Questions on finding integer/rational solutions of equations.

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46 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 ...
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13 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 = ...
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2answers
36 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.
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1answer
25 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. $$f(n) = m \Leftrightarrow P(n,m) = Q(n,m)$$ ...
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2answers
106 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)] = ...
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2answers
80 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} ...
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145 views

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

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

No of unique values [on hold]

How to find the number of uniques values AX+BY can take under various scenarios Like (A,B) being co-prime or GCD(A,B)=C,with the imposed conditions like X,Y being integers from the set (E,F).
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3answers
80 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.
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1answer
22 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 ...
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0answers
29 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: ...
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40 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. ...
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57 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 ...
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2answers
83 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 ...
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2answers
252 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.
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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)\ ...
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34 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 ...
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2answers
33 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 ...
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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 ...
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31 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
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1answer
56 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 ...
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18 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 ...
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1answer
90 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 ...
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28 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]}, ...
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45 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 ...
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3answers
75 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
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48 views

To find the polynomial.

On adjacent forum hate formula. But the question is interesting and would like to have it clear. Theme there: ...
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2answers
25 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?
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3answers
90 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 ...
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1answer
33 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: ...
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128 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 > ...
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1answer
54 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 ...
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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 ...
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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 ...
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2answers
65 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 ...
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212 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 ...
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69 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 ...
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3answers
109 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 ...
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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 ...
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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 ...
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63 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 ...
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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 ...
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53 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) $\dfrac 1{x^2}+\dfrac 2{y^2}+\dfrac 3{z^2}=\dfrac 23$ ...
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1answer
80 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 ...
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1answer
55 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?
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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 $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
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59 views

How can I show the complete symmetric quadratic form has no zeros?

The quadratic complete symmetric homogeneous polynomial in $n$ variables $t_1,\ldots,t_n$ is defined to be $$h_2(t_1,\ldots,t_n) := \sum_{1 \leq j \leq k \leq n} t_j t_k = \sum_{j=1}^n t_j^2 + ...
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1answer
67 views

How prove this diophantine equation $3^x-2^y=k$ have finitely many integral solutions

For any $\forall k\in N^{+}$ show that the diophantine equation $3^x-2^y=k$ have finitely many integral solutions. My try: if $k=2m$,then $$3^x=2^y+k=2^y+2m$$ It is clear there is no integer ...
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1answer
31 views

Solve equation. sum of negatie powers of two equal to one. Diaphantite.

Is the following correct? Let $\sum_{i=1}^n \frac{1}{2^{x_i}}=1$ where $x_i \in \mathbb{N}_0$ for $i \in \{1,\ldots,n\}$ than the only solutions is $$x_i=n-1, \quad \forall i \in \{1,\ldots,n\}.$$ ...
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18 views

Diophantine equations,is that what I have done right?

I have solved the following diophantine equations: $14x+35y=93$ $56x+72y=40$ That's what I have tried: $gcd(35,14)=7$ , but $7 \nmid 93,$ so the first diophantine equation has no solution. ...