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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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Solve Quadratic diophantine equation in two unknowns.

Determine all $m,n \in \mathbb{Z}^+$ such that $m^2+1$ is a prime number and $10((m^2)+1)=n^2+1.$ Please provide complete explanation with solution. I have made an excel sheet with all the prime numbers from $2$ to $200$ as $= m^2+1;$ and I have…
Math Tise
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Find all solutions to the diophantine equation

QUESTION : Given integers $k_1, k_2, r,s$ such that they can take any integer (negative, positive or zero) in any of the given equations, find all possible values of $a,b,c,d$ that satisfy that criteria. Equations : $$ ar+cs=k_1, br+ds=k_2$$
tomtomtom
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Show: if $k = x^3 - y^3$, where $x, y \in \mathbb{Q}_{>0}$, then $k = u^3 + v^3$ with $u, v \in \mathbb{Q}_{>0}$

Suppose we have a $k \in \mathbb{Q} \ $ ($k > 0$) such that $$k = x^3 - y^3, \textrm{ where } x, y \in \mathbb{Q} \ \textbf{ and } \ x > y > 0.$$ That is, some (rational) number $k$ is the difference of two cubes of positive rational numbers. I…
user404596
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Theorem on diophantine equations in $\mathbb Z_n$ and in $\mathbb Z$

I have an exercise sheet without solutions, and I'm stuck on the last question. We consider the following diophantine equations : $(E11)\quad :\quad x^2-5y^2=11$ and $(E6)\quad :\quad x^2-5y^2=6$ Both equations have a solution in $\mathbb Z_5$ (they…
user493048
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Find all integer solutions to $y^3 = x^6 + 6x^3 + 13$.

Find all ordered pairs of integers $(x, y)$ that satisfy $$y^3 = x^6 + 6x^3 + 13.$$ I've found the solutions $(-1, 2)$ and $(2, 5)$. I believe that these are all the integer solutions, but I don't know how to prove it. Could someone please help?
Anon
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Integer solutions of : $u+pv=(x+py)(z+pt)$

Let $p$ be an odd prime and $u,v$ be integers. Then, there exist integers $x,y,z,t$ such that $$u+pv=(x+py)(z+pt)$$ How can I find all integers $x,y,z,t$ knowing $u,v$? What I have done: $$p^2(yt)+p(yz+xt-v)+xz-u=0$$ If $yt=0$ then it's trivial. So…
user97615
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When can we solve a diophantine equation with degree $2$ in $3$ unknowns completely?

The diophantine equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ can be solved completely : for every sixtupel $(A,B,C,D,E,F)$ we can determine the complete set of integer pairs satisfying the equation. What about the more complicated diophantine equation…
Peter
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Solve $2a + 5b = 20$

Is this equation solvable? It seems like you should be able to get a right number! If this is solvable can you tell me step by step on how you solved it. $$\begin{align} {2a + 5b} & = {20} \end{align}$$ My thinking process: $$\begin{align} {2a +…
Hobbs
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Finding value in diophantine equation

A professor returned to his home USA after attending a conference in paris and london. He have some money left in euros and pounds sterling. he want to traded both of them into dollars. he received 117.98 dollars with the exchange rate of 1 euro is…
Karumi
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Why is this number an integer?

I didn't understand why this $\alpha$ is an integer. I know that $(x+y\sqrt D)(x-y\sqrt D)=4z^n$, but I don't know what to do with this information. Note: I don't know if it's relevant but $K$ is a quadratic number field.
user42912
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Find integer solution(s) to $(n-2)(p^2+n)^{14}=z^{13}$.

Find integer solution(s) to $(n-2)(p^2+n)^{14}=z^{13}$. Background. Just a spinoff from my second answer to this question, Does the equation $a^{2} + b^{7} + c^{13} + d^{14} = e^{15}$ have a solution in positive integers where I found parametric…
Old Peter
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$u^3+v^3+3^{5}w^3=2\cdot3^{2}uvw$ has no non- trivial solutions if $u,v,w$ are pairwise co-prime.

Let $u,v,w$ be 3 pairwise coprime integers. Then $$u^3+v^3+3^{5}w^3=2\cdot3^{2}uvw$$ has no non-trivial solutions. How can I prove this? I have tried to consider many individual cases such as $uvw>0$,$uvw<0$, $max(u,v,w)=u$ etc. a pretty tedious…
user97615
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Does the equation $x^3+y^3+2z^3=0$ have any non-trivial pairwise co-prime integer solutions?

I know for any pairwise co-prime integers $x,y,z$ that $$x^3+y^3+z^3\neq 0$$ $$x^3+y^3+3z^3\neq 0$$ Do we also have $$x^3+y^3+2z^3\neq 0?$$ Any Hints?
user97615
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For which integers values of $r,s,t$ does this expression equal a square number?

Context: I am designing an online test for students where they shall construct a (normalized) polynomial $p$ with $\deg(p)=3$ $$p:\mathbb R\rightarrow\mathbb R,~x\mapsto x^3+bx^2+cx+d$$ where $b,c,d$ should be integers; they are given information…
Hirshy
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Solve this equation for the largest value of $n : n^2 + 2016n = x^2$

Solve this equation for the largest value of $n : n^2 + 2016n = x^2$ My attempt: $n^2 + 2016n = x^2$ Completing the square $n^2 + 2016n + 1008^2 = x^2 + 1008^2$ $(n + 1008)^2 = x^2 + 1008^2 $ Difference of two…
bigfocalchord
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