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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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Two Diophantine equations

What is known about solutions in integers of the equations; $x^4 - y^2 = z^6$ I got $x=4st(s^4 - t^4)$ , $z=4st(s^2 - t^2)$ , $y=(4st(s^2 - t^2))^2 (s^4 + t^4 - 6 (st)^2) $ and, the equation $x^2 - y^4 = z^6 $ I got $y=4st(s^4 - t^4) , z=4st(s^2 +…
Souvik Dey
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For what $n$ does $x^4+y^4\equiv 0\pmod n\implies x\equiv y\equiv 0\pmod n$

For what $n\in\mathbb N^*$ does it hold, for $x$ and $y$ integers, $$x^4+y^4\equiv 0\pmod n\implies x\equiv y\equiv 0\pmod n$$ I'm after a characterization of $n$ that can be efficiently tested for $n$ in the millions. A sufficient condition with…
fgrieu
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Solving this $4$ variable diophantine equation

Is there a way to solve this Diophantine equation in $a,b,c,d$? $$19a^3-33a^2b+3a^2c+30a^2d+21ab^2+24abc-12abd-15ac^2-54acd-30ad^2+ $$ $$2b^3-12b^2c-6b^2d+42bc^2+108bcd+60bd^2-7c^3-51c^2d-99cd^2-56d^3=0$$ Wolfram Alpha unfortunately cannot…
Frank
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Solve $2015+x^2=3^y4^z$

If $x$, $y$ and z are positive integers, I want to solve $$2015+x^2=3^y4^z$$ What I tried: I found that $x \equiv 1$ (mod $24$). So there exist an integer $p$ such that $24p+1=x$. Replacing that on the equation, we have $$48(12p^2+p+42)=3^y4^z$$
Dinen
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Diophantine equation of $a^2+b^2=c^2+d^2$

Is there a known solution to: $$a^2+b^2=c^2+d^2$$ Hopefully the question is clear, if not I do apologize.
user356448
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diophantine equation: $ (1-ab^3)(a^3b-1)=c^2$

I wonder if the Diophantine equation $(1 -ab ^ 3 ) (a ^ 3b -1) = c^2$ admits rational solutions we must choose the numbers $a$ and $b$
curieux_2014
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Fastest way of finding solution to n*const1+ const2 = x^2

I am trying to solve the following equation: n*const1 + const2 = x^2 Where n, const1, const2 and x are integers > 0. Const1, const2 are known, n and x are variables. The naive solution to this problem is to iterate all n's and check if the square…
Vojtěch
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Prove (or disprove) 2 equation of 6 variables

Given $$(s-x_{1})(s-y_{1})(s-z_{1}) = (s-x_{2})(s-y_{2})(s-z_{2})$$ Where $s=x_1+y_1+z_1$ and all variables are positive non-zero integers. I need to prove that such values of $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2}$ can or can't exist. I have…
NoChance
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Find all positive integers that solve Mordell's equation $y^2=x^3+37$

Find all Mordell's equation: $$y^2=x^3+k$$ where $k=37$ positive integer numbers,I can't find the when $k=37$ the mordell equation solution with some result,and we can known this equation have solution, such $x=3$ then $3^3+37=64=8^2$ so…
math110
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Solving Diophantine Equation $xB=(2^N)-1$

If given a value for $x$, does anyone have a way to solve the diophantine equation below? $xB=(2^N)-1$ where $x,B,N\in\mathbb Z$ Where presumably a smaller $N$ is better, but any way to find a value of $N$ (and $B$) is a good solution. All i can…
Alan Wolfe
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how to solve $x^2-y^2=(2n-1)^2$

If $n$ is known,and $x,y,n$ belong to $\mathbb{N}^+$. What is $x$ and $y$? I know there exists a answer, for example,when $n=111$, $x=6161$, $y=6160$, but I do not know if the answer is unique.
liam xu
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Diophantine equation: $\frac1x=\frac{a}{x+y}-\frac1y$

For how many different natural values of $a$ the Diophantine equation: $\frac1x=\frac{a}{x+y}-\frac1y$ has natural roots? I rearranged the equation as: $xy+x^2+y^2=(a-1)xy$ , hence I said we must have: $(a-1)>2$ I tried another idea: reform the…
Hamid Reza Ebrahimi
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Proof that $ax+by+cz=0$ has infinitely many solutions.

For all non-zero integers $x,y,z$ clearly there exist infinitely many non-zero integers $a,b,c$ such that $$ax+by+cz=0$$ How can I prove this simple statement?
user97615
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Find the diophantine equation $x^2(y^2-1)=z^2-1$ solution

How can I solve (find all the solutions) the nonlinear Diophantine equation Let $x,y,z$ be postive integers ,and $x,y,z\ge 2$,find this following equation solution $$x^2=\dfrac{z^2-1}{y^2-1}$$ I included here what I had done so far. such …
user237685
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Proof that $x^2 + D = AB^y$ has in every case of $D,A,B$ a finite amount of solutions $x,y$

Could somebody please find me a proof that $$x^2 + D = AB^y$$ has in every case of $D,A,B$ a finite amount of solutions $x,y$. I forgot how this is called and would greatly appreciate it if someone could link me the proof
redelectrons
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