Questions on finding integer/rational solutions of equations.

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8
votes
4answers
284 views

Math contest integer triplet problem

Can any one help me with this? Determine all integer triples (x,y,z) such that 1 ≤ x ≤ y ≤ z and x + y + z + xy + yz + xz = xyz − 1. I thought of Vietta's formula but don't let me lead you into a ...
4
votes
1answer
91 views

On seventh powers $x_1^7+x_2^7+\dots+x_n^7 = 2$?

We have, $$(-6m^3 + 1)^3 + (6m^3 + 1)^3 + (-6m^2)^3 = 2$$ $$(-8m^5 + 1)^5 + (8m^5 + 1)^5 + (-8m^6 + 2m)^5 + (-8m^6 - 2m)^5 + 2(8m^6)^5=2$$ The first identity has been long known, while the second ...
4
votes
0answers
153 views

Modified Pell equation: $x^2-D y^2 = m$, $m\neq1$.

How does one solve the Diophantine equation $$ x^2-Dy^2=m, $$ where $m$ is some fixed arbitrary integer? I understand that given the fundamental solution to $r^2-D s^2=1$, and any solution to the ...
4
votes
2answers
91 views

Is the assertion about the form $\alpha x+\beta xy+\gamma y$ true?

In my answer, I was led to conjecture the following: Statement: If $\gcd(\alpha,\beta,\gamma)=1,$ then every integer can be written as $\alpha x+\beta xy+\gamma y$ for integer $x$ and $y$. ...
0
votes
1answer
56 views

A diophantine equation related to primes.

I have $2$ prime numbers $p_1$ and $p_2$. I have to find the solution of $\large{p_1t_1+p_2t_2=1}$ where $t_1$ and $t_2$ are integers. How do I do this?
14
votes
2answers
230 views

Solve $(a^2-1)(b^2-1)=\frac{1}4 ,a,b\in \mathbb Q$

Does the equation $(a^2-1)(b^2-1)=\dfrac{1}4$ have solutions $a,b\in \mathbb Q$? I search $0<p<1000,0<q<1000$, where $a=\dfrac{p}q$, but no solutions exist. I wonder is this equation ...
2
votes
4answers
88 views

Is there any solution to the following system of equations?

Is there any solution to the following system of diophantine equations? $$ \left\{\begin{array}{l} 2.a^2 = b^2+c^2+d^2 \\ a^2 = e^2+f^2+g^2 , & \mbox{with }((a,b,c,d,e,f,g)>2)\in N\mbox{ and ...
4
votes
1answer
138 views

Does $a^6+b^6 = c^6+d^3$ have a non-trivial solution?

It is conjectured that, $$x_1^8+x_2^8+x_3^8 = y_1^8+y_2^8+y_3^8\tag{1}$$ has no non-trivial solutions. However, if we relax it a bit then, $$x_1^8+x_2^8+x_3^4 = y_1^8+y_2^8+y_3^4\tag{2}$$ can be ...
2
votes
0answers
49 views

Abelian SubGroup Variant:

Consider the following problem: Find integers $x_1, x_2, x_3,\dots, x_n$ Such that: $$P(x_1,x_2,\dots, x_n) = Q$$ for some integer $Q$ and polynomial $P$ where for all permutations of any set of ...
12
votes
2answers
689 views

Diophantine equation involving prime numbers : $p^3 - q^5 = (p+q)^2$

Find all pairs of prime nummbers $p,q$ such that $p^3 - q^5 = (p+q)^2$. It's obvious that $p>q$ and $q=2$ doesn't work, then both $p,q$ are odd. Assuming $p = q + 2k$ we conclude, by the equation, ...
8
votes
2answers
185 views

Solve $x^2+y^2=2$ for $x,y\in\mathbb Q$.

Solve $x^2+y^2=2$ for $x,y\in\mathbb Q$. I think the answer should be in terms of 1 integer variable $\in\mathbb Z$ only. I rewrite the equation to $(x+y)^2+(x-y)^2=2^2$, then by the formula of ...
5
votes
1answer
915 views

The Diophantine equation $x^2 + 2 = y^3$

How to solve the Diophantine equation $x^2 + 2 = y^3$ with $x,y>0$ ? ($x,y$ are integers.)
0
votes
1answer
52 views

Intersection of two series

I'm looking for the solutions to $$\begin{eqnarray} z &=& x_0 + i \Delta_x \\ z &=& y_0 + j \Delta_y \\ i &\ge& 0\\ j &\ge& 0 \end{eqnarray}$$ Well, I know that ...
4
votes
1answer
383 views

On an exponential diophantine equation

I am trying to find all integer solutions of $5^x + 12 ^y$ = $13^z$. The obvious (and pursued) solution is $(2, 2, 2)$, and no others. I've tried to use an appropriate modular arithmetic, but to no ...
7
votes
1answer
105 views

On the Pell-like $Ax^2-By^2 = 1$

This is connected to the post, Mere coincidence? (prime factors). I was looking at NeuroFuzzy's dataset and noticed the line, {{{1, {4, 2}}, {1, 4, 2, 4, 2}, 23762}} It seems this could be ...
1
vote
1answer
231 views

Linear equation with prime coefficient.

Suppose we have a linear equation with two variables say $x$ and $y$ and three integer coefficient $a , b$ and $c$ (constant), where $a$ and $b$ are prime all are greater than zero. $ax+by=c$ how ...
8
votes
3answers
246 views

Prove that there are exactly 16 solutions to this problem.

Show that are are only 16 integer solutions to the following equation: $$11x + 8y + 17 = xy$$ What I tried: I took a modulo 2, and I got that $y$ must be even and $x$ must be odd. But beyond that, I ...
2
votes
2answers
236 views

Find all integer solutions to Diophantine equation $x^3+y^3+z^3=w^3$

Compute all integer solutions to the equation $$x^3+y^3+z^3=w^3$$
2
votes
2answers
43 views

How to solve parameters given 3 different types of information?

How Do I solve this eqns? $$x+y+z = A$$ $$xyz = B$$ $$x^2+y^2+z^2 = C$$ I have tried it in this way,,, $$yz = B/x = P$$ $$y+z = A-x = Q$$ $$y(Q-y) = P$$ $$\implies y^2-Qy+p = 0$$ I can't figure ...
3
votes
1answer
114 views

Prove $\forall a,b,k \in \Bbb Z^+$ such that $a \equiv -1 \bmod 3$ and $b \equiv 1 \bmod 3$, $2^{2k-1}a,2^{2k}b$ are non-trivial polygonal numbers

Below is my original question, which has since been modified to a more general form. Prove that $\forall p,q \in \Bbb P$ and $k \in \Bbb Z^+$ such that $q \equiv -1 \bmod 3$ and $p \equiv 1 \bmod 3, ...
1
vote
1answer
92 views

4-dim. generalization of $ab+ac+bc=0$

The equation $ab+ac+bc=0$ can be parameterized by $(a,b,c)=\lambda(-pq, p(p+q), q(p+q))$. Is there a (similar) parameterization for $ab+ac+ad+bc+bd+cd=0$? What about the 5-dimensional case? Edit: ...
1
vote
0answers
56 views

A diophantine definition of the Kleene star

Let $f(x \, | \, y_1, \dots, y_n)$ be a Diophantine polynomial that generates the Diophantine set $F$. By Matiyasevich, the set $F^*$ (Kleene star of $F$) is also Diophantine. My question: how can ...
4
votes
2answers
158 views

Number theory problem, 3rd degree diophantine equation

How many positive integers are there that can be written in the form $$\frac{m^3+n^3}{m^2+n^2+m+n+1}$$ where $m$ and $n$ are positive integers. I invented this problem and was stuck with it for a ...
4
votes
1answer
194 views

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$

If $x,y,z \in \mathbb{Z}$ such that $x^4+y^4+z^4 \equiv 0 \pmod{29}$, prove that $x^4+y^4+z^4 \equiv 0 \pmod{29^4}$. I have no idea where to start, but this is my abstract algebra homework, so I ...
1
vote
3answers
153 views

Unique Integer solution of a non-linear equation

How to find the integer solution of the equation $$\frac{m^2 + 2mn + n^2 -3m -n+2}{2}=2$$ I know that there is a unique solution
4
votes
1answer
157 views

Find positive integer that can't be expressed by $\lfloor x/2 \rfloor + y + xy$

Consider the expression $$\lfloor x/2 \rfloor + y + xy$$ where $x$ and $y$ are positive integers, $\lfloor x/2 \rfloor$ means rounding down to integer, for example, $\lfloor 3/2 \rfloor = 1$. Some ...
1
vote
3answers
106 views

algebraic geometry and elliptic curves

Does $ax^2+by^2=cz^2$ have positive integer solutions? I know that the solution exists when $(a,b,c)=(1,1,1)$ or $(1,1,n^2+1)$, but I failed to produce a general formula. Any help would be ...
0
votes
1answer
72 views

Diophantine equation and cyclicity of $\mathbb{F}_p^*$

I am trying to prove that the diophantine equation $$1998^2x^2+1997x+1995-1998x^{1998}=1998y^4+1993y^3-1991y^{1998}-2001y$$ has no solution in integers (given that $1997$ is a prime). To do so, ...
0
votes
1answer
150 views

Given $p, m$, how many $r, k$ exist such that $\sum_{i=0}^k{m+i \choose p} = {m+r \choose p}$?

I know that ${m+1 \choose p+1} = {m \choose p} + {m \choose p+1}$, does this identity extend further out? My guess is that there exist certain $k$ such that there exists $r > k$ where the title ...
5
votes
2answers
232 views

Pythagorean Quadruples:

Consider the set of integers $x_1, x_2, x_3, x_4$ Such that: $$x_1^2 + x_2^2 + x_3^2 = x_4^2$$ How does one compute all the solutions to this system? I have the following method in place for ...
5
votes
2answers
111 views

Step in a solution of $y^2 = x^3 - 2$

I am reading Algebraic Number Theory notes here by Keith Conrad. In page 9, there is a solution of $y^2=x^3-2$ using unique factorization in $\mathbb{Z}[\sqrt{-2}]$. We start by writing ...
1
vote
0answers
44 views

Gap:$\;\;L(90,28) := {\{n90 - m28 ∈ N, n, m ∈ N, n < 28\}}$

Which elements of the sets Gap:$$L(90,28) := {\{n90 - m28 ∈ N, n, m ∈ N, n < 28\}}$$ $$$$What would be a quick way to resolve?
1
vote
1answer
59 views

two questions about Diophantine Equation

I am reading an article Modular Arithmetic by Richard Taylor. I have 2 questions: For which $n$, $x^2+y^2=nz^2$ has nontravial solutions? What are the solutions? A beautiful theorem of Hermann ...
7
votes
1answer
499 views

Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number

Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number, where a number $q$ is practical if and only if every integer less than or equal to ...
2
votes
1answer
83 views

A diophantine equation

I want to understand why the equation $U^2-(m^2-4)V^2=-4$ (when $U,V,m$ are odd number and $m > 3$) is impossible. (This came from a post I was reading here)
1
vote
0answers
54 views

Is this a fruitless approach to solving diophantine equations?

Let $P(X, Y, Z)$ be a polynomial over $Q$. Let's be concerned with integer solutions. Namely that there are no solutions $(X,Y,Z)$ such that $\gcd(X,Y) = 1$. So let $X,Y$ be coprime and arbitrarily ...
1
vote
1answer
189 views

Number of Solutions to Diophantine Equation

$(a)$ Let $c < 2\pi$ be a positive real number. Show that there are infinitely many integers $n$ such that the equation $x^2 + y^2 + z^2 = n$ has at least $c\sqrt n$ integer solutions. $(b)$ Find ...
4
votes
1answer
341 views

Number of teeth in gears

I'm building something with an engine that uses gears to reduce/increse movement. The motor has itself some gears, and it's a stepper motor (it gives discrete steps), now the number of steps per ...
3
votes
1answer
416 views

A natural number equation

For what values of $n$ the equation $x^2 - (2n+1)xy + y^2 + x = 0$ has no solution in natural numbers ? (for $n=1$ it has a trivial solution).
1
vote
2answers
122 views

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

Does anyone know if a formula exists to obtain all solutions of the above Diophantine equation? All numbers integers. Addendum: After seeing the answer from Tito Piezas III, I reconsidered the above ...
4
votes
4answers
138 views

The values of $N$ for which $N(N-101)$ is a perfect square

For how many values of $N$ (integer), $N(N-101)$ is a perfect square number? I started in this way. Let $N(N-101)=a^2$ or $N^2-101N-a^2=0.$ Now if the discriminant of this equation is a ...
4
votes
3answers
610 views

Infinite solutions of Pell's equation $x^{2} - dy^{2} = 1$

Let $d > 1$ be a squarefree integer. Prove that the equation $x^{2} - dy^{2} = 1$ has infinitely many solutions in $\mathbb{Z} \times \mathbb{Z}$. What I have done: let $ \ \mathbb{K} = ...
2
votes
0answers
125 views

Count number of positive integer solutions of $x^2(8x-3)=y^2z$?

Given the Diophantine equation $$ x^2(8x-3)=y^2z, $$ is there a way to efficiently count the number of solutions that satisfy $x+y+z\leq n$, where $n$ is a fixed given integer? Also, for any fixed ...
3
votes
2answers
109 views

The complete solution to $x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k$, for $k=1,2$?

It's quite easy to give the complete rational solution to, $$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k,\;\; \text{for}\; k=1,2\tag{1}$$ One can express it in the form, ...
3
votes
4answers
134 views

Find $a,b\in\mathbb{Z}^{+}$ such that $\large (\sqrt[3]{a}+\sqrt[3]{b}-1)^2=49+20\sqrt[3]{6}$

find positive intergers $a,b$ such that $\large (\sqrt[3]{a}+\sqrt[3]{b}-1)^2=49+20\sqrt[3]{6}$ Here i tried plugging $x^3=a,y^3=b$ $(x+y-1)^2=x^2+y^2+1+2(xy-x-y)=49+20\sqrt[3]{6} $ the right ...
2
votes
2answers
134 views

Equation representing all numbers

Joe Roberts writes, in Lure of the Integers, that Matijasevič showed that "every integer has a representation in the form $a^2+b^2+c^2+c+1$". The citation he gives is Ju. V. Matijasevič, A ...
2
votes
0answers
123 views

Solving a particular system of Diophantine equations in $n$ variables (Frobenius equations)

I have a particular system of linear Diophantine equations in $n$ variables for which I need to find all nonnegative integer solutions. Specifically, they are Frobenius equations, meaning the ...
11
votes
1answer
272 views

integer solutions of $a^3 + b^2 = 100000$

Find all integer solutions of $a^3 + b^2 = 100000$ ? I'm looking for one solution and get idea from that to write an analytic solution, but I've not found yet. Is it a good idea or I should start it ...
1
vote
1answer
96 views

How many solutions are possible to the equation $a^x-b^y=c$?

If $a,b,c\in \mathbb Z$ are known and $a>b>1,(a,b)=1$, how many integer solutions are possible to the equation $$a^x-b^y=c~?\tag1$$ Can $(1)$ has more than $4$ integer solutions ?
6
votes
3answers
1k views

Erdös-Straus conjecture

I'm reading a lot about the Erdös-Straus Conjecture (ESC), a conjecture that states that for every natural number $p \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following ...