Questions on finding integer/rational solutions of equations.

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1answer
29 views

Solving diophatine equation of form $x^2+y^2=25$

How would you solve diophatine equations of the form $x^2+y^2=25$? I know how to solve linear diophatine equations but I have not done any of quadratic form before. I could use trial and error because ...
1
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1answer
40 views

Solve a system of diophantine equations

I have a problem with elemental number theory. I started with the expression $$ (a - \frac{1}{b})(b - \frac{1}{c})(c - \frac{1}{a}) $$ and task to find all natural $a,b,c$ so that the result of the ...
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1answer
22 views

System of Diophantine equations.

Quite interesting are there any ideas on solving systems of equations like these? $\left\{\begin{aligned}&a^2+b^2=c^2\\&(a+k)^2+(b+k)^2=q^2\end{aligned}\right.$ Although I recorded such ...
3
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0answers
47 views

diophantine equation $ |x^2-py^2|=\frac{p-1}{2} $

Prime $p\equiv3\pmod4$, then the equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers obviuosly, $x^2-py^2=-1$ has no solution in integers. How about this problem? Thanks a lot!
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3answers
81 views

How find this $x^3-5x+10=2^y$

let $x,y$ is positive integer,and such $$x^3-5x+10=2^y$$ find all $x,y$. since $$x=1\Longrightarrow 1^3-5+10=6$$ can't $$x=2,2^3-5\cdot 2+10=8=2^3$$ so $x=2,y=3$ $$x=3,LHS=27-15+10=22$$ ...
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0answers
34 views

Find the generating function for a series , given a recurrence relation

I am solving a problem on an Online Judge. The problems solution boils down to find the solutions to the following recurrence relation: ...
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1answer
48 views

Find two triangles of longest side length 25?

I'm using the quadratic Diophantine equations to solve for two integer triangles of longest side $25$. It's been shown that for $a^2+b^2=c^2$, which goes to $x^2+y^2=1$ where $x=\frac ac, y=\frac bc, ...
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2answers
28 views

The sum of two triangular numbers.

When triangular number is the square of an elementary formula is obtained. Sam got a couple of pieces, but I wonder how the formula looks opisyvayushaya sum of two triangular numbers is the square of ...
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3answers
34 views

Triangular numbers for numbers.

Interestingly for triangular numbers: $X(X+1)+Y(Y+1)=Z(Z+1)+a$ $a$ - this number is determined by the condition of the problem. Are all numbers equation has a solution? And what kind of formula in ...
1
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1answer
42 views

How find this integer $x,y$ such $1+5^x=2\cdot 3^y$

Find this equation integer solution $$1+5^x=2\cdot 3^y$$ I know $$x=1,y=1$$ is such it.and $$x=0,y=0$$ This problme is Shanghai mathematics olympiad question in 2014 I think this equation have no ...
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1answer
37 views

Diophantine equation with condition

The question is to find the general solution in integers $x,y,z$ to $$2x+3y+5z=7$$ where none of $x,y$ or $z$ are divisible by $7$. Without the divisible by $7$ condition I found that the general ...
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2answers
16 views

Defining and expressing as a system of two equations. Is my answer good?

We wish to spend $\$164.00$ by purchasing $10$ books, some costing $\$15.00$ and other $\$17.00$. How many books of each price do we buy? My answer: let $x$ = number of books costing $\$15.00$ and ...
0
votes
1answer
29 views

What is the value of $a + b + c + d$ if the following equation holds?

If $a, b, c$ and $d$ are positive integers less than $7$ and $$a(7)^3 + b(7)^2 + c(7) + d = 901$$ What is the value of $a + b + c + d$? Is it related to consum of roots and product of roots?
1
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1answer
20 views

General form of Bezout numbers

Bézout's lemma can be generalized to $n$ co-prime integers $a_1, \dots a_n$ : there exists integers $x_1, \dots, x_n$ such that $$a_1 x_1 + \dots + a_n x_n = 1$$ For the case $n = 2$, one can show ...
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2answers
69 views

p odd prime. Prove that if $a\equiv b~(mod~p)$ then $a^p\equiv b^p~(mod~p^2)$. Hence show $x^5+y^5=z^5$ has no integer solutions with $5\not\mid xyz$

Question: Let $p$ be an odd prime. Prove that if $a\equiv b~(mod~p)$ then $a^p\equiv b^p~(mod~p^2)$. Hence show the Diophantine equation $x^5+y^5=z^5$ has no integer solutions with $5\not\mid xyz$. ...
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0answers
11 views

Diophantine equation with 6 variables.

In this equation: $aX^2+bY^2+cZ^2=abc+2XYZ+F$ $F$ - integer number given by the condition of the problem. A rather Tran decision: $a=(2pk-p^2+p-k^2)((t-s)^2-1)+2tsk+p(1-t^2)-(2k-p+1)s^2+F$ ...
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5answers
58 views

Finding $7$ inverse modulo $11$

I'm trying to find the inverse of $7$ modulo $11$. From what I understand, the steps are: \begin{align} &11 = 1(7) + 3 \\ &7 = 2(3) + 1 \\ \end{align} From here, you work backwards ...
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3answers
48 views

Solving the congruence $7x \equiv 41 \mod{13}$

I have to solve the following linear congruence: $$7x \equiv 41 \mod{13}$$ The question where I got this from comes in two parts. The first is that it asks to find the set of the inverses of $7 ...
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4answers
59 views

solve and explain the Diophantine equation [closed]

Solve Diophantine equation and find the value of $x$ and $y$. For the value of $x$ and $y$ we solve through Diophantine equation. $$199x -98y = -5 $$
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1answer
23 views

Linear diophantine equation word problem

I have the following word problem: A small clothing manufacturer produces two styles of sweaters: cardigan and pullover. She sells cardigans for $\$31$ each and pullovers for $\$28$ each. If her ...
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0answers
19 views

Find the general solution to diophantine equation $-221x + 187y - 493 = 0$

I have to find the general solution to $$-221x + 187y - 493 = 0$$ The main issue, I'm figuring out if I have found the general solution or not. Below, are my steps: The $\gcd{(-221,187)} = 17$ and ...
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0answers
14 views

can someone help me understand how to solve linear diophantine equations

I'm having a lot of trouble solving linear diophantine equations. i understand how to find the GCD using the euclidean algorithm but after that i'm so confused about how you are supposed to reverse it ...
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2answers
36 views

Reminder of equation

I have a simple(maybe too simple) question - how to find the reminder of equation? For example: $(85^{74}+17^{95})^{15} \equiv \ ? \ (mod\ 13)$ I know that it is something simple, but I couldn't ...
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1answer
9 views

Solve the comparison

I have difficulties with these type of problems: Solve the comparison: $\displaystyle67x + 17 \equiv 0\pmod{28}.$ I'm sure it is something very simple but I'm stuck on it more than $2$ hours :( . ...
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1answer
86 views

Solutions to $x^p+y^q = z^r$

Is there any $(p,q,r)$ with $\gcd(p,q,r) = 1$ and $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} < 1$ for which we know that the only integer solutions (not necessarily primitive) to the equation ...
1
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1answer
27 views

Pythagorean triples,consecutive terms of an arithmetic progression

I am looking at the exercise: Find all the positive Pythagorean triples that are consecutive terms of an arithmetic progression. $$$$ So,according to the solution that I saw in my notes,we want to ...
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1answer
47 views

Random Algebra Problem

Prove that if a, b, c, x, y, z, and $\alpha$ are natural numbers. For every given set of x, y, z, the number $\alpha$ obtained from the following equation: $$\frac{a^2}{x^2} + \frac{b^2}{y^2} + ...
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2answers
152 views

number of solutions of x*y<N [closed]

How many solutions (unique pairs (x,y) ) exist for equation $xy < N$ ? constraints : $x >1 , y>1 , N<=50000$ I tried following method , but it fails for say N=24 , in which i calculate ...
0
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1answer
26 views

The trivial solutions of the diophantine equation $x^2+y^2=z^2$

The trivial solutions of the diophantine equation $x^2+y^2=z^2$ are the following: $$x=0, y=n, z=\pm n, n \in \mathbb{Z}$$ $$x= \xi , y=0, z= \pm \xi, \xi \in \mathbb{Z}$$ $$$$ My question is, why is ...
0
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1answer
40 views

What is the best way of solving for $x,y\in\mathbb N$ given the conditions $\begin{cases}x\mid y+a\\y\mid x + b\end{cases}$?

What is the best way of solving for $x,y\in\mathbb N$ given the conditions $\begin{cases}x\mid y+a\\y\mid x + b\end{cases}$? The letters $a,b\in\mathbb N$ denote constant known numbers. The ...
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3answers
97 views

Solutions to $ax^2 + by^2 = cz^2$

The integer solutions to the equation $x^2 + y^2 = z^2$ are very well studied. I'm wondering if there's any literature about the integer solutions to the equation $ax^2 + by^2 = cz^2$ where a,b,c are ...
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0answers
24 views

prove that $ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$

Let $a,b,c,d\in \mathbb Z$ prove that: a)$ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$ b)$ax+by=c$ has solution in $\mathbb Z$ if and only if ...
1
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2answers
50 views

Equation in rational numbers?

Is it true that this equation $6=\frac{x^2}{y^2+1}$ has no solutions in rational numbers? If so, why? It is quite evident that it has no solutions in integers (because $y^2+1$ never divides $3$).
11
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1answer
212 views

Prove that if $n(a^2+b^2+c^2)=abc$ then $2\mid n$

Is it true that if $n\in\mathbb N$ and the diophantine equation $$n(a^2+b^2+c^2)=abc,\\(a,b)=(b,c)=(c,a)=1\tag1$$ has positive integer solutions $a,b,c$, then $2\mid n$? I can prove that ...
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1answer
22 views

How many n-tuple to genereate zero from some random several variable equation that use constant power and a variable as the base?

Define algebraic number tuple as the aphabetical order sequence of variables that use in the equation. How many algebraic number n-tuple (x,y,z,...) are able to genereate zero to input into a several ...
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2answers
67 views

Proof that $y^2=x^3+x$ has a unique integer solution

Prove that the equation $y^2=x^3+x$ has only one integer solution, namely $x=y=0$.
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2answers
67 views

Solve and explain diophantine equation

A Diophantine equation ax+by = c always has a solution whenever a and b are relatively prime. Find x ,y such that $$93x-81y=3 $$
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0answers
37 views

The solutions to $x^2+y^2=5$ in $\mathbb{Q}$. [duplicate]

Consider the following equation: $$x^2+y^2=5.\tag{1}$$ What are the solutions to this equation if $x,y\in\mathbb{Q}$, where $\mathbb{Q}$ is the set of all rational numbers? My attempt: Because ...
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0answers
46 views

Diophantine Equation $2x^2+25=y^3$

I'm trying to find integer solutions to: $2x^2+25=y^3$. Here's what I've managed to do so far: y is odd. y and x are co-prime. In $\mathbb{Q}(\sqrt{2},i)$ we can write: ...
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1answer
29 views

Is every unit vector in $\mathbb{Q}^n$ the first column of a rational orthogonal matrix?

Equivalently, does every unit vector in $\mathbb{Q}^n$ belong to some orthonormal basis for $\mathbb{Q}^n$? This is clearly true for $\mathbb{Q}^2$, and for $\mathbb{Q}^3$ it seems to be true for ...
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4answers
103 views

The solutions to $x^2+5=y^2$.

Consider the equation $$x^2+5=y^2.\tag{1}$$ If $x,y\in\mathbb{Z}$, what are solutions to (1)? If $x,y\in\mathbb{Q}$, what are solutions to (1)? Note: $\mathbb{Z}$ is the set of all integers and ...
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4answers
57 views

solve this problem with diophantine equation

A man arrives in a bank to cash a cheque. for some stated amount. The teller on the counter makes a mistake and interchanges dollars and cents. I donated 5 cents to a charity box at the bank. Later, I ...
4
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2answers
226 views

solve for all integer solutions to the diophantine equation

$$1/x+1/y=1/14$$ Find all integer solutions for x and y. I can solve linear diophantine equations without a problem normally but this has me stumped.
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0answers
28 views

Evaluation of certain trigonometric sums

In trying to approximate the number of solutions to the equation $3^n - 2 = k^2$ for positive integers $n, k$, I tried to use the circle method. In doing so, I had to bound the trigonometric sum for ...
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2answers
96 views

Integer solutions of $800000007 = x^2+y^2+z^2$

Prove that the equation, $800000007 = x^2+y^2+z^2$ has no solutions in integers.(That is $8$ followed by $7$ zeroes, with a $7$ at the end). I tried checking modulo $3$, $5$, $7$, and $10$, but ...
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4answers
111 views

One Diophantine equation

I wonder now that the following Diophantine equation: $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ have only this formula describing his decision? $a=-(k^2+2(p+s)k+p^2+ps+s^2)$ $b=2k^2+4(p+s)k+3p^2+3ps+2s^2$ ...
1
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1answer
33 views

On representing the general solution for the diophantine equation $a_1x_1+\dotsb+a_nx_n=c$

On representing the general solution with the special solutions for the diophantine equation $$a_1x_1+a_2x_2+\dotsb+a_nx_n=c$$ here $a_1 ,a_2, \dotsb,a_n,c\in\Bbb Z,(a_1 ,a_2, \dotsb,a_n)=1$. Can ...
0
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0answers
30 views

Divisors of Pell Equation Solutions

Let $d > 0$ be square-free. Let $\epsilon = x_0 + y_0 \sqrt{d}$ be the minimal solution to the Pell's equation $x ^ 2 - d y ^ 2 = 1$. Let $x + y \sqrt{d} = \epsilon ^ l, l \geq 1$ be a solution. ...
2
votes
3answers
85 views

Show that $0 = 2a^3-5ab^2+25b^3$ has no other integer solutions than $a = b = 0$.

I am trying to solve the following problem: I have the equation $0 = 2a^3-5ab^2+25b^3$, where $a,b \in \mathbb Z$. Obviously, $a = b = 0$ is a solution of this equation. But how can I show that there ...
-1
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0answers
32 views

Checking if integer solutions exist

I have a linear equation $\alpha_1a_1+\alpha_2a_2+\dots=\beta$. I only need to check is there exists α1,α2... such that all are greater than or equal to zero. I am a computer science student , i got ...