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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.

5324 questions
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Find all prime numbers p,q

Find all prime numbers $p,q$ so that $pq$ divides $(5^p-2^p)(5^q-2^q)$. I expanded the factors and got the result that $pq$ divides $5^p$ and $2^q$, which gives that the must be either $2$ or $5$, but that's not a solution.
MathFanatics
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Unique solution of a diophantine equation

Suppose $m_{1}^{h}+\cdots m_{k}^{h}=n_{1}^{h}+\cdots n_{k}^{h}$ for $h=1,\dots ,k$, where $0
ericc
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About the diophantine equation $y^3=8x^6+2x^3y-y^2$

How I can solve the equation $y^3=8x^6+2x^3y-y^2$ in integers? I made the substition $x^3=z$ and got the equation $8z^2+2yz-y^3-y^2=0$. So I decided to apply the general formula for quadratic equations and thus I got $$z=\frac{-2y\pm…
Xam
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Find 7-tuples of pairwise distinct positive integers such that the sum of squares of first 4 equals sum of squares of last 3

As already stated in title, find 7-tuples ($a_1,a_2,a_3,a_4,b_1,b_2,b_3$) of pairwise distinct positive integers such that $$a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$ This came in RMO 2016 Delhi paper where one was asked to prove that infinite…
ghosts_in_the_code
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solutions of the equation $x^3-y^3=z!-18$

What are the solutions of the equation $x^3-y^3=z!-18$? Here $x,y,z$ are non-negative integers. I have tried brute force but is there a better method?
Anshul
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Diophantine equation $10^x=yzwt-3$

I have resolved, brute force, the following problem someone asked me: Solve the Diophantine equation $$10^x=yzwt-3\space \text{where}\space \space y,z,w,t \space \text {are distinct primes}$$ $$ **********$$ I found the solution…
Piquito
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Find all positive integers solutions for $x^2-xy-y^2=1$

Find all positive integers solutions for the following diophantine equation $$x^2-xy-y^2=1$$ My work so far 1)$$x^2-xy-y^2-1=0$$ $$D=y^2+4(y^2+1)=5y^2+4=k^2, k \in \mathbb Z$$ 2)$$ y^2+xy-x^2+1=0$$ $$D=x^2+4x^2-4=5x^2-4=m^2, m\in \mathbb Z$$
Roman83
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Find the integer solution of $ a^b = 2^{2 c + 1} + 2^c + 1 $

Find the possible number of integer solution for this equation, such that $ b>1$ $$ a^b = 2^{2 c + 1} + 2^c + 1 $$ From $1$ to $1000$, $ {a = 2, b = 2, c=0} $ and $ {a = 23, b = 2, c=4} $ computationally. Are there any other possible solutions? How…
S L
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$x^5 - y^2 = 4$ has no solution mod $m$

A common technique for proving that a diophantine equation does not have a solution is to prove that it does not have a solution mod $m$ for a suitable modulus $m$. This technique works with $m=11$ for the equation $x^5 - y^2 = 4$ mentioned in this…
lhf
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Parametrization of $a^2+b^2+c^2=d^2+e^2+f^2$

Is there an existing parametrization of the equation above that is similar to Brahmagupta's identity for $a^2+b^2=c^2+d^2$? I need either a reference to look it up or a hint to solve it. Thanks.
user97615
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Can only the middle school math knowlegde help to find solutions for $2013 y^2 -xy -4026 x=0$?

I found the following equation form an answer written for a question. $$2013 y^2 -xy -4026 x=0$$ But I'm confused that can I really learn how to find the positive integer solutions for $x,y$ with having only the middle school mathematics knowledge.…
Tharindu Sathischandra
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Equation $x^3+2x+1=2^n$ in positive integers

Determine all pairs of positive integers $(x,n)$ which satisfy the condition $$x^3+2x+1=2^n.$$ My work so far: No solution exists for $n=1$. For $n=2$ we get $x=1$. We show that no solutions exist for $n\ge3$. Suppose that $n \ge 3$. Obviously, $x$…
Roman83
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What kind of Diophantine equations are these?

Assume that the $a$'s in the following equations must be limited to a binary values of $\{0,1\}$ and the $c$'s are integer constants: \begin{align} 2a_{34}a_{23}a_{24} &= c_1\\ a_{23} + a_{24} + a_{34} &=…
Hooked
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Solve this equation $xy-\frac{(x+y)^2}{n}=n-4$

Let $n>4$ be a given positive integer. Find all pairs of positive integers $(x,y)$ such that $$xy-\dfrac{(x+y)^2}{n}=n-4$$ What I tried is to use $$nxy-(x+y)^2=n^2-4n\Longrightarrow (n-2)^2+(x+y)^2=nxy+4$$
user246688
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Diophantine with Gaussian Integer

I'm trying to find the set of solutions to a specific diophantine equation over $\mathbb{Z}[i]$. The equation is the following: $$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$ with $ z_1$ (resp $z_2$) such that $\exists a,b \in \mathbb{Z} , z_1$ (resp $z_2$)…
Mitch
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