# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### is A an even number?

Let $a,b,c,d$ be positive integers such that $(3a+5b)(7b+11c)(13c+17d)(19d+23a)=2001^{2001}$ hence, prove that $a$ is even. I tried to approach this problem reducting it modulo 6. From which we ...
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### $N=(x^2-1)(y^2-1)$ has more than one solution

Given that $N=(x^2-1)(y^2-1)$ where $N,x,y,a,b$ are positive integers, find with proof the smallest value of $N$ such that $N=(x^2-1)(y^2-1)=(a^2-1)(b^2-1)$, where $a$ is not equal to either $x$ or $y$...
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### If $\frac1x-\frac1y=\frac1z$, $d=\gcd(x,y,z)$ then $dxyz$ and $d(y-x)$ are squares

Let $x, y, z$ be three non negative integer such that $\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{1}{z}$. Denote by $d$ the greatest common divisor of $x, y, z$. Prove that $dxyz$ and $d(y-x)$ are squares ...
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### Find minimum of the $n$ such $x+11y+11z=n$ has $16653$ triples of postive integers solution

I wish to solve following problem $$x+11y+11z=n(n\in N^{+})$$ has $16653$ triples $(x,y,z)$ of postive integers. Find $n_{\min}$ Of course, I can't solve it by Now, so there any solution? Problem 2:...
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### Find all integer solutions for $x*y = 5x+5y$

For this equation $x*y = 5x + 5y$ find all possible pairs. The way I did it was: $x=5y/(y-5)$ And for this I wrote a program to brute force a couple of solutions. If it helps, some possibilities ...
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### Integral solutions $(a,b,c)$ for $a^\pi + b^\pi = c^\pi$

We know that $a^n + b^n = c^n$ does not have a solution if $n > 2$ and $a,b,c,n \in \mathbb{N}$, but what if $n \in \mathbb{R}$? Do we have any statement for that? I was thinking about this but ...
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### Can only the middle school math knowlegde help to find solutions for $2013 y^2 -xy -4026 x=0$?

I found the following equation form an answer written for a question. $$2013 y^2 -xy -4026 x=0$$ But I'm confused that can I really learn how to find the positive integer solutions for $x,y$ with ...
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### Looking for the most elementary proof that $48X^4+12X^2+1=Y^2$ has no non-trivial integer solution.

As relayed in this question of mine (which is more general in scope), I believe I have found a relatively easy, and completely elementary, way to show that the equation $$48X^4 + 12X^2+1 = Y^2$$ has ...
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### Weight 2 Newforms of large level computations.

I am stuck with some weight $2$ newform computations of large level. For example I want to compute newforms of level $11520$. Can anyone suggest me a way to do it? I need it to solve some diophantine ...
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### integer points on an ellipse

I have the equation $1=x^2-xy+y^2 = \frac 1 4 (x+y)^2 + \frac 3 4 (x-y)^2$ where I am looking for integer solutions $x,y \in Z$. When you draw this ellipse it is quite obvious that the integer points ...
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### Integer solutions for $x^4 + 4xy^3 = z^2$

Find all triplets $(x,y,z)$ of integers so that $$x^4 + 4xy^3 = z^2.$$ What I've done: Suppose $x=0$. Then we see $z=0$ and hence $(0,y,0)$ is a solution. Suppose $y=0$. Then we see $x^4 = z^2$ ...
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### Alternative derivation of the second Frobenius number

Let $a,b$ be integers with $\gcd(a,b)=1$. What is then the largest integer $N$ which cannot be written as a linear combination with non-negative integer coefficients of $a$ and $b$? A few days ago, I ...
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### Is there a test to determine whether a Diophantine equation has a solution in the non negative integers?

Is there a simple test to determine whether the Diophantine equation, $\sum_i a_i x_i = c$ with $a_i, x_i, c$ as integers and $a_i > 0, x_i \geq 0, c > 0$ has a non-negative integer solution?
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### Solving $2(n-1)n(n+1)(n+2)=(m-3)(m+3)$

The question is: Find all pairs $(n,m)\in\mathbb{N}^2$ such that $$2(n-1)n(n+1)(n+2)=(m-3)(m+3)$$ I checked all $n<10000$ and only got $n=1$ and $n=4$ with their corresponding $m$, so I ...
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### When is $991n^2 +1$ a perfect square?

What should be the value of $n$ so that the number obtained after adding $1$ to $991$ times its square is itself a perfect square? Can you please give me a few hints on this topic with a few specific ...
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### $x^3 + 5x + 6 = 3\cdot 2^{1+x-k}$
Does anyone know how to solve $$n^3 + 5n + 6 = 3\cdot 2^{1+n-k}$$ where n,k are natural numbers? I was told that there are prime number arguments that can be used but I am totally stuck. It is a ...