Mathematics Stack Exchange
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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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How would you solve the diophantine $x^4+y^4=2z^2$

I would like to know any way of solving the diophantine equation $x^4+y^4=2z^2$. Or ideas that seem worth trying out. By solving I mean fining all solutions and proving there are no more. Keith Conrad showed how to reduce this equation to a…
quanta
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solution for equation

For $a^2+b^2=c^2$ such that $a, b, c \in \mathbb{Z}$ Do we know whether the solution is finite or infinite for $a, b, c \in \mathbb{Z}$? We know $a=3, b=4, c=5$ is one of the solutions.
bsdshell
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$\sqrt{x} + \sqrt{y} = \sqrt{333}$ Better way to solve than trial and error

Question: $x$ and $y$ are positive integers and $\sqrt{x} + \sqrt{y} = \sqrt{333}$. Find the numerical value of $x + y$. I know the answer to the question is $185$ and $x$ and $y$ being $37$ and $148$ I got this through making a program that simply…
SomeGuyDoingMaths
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Integer solutions of $a^6 + 4 b^3 = c^6$.

I assume that the $$a^6 + 4 b^3 = c^6$$ has no solution in integers. I think this can be solved trivially, but no success so far. I tried to treat this as a $$(a^3)^2 + 4 b^3 = (c^3)^2 \\ (a^2)^3 + 4 b^3 = (c^2)^3$$ But no success. Could you…
Gevorg Hmayakyan
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Find integral solutions - rational points of $4p=x^2+27y^2$ for $p>3$ prime

Let $p\geqslant 5$ be a prime. I want to solve $$4p=x^{2}+27y^{2} \tag{1}\quad x,y\in \Bbb N$$ This comes when considering the discriminant of a cubic polynomial $T^3 -pT-yp$ with cyclic group. Are there only elementary ways to find solutions? I…
NevD
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solve this diophantine equation: ${\rm lcm}[x,y]+{\rm lcm}[y,z]+{\rm lcm}[z,x]=3(x+y+z)$

I am almost certain it is a duplicate question but I am looking for a reference regarding how solve the diophantine equation，Find the postive integer $x,y,z$ such $${\rm lcm}[x,y]+{\rm lcm}[y,z]+{\rm lcm}[z,x]=3(x+y+z)$$
math110
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Integer solutions for $a+b+c+d=0$, $a^3+b^3+c^3+d^3=24$

How to get all integer solutions for $$a+b+c+d=0,$$ and $$a^3+b^3+c^3+d^3=24$$ So far I've put $a=-b-c-d$ into 2nd equation and try to factorise it, but didn't find anything useful.
athos
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Solutions for diophantine equation $3^a+1=2^b$

I am looking for solutions for the diophantine equation $3^a+1=2^b$ where $a\in \Bbb N$ and $b\in \Bbb N$. Is there a power of $3$ that gives a power of $2$ when you add $1$? Two solutions are easy to find: $3^0+1=2^1 \rightarrow…
Hubert Schölnast
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how many natural numbers on a sphere

how many natural solutions are there to the following equation: $$ \sum_{i=0}^k x_i^2 = n$$ where $n,k \in\ \Bbb{N}$ i well like to get a answer for every n and k, but could do with just $k=2,3$.
elyashiv
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Diophantine equation $2y^4-2y^2 +1=z^2$

How to solve the diophantine equation $2y^4-2y^2 +1=z^2$, where $(y,z) \in \mathbb{N}^2$ ? Thanks, W
Watsoon
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Diophantine equations $x^n-y^n=2016$

Solve equation $$x^n-y^n=2016,$$ where $x,y,n \in \mathbb N$ My work so far: If $n=1$, then $y=k, x=k+2016, k\in \mathbb N$ If $n=2$, then $2016=2^5\cdot 3^2 \cdot 7$ $x-y=1; x+y=2016$ $x-y=2; x+y=1008$ ... If $n=3$, then…
Roman83
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Solve in positive integers the equation $a^3+b^3=9ab$

Solve in positive integers the equation: $$a^3+b^3=9ab$$ I try to: $$\dfrac{a^2}{b}+\dfrac{b^2}{a}=9\Longrightarrow a^2<9b,b^2<9a$$ Of course, I can't solve it. Can anyone help?
math110
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Natural numbers $a,b,c$ satisfaying $abc=2(a+b+c)$

How can one find all natural numbers such that: $a≤b≤c$ $$abc=2(a+b+c)$$ I tried this : $abc-2c=2a+2b$ so $c=\frac{2(a+b)}{ab-2}$
user233658
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Solve $x^n+y^n=2015$

Determine the natural numbers $x,y,n$ matching equality $$x^n+y^n=2015.$$ I noticed for $n = 1$ the equation has solutions $(x, 2015-x), x$ integer. For $n = 2$, given that $x$ and $y$ are different parities taking $x = 2k$ and $y=2m + 1$ we come…
medicu
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A diophantine equation $x^3+y^3-xy^2=1$

What kind of methods there are to find integer solutions of $x^3+y^3-xy^2=1$? I tried some inequalities and congruences without success. I also found on Wikipedia that this might be a Thue equation but I have no idea what is a bivariate form.
BeginnerAlgebraist
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