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Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

Use this tag for questions about finding integer, or perhaps rational, solutions to polynomial equations.

Diophantine equations are named after Diophantus of Alexandria, a third century Greek mathematician.

An example of a Diophantine equation is to find all quadruples of integers $(w,x,y,z)$ such that $$w^2+x^2=3(y^2+z^2).$$

Solving Diophantine equations often involves other areas of mathematics such as congruences, linear algebra, inequalities, forms (e.g., binary quadratic forms), and elliptic curves. Special solution methods include comparing divisors, considering orders of magnitude, Fermat's method of descent, and finding intersections of curves with lines of rational slope through a known rational point.

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Prove this diophantine equation $2^a-3^b=5~，a,b\in N^{+} $ has no postive integers solution

show that the diophantine equation $$2^a-3^b=5~~~~,a>5,b>3，a,b\in N^{+} $$ has no postive integers solution maybe is old problem,But I try somedays,can't solve it by now
math110
  • 93,304
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Bilinear diophantine equations

Is there a fast way ($O((\log n)^c)$) to solve $$ax+by+xy=n$$ over integers when $a,b$ are known and $0
user257494
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Solve Linear Diophantine $12x+18y = 54$

What is asked? As the title suggests I'm trying to solve a very simple Linear Diophantine Equation: $$12x + 18y = 54$$ Also find an expression for all integer solutions What have I done? Firstly, I know that $ax+by = c$ has a solution $\iff$…
Rubicon
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Diophantine equation $n^2+n+1=m^3$

Is there an elementary method for solving Diophantine equation $n^2+n+1=m^3$ for integers $m$ and $n$? There is a similar one, which I could solve:$$p^2-p+1=q^3,$$where $p$ and $q$ are prime numbers. But the technique that I used for solving this…
Ghartal
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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 are: [4,-20], [6, 30], [10, 10] So my question is:…
OhOkay
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Is there a general formula to solve $a^2+b^2+c^2=d^2+e^2+f^2$?

Is there a general formula to solve the Diophantine equation $$a^2+b^2+c^2=d^2+e^2+f^2?$$ If so, can I please have the reference? Thanks.
user97615
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Under what situations does $x+1$ divide $4n^2-x$?

I am looking at the equation $$\frac{4n^2-x}{x+1} = y$$ for even $x$ and $y$, both positive. Under what situations does $x+1$ divide $4n^2-x$?
MyNameIsKhan
  • 908
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Linear Diophantine equation solving

Find all the positive solutions in integers of $x+y+z=31$ $x+2y+3z=41$ For the first, I have subtract second equation from first , and by that I have found that $y=2k-10$ ,$z=k$,is it possible?
Function
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Find all solutions to the diophantine equation $(x+2)(y+2)(z+2)=(x+y+z+2)^2$

Solve in postive integer the equation $$(x+2)(y+2)(z+2)=(x+y+z+2)^2$$ It is rather easy to find several parametric solutions, (such $(a,b,c)=(2,1,1),(2,2,2)$).but it seems harder to find a complete enumeration of all the solutions. and I have proved…
user237685
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Can we find $x_{1}, x_{2}, ..., x_{n}$?

Consider…
esege
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Describe the integral solutions to $y^2 = 12x^3 - 39$

Does the above Diophantine equation have infinitely many integer solutions ? One such solution is $(x,y) = (4,27)$.
Q_p
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For any coprime integers $(x, y)$, $m\geq 2$, $\delta\geq 1$, is it true that $\mid x^m - y^{m+\delta} \mid \geq \delta$?

Is it true that if $x,y,m,\delta$ are integers, $\gcd(x,y)=1$, $m\ge2$, $\delta\ge1$, then $$|x^m-y^{m+\delta}|\ge\delta?$$ Any proofs or references will be most welcome.
Q_p
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Describe the nonzero integer solutions to the equation $a^3 + b^3 + c^3 + d^3 + e^3 + f^3 + g^3 =0$

Can someone describe all the integer solutions to the above equation such that $abcdefg\neq 0$ ?
Q_p
  • 919
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Integer solutions $d = \frac{ab}{a + b + 2\sqrt{ab}}$

$$d = \frac{ab}{a + b + 2\sqrt{ab}} = \frac{ab}{(\sqrt{a} + \sqrt{b})^2}$$ What are the positive integer solutions? The majority of solutions are when $a=b$, so that $a = 4d$.
qwr
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Solve $z^3=(sr)^3$ where $r,s,z$ are integers?

Let $x,y,z$ be 3 non-zero integers defined as followed: $$(x+y)(x^2-xy+y^2)=z^3$$ Let assume that $(x+y)$ and $(x^2-xy+y^2)$ are coprime and set $x+y=r^3$ and $x^2-xy+y^3=s^3$ Can one write that $z=rs$ where $r,s$ are 2 integers? I am not seeing…
user97615
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