# Tagged Questions

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### Solution of a simple linear diophantine equation

I'm having a slight problem with a simple equation of the sort $a_1+a_2+a_3...=n$. Where $n,a_1, a_2, a_3... \in N$. I do know how to find the number of solutions to these equations when they are of ...
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### Cubic diophantine equation

How can I solve the equation $x^3+x-1=y^2$ in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ...
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### Formulas for the solution of Diophantine equations [closed]

Do not like these formulas. But this does not mean that we should not draw them. To start this equation zayimemsya, well then, and others. $aX^2+bXY+cY^2=jZ^2$ Solutions can be written if even a ...
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### $x^2-5py^2=-1$,If Diophantine equation is solvable, what's the set p [closed]

If pell equation $x^2-5py^2=-1$ has the integers solution, what's the set p ,with p a prime.if p is a Fermat prime, the equation has solution. relevant diophantine equation ...
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### solving $x^3-2y^3=1$ using cubic number field

I am trying to solve the diophantine equation $x^3-2y^3=1$ using $\mathbb{Q}(\sqrt[3]{2}).$ I've read this link: Solve $x^3 +1 = 2y^3$ The following is what i have tried: Finding all integer ...
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### 15th power Diophantine equation

I'd appreciate some help (a hint) for the following. If $x,y>1$ are so that $2x^2-1=y^{15}$ then $x$ is a multiple of $5$. Don't know if this helps but the equation can be rewritten as ...
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### $px^2-2y^2=1$,for what p,the pell equation has the result

$p$ is a odd prime ,If $px^2-2y^2=1$ is solvable,we can get Jacobi symbol $(\frac{-2}{p})=1$ ,so $p=8k+1,8k+3$ but when $k=12,p=97$, the pell equation $97x^2-2y^2=1$ is unsolvable.I think it's ...
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### Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$.

The problem is to find all positive integers $a$ and $b$ such that $a^2(2^a-a^3)+1=7^b$. I found a=10, and my intuition tells me there are no more solutions. I've also shown that $a=42k+10$ for some ...
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### Co-primality of coefficients of coprime integers

Given that $a,b$ are co-prime, we have infinitely many solutions for $x,y$ to the equation $$ax+by=c.$$ Furthermore, solutions have the form: $x=ca^{-1}+tb,y=cb^{-1}-ta$. Given that $c$ ...
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### How to solve diophantine equation $\frac{x^p-y^p}{x-y}=n$

$$\frac{x^p-y^p}{x-y}=n$$ whit $p$ a prime greater than or equal to $3$,for what value to $n$, it's solvable and how to solve,and whether $\frac{x^p-y^p}{x-y}=q_1$ $\frac{x^p-y^p}{x-y}=q_2$ is ...
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### If $(m,n)\in\mathbb Z_+^2$ satisfies $3m^2+m = 4n^2+n$ then $(m-n)$ is a perfect square.

I came across this question on another forum. The question is: $$\text{If m,n\in \mathbb{Z}_+ such that 3m^2+m=4n^2+n, then (m-n) is a perfect square.}$$ I have managed to partially prove ...
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### nonlinear diophantine equation $x^2+y^2=z^2$

how to solve a diophantine equation $x^2+y^2=z^2$ for integers $x,y,z$ i strongly believe there is a geometric solution ,since this is a pythagoras theorem form or a circle with radius $z$ ...
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### How to solve equations of type $x^x = k \pmod{n}$

Consider the equation $x^x = k \pmod{n}$, where $n$ is an integer, and $k \in \mathbb{Z}_n$ is fixed. Is there an algorithm to solve this equation for $x \in \{1, \ldots, n-1 \}$? Trivial example: ...
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### Number theory problem assignment

$x^2 + x - 3 \equiv 0 \pmod {p^2}$. $p$ is a prime number and satisfies $13^{(p-1)/2} \equiv 1 \pmod p$. I need to find positive value of $x$. Please help me.
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### Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone prove the special case of Fermat's Last Theorem for $n=3$, i.e., that $$x^3 + y^3 = z^3,$$ has no positive integer solutions, as simply as possible? I have seen some good proofs, but ...
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### Diophantine equation problem

I want to solve the equation $x^2 - (4a^2)y = 13$ Here $a$ is a positive integer. Is it possible to get a parametric equation?
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### parametric solution for the sum of three square

Is there a parametric integer solution for $x,y,z,t$ when the sum of three square is equal to a square, i.e, $$x^2+y^2+z^2=t^2$$?
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### Prove that $2^p+3^p = a^n$ has no solutions

In this equation p is a prime number and $a,n > 1$ and they are whole numbers. The only thing that I proved is that $2^p+3^p$ is divisible by $5$ or that it gives a reminder $5$ when divided by $p$ ...
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### Linear Diophantine equation in two variables with additional constraints

Given, $$aX + bY = c$$ where, $$c > b > a > 0;\quad X, Y > 0;\quad b\nmid c, a\nmid c$$ I want to find out if a solution exists as efficiently as possible (I'm not interested ...
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### Finding all positive integers $x,y,z$ that satisfy $3^x - 5^y = z^2$

Find all positive integers $x,y,z$ that satisfy: $$3^x - 5^y = z^2.$$ I think that $(x,y,z)= (2,1,2)$ will be the only solution. But how to prove that?
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### Diophatine equation $x^2+y^2+z^2=t^2$

Probably duplicate but I don't find: I'd like to solve the diophantine equation $$x^2+y^2+z^2=a^2$$ which has solutions, by exemple $1^2+2^2+2^2=3^2$ or $2^2+3^2+6^2=7^2$. Every such solution gives ...
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### If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $x^3-y^5=(x+y)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this ...
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### Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that ...
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### Is $x^3 + y^3 = z^3$ possible?

Is $x^3 + y^3 = z^3$ possible when $x$, $y$ and $z$ are integers? If not, how to prove that they are not possible? (I am a grade 10 student so please answer in a simple way)
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### rational solutions of Pell's equation

1) $D$ is a positive integer, find all rational solutions of Pell's equation $$x^2-Dy^2=1$$ 2) What about $D\in\Bbb Q$ ?
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### Find all positive integers m, n, p such that $(m+n)(mn+1)=2^p$

Find all positive integers m, n, p such that $$(m+n)(mn+1)=2^p$$ Please give me some hints Thanks
### How to solve the diophantine equation:$xa^3+yb^3=c^3$
Let $a,b,c,x,y \in \mathbb{Z}> 1$. Any hint on how to solve of the diophantine equation $xa^3+yb^3=c^3$?
### Find all Integers ($n$) such that $n\neq 6xy\pm x\pm y$
I am interested in proving that there exist an infinite number of positive integers ($n$) which are not of the form $$n=6xy\pm x\pm y$$ for $x,y\in\Bbb Z^+$. [Note: The $\pm$ signs above are ...