Questions on finding integer/rational solutions of equations.

learn more… | top users | synonyms

3
votes
3answers
130 views

$x^2+1$ is almost always square free

It seems like $x^2+1$ is almost always square free. Any research or heuristics why? I tried breaking the problem into solving $$x^2-ky^2=1$$ For various $k$, and I conjecture that for every $k$ there ...
1
vote
2answers
38 views

Diophantine solution for a quadratic in two unknown variable

How do we determine integral solutions to the following equation: $$324x^2-8676x + 56700 = y^2$$ Where $x$ and $y$ are positive integers.
0
votes
3answers
33 views

number of ordered pairs of integers (x,y) satisfying the equation

i need to find number of ordered pairs of integers(x,y) satisfying below equation. $$x^2 + 6x + y^2 = 4$$ i have tried and i think x<0 . is there a specific way to solve such equations?
1
vote
3answers
29 views

number of solution to the given equation.

a,b,c, are all non-negative integers such that a + b + c=100 and 1000a + 300b + 50c = 10000 How many such triplets are possible? i have tried to reduce ...
3
votes
1answer
42 views

How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
2
votes
3answers
78 views

solve for three unknowns with two equations

Apple cost 97 dollars. Orange cost 56 and lemon cost 3. The total amount spent is 16047 dollars and total fruits bought is 240 and each one is bought atleast one. Calculate how many of each have been ...
1
vote
2answers
223 views

Solutions to the Mordell Equation modulo $p$

It is well known that the Mordell Equation $x^2 = y^3 + k$ has finitely many solutions, but has solutions modulo $n$ for all $n$. One proof of this involves using the Weil Bound to show that $x^2 = ...
1
vote
2answers
47 views

Diophantine solution to a fraction

How can we find solutions to the following equation: $$ y=\dfrac{x^2-1085}{14718-2x}$$ where $x,\ y$ are integers.
2
votes
1answer
33 views

Primitive-recursive functions and polynomial equations

I am looking for examples of primitive-recursive functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that can not be written as a pair of polynomials, i.e. $$f(n) = m \Leftrightarrow P(n,m) = Q(n,m)$$ ...
3
votes
2answers
116 views

Find all integer solutions of $1+x+x^2+x^3=y^2$

I need some help on solving this problem: Find all integer solutions for this following equation: $1+x+x^2+x^3=y^2$ My attempt: Clearly $y^2 = (1+x)(1+x^2)$, assuming the GCD[$(1+x), (1+x^2)] = ...
3
votes
2answers
91 views

A transcendental number from the diophantine equation $x+2y+3z=n$

Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation $$x+2y+3z=n$$ Prove that $$ \sum_{n=0}^{\infty} ...
0
votes
2answers
63 views

Solving $4a+5b=27$ where $a,b$ are two different positive integers

This problem is making my head spin: The costs of equities of symbol A and symbol B (in dollars) are two different positive integers. If $4$ equities of symbol A and $5$ equities of symbol B together ...
-2
votes
3answers
160 views

Equation $a^{n}+b^{n}=2008$ has no integers solutions. [closed]

Prove that the equation $a^{n}+b^{n}=2008$ has no solutions for $a,b,n\in\mathbb{Z}, n\geq2.$
3
votes
4answers
128 views

Diophantine equation abc + abd + acd + bcd= 1

Is there a reference which classifies or at least gives an infinite family of integer solutions to the above equation? A solution to the problem would also be great obviously.
0
votes
1answer
27 views

integral point on conics

Suppose we have a conic $ax^2 + bxy + cy^2 + dx + ey + f = 0$ where $a,b,c,d,e,f \in \mathbb{Q}$. Is there a way of computing the integer points on this curve. Since it is affine an not projective we ...
0
votes
0answers
32 views

Solving a general Diophantine Equation

For "normal" equations in one variable we have several techniques for solving equations, such as $\sin(5x) = 5\pi\cos(5x)$ or $\ln(x + 2) = 4$. However, imagine we have the following equations: ...
0
votes
0answers
43 views

Looking for solutions to $xy^2 = (1 + z)^2 (5 + 8z)$ in integers

I've been reading about Weierstrass equations and shifted Weierstrass equations and Mordell curve and elliptic curves, but so far I haven't been able to transform my equation to any of this type. ...
1
vote
0answers
58 views

How to solve the diophantine equation $x^y = y^x $? [duplicate]

I know this might be an obvious question as we all know that the answers (beside $(x,y)=(1,1)$) are $(x,y)=(2,4)$ but the problem is, how is this exactly solved? Tags might be inaccurate so feel free ...
2
votes
2answers
88 views

Integer solutions of the equation: $x^2+y^2+z^2=kxyz$

Given the equation: $$x^2+y^2+z^2=kxyz$$ with: $(k,x,y,z)\in\mathbb{N}$, the only solution for $k=2$ is: $x=0,y=0,z=0$. For what values of $k$ the equations has solutions in which $x,y,z$ are ...
7
votes
2answers
267 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
0
votes
0answers
29 views

Polynomials that represent a function

Let $D(x,n_1,\dots,n_k) \in \mathbb{Z}[x,n_1,\dots,n_k]$ be a polynomial. Every such polynomial represents a semi-decidable property of natural numbers by $$P(x) :\equiv (\exists n_1,\dots,n_k)\ ...
0
votes
0answers
35 views

Matrix Multiplication Integer Solution

Given a matrix multiplication and a vector addition. (A,b has rational entries) $$Ax+b$$ how do i get an $x$ for that $Ax+b$ is integer or show that there is not such a solution? $x$ has no ...
0
votes
2answers
34 views

Finding Solutions to a Diophantine Equation with Factorials

How many ordered pairs of positive integers $(a, b)$ are there such that $a!+\dfrac{b!}{a!}$ is a perfect square? Is the number of solutions finite? Source: Ran into it on Facebook. I have plugged ...
3
votes
1answer
88 views

An operation with respect to which the set of prime numbers is closed

Like every (semi-)decidable set of natural numbers the set $P$ of prime numbers is diophantine, i.e. there are two polynomials $p(x)$, $q$ with natural coefficients and exponents – the first of ...
1
vote
2answers
32 views

Rational Number of a given fraction

Find all rational numbers $\frac pq$ such that $\frac pq=\frac {p^2 +30}{q^2 +30}$. How can I go about it. If I substitute p and q by real values $\frac pq$ gets innumerable rational numbers
1
vote
1answer
66 views

Has anyone solved this general Diophantine Equation?

I know that Pythagorean triples have been parameterized, I also know that Andrew Wiles has proved that there are no distinct integer solutions for $ a^n + b^n = c^n$, when $ n \ge 3 $. However we may ...
0
votes
0answers
21 views

Diophantine equations (Mordell theorem)

I have a really serious problem with this exercise, I don`t know how I can resolve it. Could you help me? I study in Spanish, so if you don't understand my translation, please ask me... We have the ...
2
votes
1answer
93 views

Solutions to a diophantine equation

I tried to find integer solutions to the following diophantine equation $$x^3 - 3y^3 + 5z^3 - 3xy^2 + 3x^2y + 9xz^2 + 7x^2z + 3yz^2 - 3y^2z + xyz = 0$$ but was unable to do so. I suspect that there ...
0
votes
0answers
29 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
0
votes
0answers
48 views

Diophantine eqn, general solution?

Here's the equation: $$ 4 \left( x^2+y^2-z^2 \right)=\left( 2k+1 \right) \left( x+y-z \right) $$ Is there a nontrivial solution for this in integers? If not, why not? If there is, can a general ...
1
vote
3answers
80 views

How to solve $b^2-a^2=d^2-c^2$

I'm looking for how to solve the equation $b^2-a^2=d^2-c^2$ where $a,b,c,d$ are naturals and $d>c>b>a>0$ , an algorithm would be appreciated Regards
0
votes
2answers
33 views

Quarters weigh 6 grams while dimes weigh 2 grams.

Quarters weigh $6$ grams while dimes weigh $2$ grams. Tiffany has $\$5.35$ worth of quarters and dimes in her pocket weighing a total of $124$ grams. How many quarters does Tiffany have?
1
vote
3answers
108 views

Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards ...
0
votes
1answer
34 views

The sum of the cubes and the amount of combinations.

Quite simply turned out to solve this Diophantine equation, when he made the assumption that the solutions of these equations symmetric. So given this equation: ...
5
votes
0answers
136 views

$(b-a)^2-2ab$ is a perfect square.

I'm in need of some help if possible, about a formula, theorems, old works, ideas, or even an existing solution are welcome. The problem is that i have two distinct natural numbers as $b > a > ...
3
votes
1answer
55 views

Special kind of a linear Linear Diophantine equation

Could any one help me to point out some literature/ papers which solves a homogenous linear Diophantine equation (one equation) of the form $a_1 \times x_1+a_2 \times x_2 + a_3 \times x_3+....+a_n ...
0
votes
2answers
41 views

$A^7 \not\equiv A(\mod 13) \Rightarrow A^{78} + 1 \equiv 0 (\mod 169)$

Let variable $A$ is integer and $A^7 \not\equiv A(\mod 13)$. Prove that $A^{78} + 1 \equiv 0 (\mod 169)$ Could someone explain, how to solve this type of problems? Any help would be greatly ...
0
votes
2answers
41 views

Two diophantine equations with lots of unknowns

Is it possible (tractable) to determine if the following system of equations has any nontrivial solutions (ie, none of the unknowns are zero) in the domain of integers? $$A^2 + B^2=C^2 D^2$$ $$2 C^4 ...
4
votes
2answers
67 views

Need help with a diophantine expression

I'm faced with this problem. Under what conditions is this expression a positive odd integer: $$\frac{2^g(x^2+y^2-z^2)}{x+y-z}$$ where $g,x,y,z$ are nonnegative integers. x and z are odd, and y is ...
14
votes
2answers
230 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
3
votes
0answers
75 views

Does the special Pell equation $X^2-dY^2=Z^2$ have a simple general parameterization?

In Carmichael's Diophantine Analysis ($\S8$), he notes that the equation $$X^2-dY^2=Z^2 \qquad(\dagger)$$ has a two-parameter solution $$x=m^2+dn^2, \quad y=2mn, \quad z=m^2-dn^2. \qquad(\star)$$ He ...
3
votes
3answers
118 views

Rational points on a surface

I am trying to find rational points on this surface $$ \left( \left( 1-x \right) ^{2}+{y}^{2} \right) \left( \left( 1+x \right) ^{2}+{y}^{2} \right) ={z}^{2}$$ I am actually only interested in ...
0
votes
1answer
27 views

Base convertion and equations

I am studying for an exam in my course, and I will certainly have a question of the kind: In what base is the equation right, for example: 42-3=36 Another ...
0
votes
1answer
36 views

Solutions to diophantine equation $m^2-12mn-3m+2=0$

I am trying to find all solutions to the relativly simple diophantine equation $m^2-12mn-3m+2=0$. I suspect that the only solutions are $n=0$, $m=1$ and $n=0$, $m=2$, but I am currently unable to show ...
1
vote
3answers
65 views

How to show $n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$ has no nonzero integer solutions?

How do we prove that $$n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$$ has no nonzero integer solutions? I know two ways to prove this by taking a geometric interpretation but I don't want such a version. How ...
1
vote
2answers
36 views

Equation over Z

Solve the equation $xy+1=3x+y$ over $\mathbb{Z}^2$ Indeed, $$ xy+1=3x+y \Longleftrightarrow (x-1)(y-3)=2 $$ or $ \textrm{Div}(2)=\{k \in \mathbb{Z}/ k|2 \}=\{-1;1;-2;2\}$ Then $(x-1)/2 \implies ...
0
votes
3answers
58 views

Solving different types of Diophantine equation [closed]

In each of the following three equations I need help in finding all solutions in positive integers : i) $\dfrac 1x+\dfrac 2y-\dfrac3z=1 $ ii) $\dfrac 1{x^2}+\dfrac 2{y^2}+\dfrac 3{z^2}=\dfrac 23$ ...
3
votes
1answer
85 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
3
votes
1answer
57 views

For what positive integers $p$ and $q$: $(p+1)!+(q+1)!=(pq)^2$

I tried this problem using brute force and got the answers as $(3,4)$ and $(4,3)$,but is there a way to solve this question?
3
votes
0answers
52 views

Solution of a equation in natural number nvolving reciprocal of prime

Let $p$ be a prime and $n$ a natural number . Solve in $\mathbb{N}$ the equation $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$