Questions on finding integer/rational solutions of equations.

learn more… | top users | synonyms

0
votes
3answers
31 views

Find a and b in quadratic equation

I have the problem to find $a$ and $b$ given $f(x)=-x^2-2ax+b, a\neq0$ $f(1)=3$ , and the maximum value of $f(x)$ is $4$ and have they key with the answer $a=-2,b=0$, but which steps do I take to ...
1
vote
2answers
65 views

To solve $1+2^mp^2=q^5$

How do we find all posible solutions of $1+2^mp^2=q^5$ for positive integer $m$ and primes $p,q$ ? $m=1,q=3,p=11$ is a solution , is there any other solution ?
0
votes
1answer
40 views

Cubic diophantine equation in 3 variables $(x+2y)(x-4y+k)(x-4y-k) - 28y^3 = 0$, $x,y,z \neq 0$

From research completely unrelated to Number Theory I stumbled onto the following equation: $$ xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3 $$ for $x, y, z$ integers, $x,y,z \neq 0$(I ...
33
votes
4answers
763 views

Conjecture: There's only one Fibonacci number that is the sum of two cubes

As the title says, I need help proving or disproving that there is only one Fibonacci number that's the sum of two (positive) cubes, $2$. I did a small brute force test with Fibonacci numbers below ...
2
votes
2answers
118 views

When is the power of a binomial equal to the sum of like powers of its terms?

Question: Under what circumstances/restrictions on $x$ and $y$ does $(x + y)^n = x^n + y^n$ given the value of $n$? That is, what can we tell about $x$ and $y$ from the value of $n$ and the equation ...
0
votes
0answers
32 views

Nature of roots of a quadratic equation

If $L+M+N=0$ and $L, M, N$ are rationals the roots of the equation $( M+N-L) x^2 +(N+L-M)x + (L+M-N) =0$ are $a)$ Real and irrational $b)$ Real and rational $c)$ Imaginary and equal ...
2
votes
1answer
45 views

Integer solutions to equations of the form $a^n+b^n+\cdots=c^n$

I shall refer to the number of terms on the left side of the equation as $m$. Suppose that all numbers in the equation are positive integers. I am wondering if anything is known about for which ...
3
votes
1answer
671 views

Fermat's Last Theorem where $n$ is a power of $2$

I have seen the proof that Fermat gives for $$x^4 +y^4 \neq z^2$$ which we know also works for $z^4$. BUT I am wondering if the same basic argument can be used for the power of $2^n$. Thinks 8,16,32 ...
1
vote
0answers
31 views

To solve $ \dfrac1m+\dfrac1n-\dfrac1{mn^2}=\dfrac34$ on all integers

Refering To solve $ \dfrac1m+\dfrac1n-\dfrac1{mn^2}=\dfrac34$ , I think it is an interesting question, if the possible solution are integers, thus How do we find all integers $(m,n)$ such that $ ...
1
vote
2answers
52 views

Prove/Dis-Prove that the set of diophantine equation is infinite

Given diophantine equation $4x^3 - 3 = y^2$ ($x > 0$). How many solutions are there ? I don't know where to start, please give me a hint
0
votes
2answers
32 views

A diophantine question about squares

I have been trying to solve the following problem: Classify triples of integers $(m,n,k)$ satisfying the following equation $2mn+m+n=k^{2}$. It is very easy to obtain some solutions. However, I am ...
1
vote
0answers
27 views

Another triple.

Solving the equation. $X^2+Y^2+Z^2=X^3$ got some solutions, but still the question remains. Below are all the decisions or not? $X=5t^2+2t+2$ $Y=11t^3+5t^2+2t$ $Z=2t^3+10t^2+4t+2$ And more. ...
1
vote
1answer
68 views

Solving Pell's equation: algorithm to converge $\sqrt n$

I'm trying to come up with an algorithm to solve the Diophantine equation $$ x^2 - ny^2 = 1 $$ for minimum values of $x$ when $ n $ is given. This equation is also known as Pell's Equation. The ...
0
votes
2answers
35 views

Find all solutions in positive integers of the Diophantine equation (proof explanation)

An example problem in my textbook asks: Find all solutions in positive integers of the diophantine equation $x^2 + 2y^2 = z^2$. The provided proof appears as follows: $2y^2 = z^2 - x^2 = (z - ...
1
vote
3answers
55 views

Diophantine equations problem/exercise 3

Find all the pythagorean tripples (x,y,z) with x=40. Well I started with the known formulas for the pythagorean tripples but got me nowhere. Or I was not able continue the thought process required. I ...
0
votes
1answer
50 views

Diophantine equation exercise [duplicate]

Prove that the diophantine equation $x^4-2(y^2)=1$ has only 2 solutions. Any hint on how to start and what to do .. I do not have a lot of experience on non linear diophantine equations and do not ...
-1
votes
3answers
25 views

Find a and b in equation given range of x

I have the problem to find $a$ and $b$ given $-ax^2+bx+4\geqslant0$, $-1/3\leqslant x\leqslant4$ and have they key with the answer $a=3,b=11$, but which steps do I take to get to that answer?
2
votes
1answer
68 views

Find all integers n such that n−2014 and n+ 2014 are both triangular numbers.

I came across this problem when searching for triangular numbers questions. I know that I need to use the equation, $$\frac {n(n+1)}{2} $$ but I don't know how to apply it to this problem.
0
votes
3answers
97 views

Problem Heron of Alexandria.

Meaning of the problem is to find two right triangles equal perimeter, but with a predetermined magnification area. That is necessary to solve a simple system of equations. ...
2
votes
1answer
97 views

Number Theory Question: $x^2-33y^3=10$ no solutions

I've been struggling to get my head around this for a while! Show that: $x^2 - 33y^3 = 10$ has no integral solutions
0
votes
0answers
20 views

Matrices and diophantine equations

Let A be $mxn$ matrix with integral elements, and let r denote the rank of A. For $1 \leq k \leq r$, let $d_k (A)$ be the gcd of the determinants of all $kxk$ matrices. This is called determinantal ...
1
vote
1answer
111 views

How to solve this Diophantus equation$(s^2=4m^2n^2+p^2$,$p^2=m^2+n^2)$?

$$s^{2}=4m^{2}n^{2}+p^{2}; p^{2}=m^{2}+n^{2}; 1<m<n<p<s$$ I think that this equation does not have positive Integer solution, but how to prove?
1
vote
2answers
35 views

Diophantine equation. Three.

Diophantine equation. $X^2+Y^2=qZ^3$ I wonder at what values ​​of the coefficient $q$ equation has a solution. And of course I wonder how she looks like a formula describing their solutions. For ...
0
votes
2answers
64 views

number of positive integer solution of inequation

Given an inequation with P,Q,R all integers, $P \cdot R \cdot b + P \cdot Q \cdot c - Q \cdot R \cdot a \geq 0$ how many positive integer solutions of $(a, b, c)$ ? Here $a \leq P, b \leq Q, c \leq ...
1
vote
2answers
58 views

Solving a problem with a diophantine equation without trial and error.

I have the following problem: A teacher bought toys for the students of an academy, every toy for a boys costs $290$ and every toy for a girl costs $330$. If he spends $24300$, how many of each ...
1
vote
1answer
61 views

An equation demanding solutions in $\mathbb{Q}^3$

Playing around the problem in the book of a library . I deduced a question to finding all $(a,b,c) \in \mathbb{Q}^3$ such that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=0$$ But now I don't know anything ...
3
votes
3answers
78 views

Divisor of $3^{2n+1}+61$

I have difficulty to show the following: If $p$ is a prime and $p^2$ divides $3^{2n+1}+61$, then $p$ must be $2$. I appreciate any help.
0
votes
1answer
84 views

Any simpler proof of Catalan's conjecture?

visit "http://mathworld.wolfram.com/CatalansConjecture.html" Does there exist any simpler or different proof of Catalans conjecture?
2
votes
1answer
68 views

Find all integer solutions of $x^4 + 2x^3 + 2x^2 + 2x +5 = y^2$

Find all integer solutions to $x^4 + 2x^3 + 2x^2 + 2x +5 = y^2$. I'm in a dead end. I've transformed the expression in the following state: $(x^2+1)(x+1)^2 = y^2 -4$ I couldn't see anyway in ...
4
votes
1answer
118 views

Positive integer solutions to $x^2+y^2+x+y+1=xyz$

The question asks for positive integer solutions to $x^2+y^2+x+y+1=xyz$ . We at first note that $x|y^2+y+1$. Now,let there exist positive integers $x,y$ that satisfy the given equation.Then ...
1
vote
2answers
45 views

integer solutions to $x^2+y^2+z^2+t^2 = w^2$

Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?
1
vote
1answer
43 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
vote
1answer
61 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 ...
0
votes
9answers
147 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 ...
8
votes
2answers
287 views

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

Prime $p\equiv3\pmod4$, then diophantine equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers en, $x^2-py^2=-1$ has no solution in integers. I'd be grateful for any help you are ...
4
votes
3answers
111 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 $x=2,y=3$ $$x=3,LHS=27-15+10=22$$ ...
2
votes
1answer
53 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: ...
1
vote
1answer
58 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, ...
0
votes
2answers
39 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 ...
0
votes
3answers
38 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 ...
2
votes
0answers
50 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 ...
0
votes
1answer
41 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 ...
0
votes
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
34 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
vote
1answer
21 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 ...
2
votes
2answers
88 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 $5\not\mid xyz$. ...
0
votes
0answers
17 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: $a=(2pk-p^2+p-k^2)((t-s)^2-1)+2tsk+p(1-t^2)-(2k-p+1)s^2+F$ ...
0
votes
5answers
63 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 ...
0
votes
3answers
66 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 ...
0
votes
4answers
67 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. $$199x -98y = -5 $$