial and error. 1) Find the smallest positive integer which, when divided by 6, ...
4
votes
0answers
79 views

Solve $(x+1)^n-x^n=p^m$ in positive integers

Solve in positive integers: $$(x+1)^n-x^n=p^m$$ $p$ is prime, $n\ge 2$. Seemingly Zsigmondy's Theorem and LTE won't work here. Though you can tell (as suggested by user barto), using ...
0
votes
0answers
13 views

Diophantine equation involving three variavles

I wonder if it is possible to enumerated all the integer solutions of the Diophantine equation $ax+bxy^2=xz^2+4z$, with $a,b\in\mathbb{N}$. Of course we may solve it for $x$ and obtain rational ...
1
vote
2answers
59 views

Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
2
votes
0answers
62 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
2
votes
5answers
61 views

Possible solutions of a diophantine equation: $p^2+pq+275p+10q=2008$

What are couples of prime integers that verify this diophantine equation: $$p^2+pq+275p+10q=2008?$$ I tried to solve this equation trough the rules of modular-arithmetic. I rewrite the equation as: ...
0
votes
1answer
29 views

Solving a Diophantine Equation of the form $N(N-1) = 2X(X-1)$ for $N, X > 0$

When working on a problem on Project Euler I came up with a formula I need to solve: $N(N-1) = 2X(X-1)$ for $N > 10^{12}, X > 0$ with $N$ and $X$ being integer numbers. After some ...
1
vote
1answer
80 views

Diophantine equation $a^3 + b^3 + c^3 = 2$

I have a pretty difficult math question that I have no idea even how to begin. Here it goes: Find the nonzero integers $a$, $b$, $c$ such that $a^3 + b^3 + c^3 = 2$? I would assume that at ...
3
votes
0answers
147 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
3
votes
5answers
84 views

Can every perfect square exist as the sum or difference of two perfect squares?

I believe this is trivial and I'm over-complicating it. But can every squared integer be expressed as the sum of two squared integers OR the difference of two squared integers? And is there a proof ...
0
votes
1answer
32 views

Solutions $3 p\sin x - (p+\sin x)(p^2-p \sin x +\ sin ^{2} x) =1$

$3 p \sin x - (p+\sin x)(p^2-p \sin x + \sin ^{2} x) =1$ has a solution for $x$. Then number of integral solutions of $p$ are ?
3
votes
2answers
64 views

Faster Sage Code for Diophantine Equation? [closed]

I'm having trouble with the computation time. Does anyone have any ideas for faster code? ...
2
votes
3answers
127 views

Integer solutions to $x^{x-2}=y^{x-1}$

Find all $x,y \in \mathbb{Z}^+ $ such that $$x^{x-2}=y^{x-1}.$$ I can only find the following solutions: $x=1,2$. Are there any other solutions?
9
votes
1answer
77 views

Solving a Diophantine equation: $p^n+144=m^2$

I found this Diophantine equation: $$p^n+144=m^2$$ where $m$ and $n$ are integers and $p$ is a prime number. I solved it but I want to know if there exist other proofs through the use of rules of ...
1
vote
1answer
41 views

How many non-negative integral solutions?

How many non-negative integral solutions does this equation have? $$17x_{17}+16x_{16}+ \ldots +2x_{2}+x_1=18^2$$ I add some conditions that bring more limitations: $$\sum_{i=1}^{17}x_{i}=20 \quad 0 ...
0
votes
0answers
28 views

Diophantine eqution with a parameter

My question is about the problem when is the number $$\frac{m^3 + n^3}{n^2+m^2+m+n+c}$$ a natural number. Here $c\in \mathbb{N}$ is a constant and $m, n \in \mathbb{N}$ are the variables. This ...
3
votes
1answer
35 views

Bounding $x^2+6x$ between consecutive cubes when solving $y^3=x^2+6x$

I am familiar with the method of bounding a polynomial between consecutive squares to prove it is not a square. For example, this method can prove $y^2=x^2+x+1$ has no solutions since ...
0
votes
2answers
50 views

The diophantine equation $z^2=a^2+bx^2+cy^2$

Is there a way to obtain (enumerate) the integer solutions $(x,y,z)$ of the following quadratic Diophantine equation $z^2=a^2+bx^2+cy^2$ where $a$ is an integer and $b, c$ are positive integers? I ...
2
votes
2answers
46 views

A simple question:

let $a,b,c,d$ be all positive integers such that $a-bc \neq 0$,and $\gcd(a,b)=1$. Under what conditions, $(a-bc)$|$(a-b^d)$? In other words, does it exist any integer $k \neq 1 $ such ...
2
votes
4answers
309 views

Sum of square patterns

Can anyone give the name of this pattern $$136^2+137^2+138^2+139^2+140^2+141^2+142^2+143^2+144^2 =\\ 145^2+146^2+147^2+148^2+149^2+150^2+151^2+152^2$$
1
vote
3answers
48 views

How can i solve this diophantine equation:$x^2-(6p-4q)x+3pq=0$?

I found this diophantine equation $$x^2-(6p-4q)x+3pq=0$$ (p and q both prime numbers) and i posted my answer but i want to know if there are other methods to find the solutions of this equation. What ...
1
vote
1answer
19 views

Number of coins using Diophantine equation

I'd like to solve this question using Diophantine equations: We have an unknown number of coins. If you make 77 strings of them, you are 50 coins short; but if you make 78 strings, it is exact. ...
3
votes
2answers
52 views

Solving the diophantine equation $p^2+n-3=6^n+n^6$

What are the pairs ($p,n$) of non-negative integers where $p$ is a prime number, such that $$p^2+n-3=6^n+n^6$$ How can I solve this diophantine equation?
6
votes
2answers
205 views
+100

diophantine equation $x^3+x^2-16=2^y$

Solve in integers: $x^3+x^2-16=2^y$. my attempt: of course $y\ge 0$, then $2^y\ge 1$, so $x\ge 1$. for $y=0,1,2,3$ there is no good $x$. so $y\ge 4$ and we have equation $x^2(x+1)=16(2^z+1)$, ...
0
votes
0answers
27 views

Two variables, one equation

I have the equation: $(x + y - 1)(xy - 1) = 2015$. I solved this breaking up 2015 into its prime factors (5, 13, and 31). After a bunch of guess and check, I got that $x = 2$, and $y = -32$. Is ...
0
votes
1answer
26 views

3 Variables, One Equation

What triples (x, y, z) will satisfy the following equation?: $x^2$ + $y^2$ + $z^2$ = $7(x+y+z)$ I tried factoring the left side as $(x+y+z)^2 - 2xyz$, and I wasn't sure how to continue from there. ...
0
votes
3answers
39 views

Linear Diophantine Equation iff statement [closed]

Let $a,b,c$ be integers. For every integer $x_0$, there exist an integer $y_0$ such that $ax_0+by_0=c$. Determine conditions on $a,b,c$ such that the statement is true iff these conditions hold.
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vote
0answers
74 views

find all integer solutions of $y^2=x^3-2$ [duplicate]

I’m blind about integer solutions of a polynomial. I have no number theory background, but I’m curious about how to figure out all integer solutions of a polynomial, for example this question. It is ...
5
votes
0answers
81 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
0
votes
0answers
36 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?
3
votes
5answers
69 views

If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$, is $a=b=c=0$?

If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$ , then is it true that $a=b=c=0$ ? I was thinking of infinite descent but can't actually proceed , please help. Thanks in advance
1
vote
3answers
124 views

Solve $x^p + y^p = p^z$ when $p$ is prime

Find the solutions in positive integers of $x^p + y^p = p^z$, where $p$ is a prime number. Particular case $p=2$: For $z=0$ there are no solutions. For $z=1$ the only solution is $x=y=1$. For ...
3
votes
2answers
56 views

The diophantine equation $x^2+y^2=3z^2$

I tried to solve this question but without success: Find all the integer solutions of the equation: $x^2+y^2=3z^2$ I know that if the sum of two squares is divided by $3$ then the two numbers ...
3
votes
1answer
56 views

Partition problem for consecutive $k$th powers with equal sums (another family)

This is the partition problem as applied to a special set, namely the first $n$ $k$th powers. Assume the notation, $$[a_1,a_2,\dots,a_n]^k = a_1^k+a_2^k+\dots+a_n^k$$ I. Family 1 The following ...
4
votes
1answer
51 views

For a given positive integer $n>1$ , how to find all positive integers $s,t$ such that $n^s-(n-1)^t=1$ ?

For a given positive integer $n>1$ , how to find all positive integers $s,t$ such that $n^s-(n-1)^t=1$ ? $s=t=1$ is clearly a solution . One more thing is clear that for any such $s,t$ we must have ...
0
votes
0answers
30 views

A bivariate quadratic diophantine equation

Given $a,b,c>0$, is there a procedure to solve $(x,y)\in\Bbb Z:ax^2+by^2=c$ in $O(\log^d c)$ arithmetic operations (either randomized or deterministic) with $d>0$ being fixed? Is there a ...
1
vote
0answers
30 views

Diophantine linear Equation Gaussian Integers

We know that $ax+by=c$ with $gcd(a,b)=1$ could be solved over $\Bbb Z$. Supposing if $a,b,c\in\Bbb Z[i]$, is there an analogous framework to find $x,y\in\Bbb Z[i]$ (at least of minimum norms)?
0
votes
1answer
98 views

On a theorem of Kronecker! [closed]

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: ...
7
votes
2answers
152 views

Solve $x^3=y^2-y+1$ in positive integers.

I recently started doing number theory and have finished with all the basic, intermediate and some of the advanced stuff with ease. However, I encountered this question and have been stuck for about a ...