Mathematics Stack Exchange
Mathematics Stack Exchange

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 to find the number of integral solutions when exponential functions are involved?

take the example $2^x+2^y= 9$ (say), in general how would I find the number of solutions that exist? Here of course the answers are $(3,0) \text{ or }(0,3)$. But how would that work for something more complicated like $2^x +2^y +2^z= 2336$?
math and physics forever
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Wrong result with diophantine equation when there is subtraction

I have equation $966x-686y=70$ and I get the wrong solution every time when there is "-" in equation $ax\textbf{-by}=c$. I don't know where I am making a mistake. And my solution seems like that: step 1) $GCD(966,686) =14\\ 966=…
DScounterGO
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single quadratic diphontine equation with 4 unknowns, all solutions?

I am trying to find a general solution for $d\left(a+b-c\right) = a^2+b^2-c^2$ with the requirement that all variables are integers. Obviously $\frac{a^2+b^2-c^2}{a+b-c}$ must an integer, but my understanding is that there must be some constraints…
Mark
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Rational points on $y^2=5(x^4+1)$

How would one find $\mathbb{Q}$ -rational points on $y^2=5(x^4+1)$. I do not think there is one but I would also love to see a proof of it. I thought about writing them in terms of fractions and got the following diophatine equation…
UnsinkableSam
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Solving the Diophantine equation $x^3+y^3+z^3=txyz$

Solve in $\mathbb{Z}^4$ $$x^3+y^3+z^3=txyz$$ (Ion Ionescu, 1931) This is the problem. What I tried to do: $t=3+k\Rightarrow$ $$(x+y+z)(x^2+y^2+z^2-xy-xz-yz)=kxyz$$ For $k = 0$ we have infinite solutions. For $k\ne0$ I have no idea. Please help, I am…
Neox
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The Diophantine equation $ax^2+bxy+cy^2=0$ has integer solutions

Let $q(x,y)=ax^2+bxy+cy^2$ is a binary quadratic form in integers $x,y$. Then my question is: Find conditions on $a,b,c$ such that the Diophantine equation: $ax^2+bxy+cy^2=0$ has integer solutions $(x,y)$ with $x>0,y>0$ and $x,y$ have a common prime…
Safwane
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How to prove if $xy$ is a cube, $\gcd(x,y)=3$ and $3$ divides $y$ only once, then $3x=r^3$ and $3^2y=s^3$ for some integers $r,s$?

Let $x$ and $y$ be nonzero integers where $xy$ is a cube. If $\gcd(x,y)=3$ and $3$ divides $y$ only once, then there are integers $r,s$, such that: $$3x=r^3$$ $$3^2y=s^3.$$ What is a simple method to prove this? Any input will be appreciated. I put…
NumThcurious
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Diophantine equation $3x^2+y^2=z$

I am currently facing a Diophantine equation $3x^2+y^2=z$, in which $x$, $y$, $z$ are integers. My major is not math and I am entirely new to Diophantine equation. I googled this but only found questions like $ax^2+by^2=z^2$, which are not the same…
Shuo
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$\frac{ (359\cdot (109+215\cdot x)-1)}{10^x}=y$

Consider the diophantine equation: $\frac{ (359\cdot (109+215\cdot x)-1)}{10^x}=y$, for x,y positive. The only solution I found is $x=2$, $y=1935$. Can it be proven that if there is a solution there are infinitely many other solutions?
Enzo Creti
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The equation $ax+bu+cs=ayz+bvw+crt$ also has no integers solutions for any integers $a,b,c$

Assuming that the system of equations: $$x=yz$$ $$u=vw$$ $$s=rt$$ has no integers solutions. How I can prove that the equation $$ax+bu+cs=ayz+bvw+crt$$ also has no integers solutions for any integers $a,b,c$
Safwane
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Find the integer solutions of $y^x=x^{50}$

I can't solve this olympiad problem, I tried with simple cases: $4^2=2^k$ And I think maybe that $y=50^k$. But I can't keep going?
user817101
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Diophantine equation with negative numbers

Good morning, I can't solve this diophantine equation through the Euclidean division: $45x - 8y = 231$ $45x'-8y' = 1$ Euclidean division $45 = -5*(-8)+5$ $-8 = -2*5+2$ $5 = 2*2+1$ $2 = 2*1+0$ $1 = 5 -2*2$ $2 = -8+2*5$ $5 = 45-5*8$ $1 =…
Shyvert
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prove or disprove that the integer solutions to equation $x^2-2xy+y^2-x+y+1=x^3-2y^3$ are only ( x=1, y=0) and (x=-2, y=3).

integer solutions to equation $x^2-2xy+y^2-x+y+1=x^3-2y^3$ are only ( x=1, y=0) and (x=-2, y=3). This question is related to another question about a system of equations. I showed how these solution can be the solution of system of equation but…
sirous
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Integer solutions of $3^x = 2 + y^2$

$$3^x = 2+y^2$$ Solve for the integer values of $x$ and $y$. Can this be solved using graph theory and calculus? I am trying out for half an hour and uses most of my preknowledge but still it is not solved, however I found two solutions for…
aman rana
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Can $3^x=2y^2-1$ be solved over the natural numbers?

In the equation $3^x=2y^2-1$, $x$, $y$ are natural numbers. I found $x=1$ or $2$ (mod $4$), and $y^2=1$ or $4$ (mod $120$) but I even don't know if the number of solutions is infinite. Is there a way to find the solution of this indeterminate…
eandpiandi
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