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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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Integer solutions of $(b^2+1)(c^2+1)=a^2+1$

As the title says, I'm interested in integer solutions of the equation $(b^2+1)(c^2+1)=a^2+1$. Is it possible to parametrize the solutions as in the case of Pythagorean triples? If yes, then how would one proceed to find a parametrization in this…
Beni Bogosel
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Integer solutions of $x+y+z=axyz$

let $x,y,z$ be 3 co-prime integers and $a$ a positive even integer. Find all the integer solutions of $$x+y+z=axyz$$ Any hints? What I have done: Suppose there exist $rst\neq 0$ where $$y+z=rx$$ $$z+x=sy$$ $$x+y=tz$$ and…
user97615
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Solve the Diophantine equation $a+2b=2ab$, where $(a,b)$ are positive integers.

Let $a$ be a positive integer. Show that $\gcd(a, a-1) = 1.$ Let $d$ be the greatest common divisor of $a$ and $a-1.$ I.e. $\gcd(a, a-1) = d$ Therefore, $d$ must divide $a-(a-1),$ following the rule that if any number, when some number d divides two…
knowledge_is_power
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$2^n+3^n= m^k$ imply $k=1$

If $m,n,k$ are natural numbers such that $$2^n+3^n= m^k$$ where demonstrate that $k=1.$ I tried to prove that divides equal amount left 5 but does not divide by 25 and we did. Maybe someone else has the idea. Thank you!
medicu
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Are there different integers so that $ab=c^2$ and $a^2+b^2=2d^2$?

I wish to find 2 integers $a,b\ \ (a\neq b)$ so that $GM(a,b)$ and $RMS(a,b)$ are integers. So I write down: $$\sqrt{ab}=c\ \ \rightarrow ab=c^2\\\sqrt{\frac{a^2+b^2}{2}}=d\ \ \rightarrow a^2+b^2=2d^2$$ By lots of trial and error I think there…
user370285
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Exactly 6 integer solutions

I am trying to find the greatest value of $"c"$ for which $7x + 9y = c$ has exactly six solutions over positive integers. My approach: Let one of the solution = $(a, b)$ General solution is $(a + 9t, b - 7t)$ For values t = 0, 1, 2, 3, 4, 5 we get 6…
Ramakrishna S
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Find the positive integer solutions for the equation $x^y-y^x=1$

Find all positive integer $x,y$ such $$x^y-y^x=1$$ It is clear $(x,y)=(2,1)$ and $(x,y)=(3,2)$ hold.
math110
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The equations $ax^2+by^2+cz^2=0$ and $ax^2+by^2+cz^2=1$

Let $a,b,c -$ a non-zero integers. It is known that the equation $ax^2+by^2+cz^2=0$ it has a non-zero integer solution. Prove that the equation $ax^2+by^2+cz^2=1$ has solution in rational numbers. My work so far: Let $a>0,b>0,c<0$ and $z<0$. Let…
Roman83
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How to find all the integral solutions of $x^2-by^2=z^k$ where $k$ is an odd integer >2 and $b>0$?

Consider the bivariate quadratic polynomial of the form: $$ x^2-by^2=z^k$$ where $k$ is an odd integer>2 and $b>0$. It's well-known that Euler's method: $$x^2-by^2=(p^2-bq^2)^k $$ provides a class of integer solutions. I am interested in finding a…
user97615
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Diophantine Equation: $4x^r = 3y^2 + 1$

If $r \ge 3$ is an integer, show that $4x^r = 3y^2 + 1$ does not have positive integer solutions $(x, y)$ except for $(1, 1)$. (I am not sure whether this is an open problem; in any case, it is a slightly stronger but probably more familiar form of…
Eric Neyman
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Integer solutions to $210y^2=(x)(x+1)(2x+1)$

I'm looking to find integer solutions for large positive $y$ values (say over 1000) to the following equation: $210y^2=(x)(x+1)(2x+1)$ What I know so far: Integer solutions include (0,0) and (7,2) Rational Solutions include (2/3, 1/9), (3/4, 1/8),…
dora
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Diophantine equation, 3 variables

How do I solve the following equation, where $x,y,z$ have to be positive integers? $$ \frac{x^2}{y} + \frac{y^2}{z}+ \frac{z^2}{x}= \frac{y^2}{x} + \frac{z^2}{y} + \frac{x^2}{z} $$ Given that $$xyz=2205$$
Higgsino
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$\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$, where $y$ is and odd integer less than $61$. Find the positive integer solutions (x,y).

$\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$, where $y$ is and odd integer less than $61$. Find the positive integer solutions (x,y).
HOLYBIBLETHE
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Solution of equation of the form $n = 1234a + 56b + 7c$

I have $n = 1234a + 56b + 7c$. Is there a way to check if a triplet $(a,b,c)$ exists, such that all three are non-negative?
Sayan Ghosh
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System of diophantine equations $x^2+3y=u^2$, $y^2+3x=v^2$

Solve the following system of Diophantine equations(the unknowns are positive integers): $$ \left\{ \begin{array}{c} x^2+3y=u^2 \\ y^2+3x=v^2 \end{array} \right. $$ I worked as follows: subtract the two equations to get:…
Hamid Reza Ebrahimi
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