# Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

3,815 questions
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### What are the possible solutions for the diophantine equation $4x^2-3y^2=1$ and is there a general formula?

Assuming that $a = x^2$ and $b = y^2$, i converted this equation to a linear diophantine equation for sake of convenience: $$4a - 3b = 1$$ where after calculating a particular solution (like $(1, 1)$...
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### Area of a digital disk

Considering the number of integer solutions of $x^2+y^2\le n^2$, a digital disk, which is obviously asymptotic to $\pi n^2$, how can we find a tight upper bound of the form $$an^2+bn+c\ ?$$ The ...
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### What property of rational solutions of a Diophantine equation makes this graph impossible?

Yesterday I read up on Diophantine equations and the property in which two rational solutions of such an equation make a line that gives a third rational solution to the equation. I was thinking about ...
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### Need help to understand this solution of $2^mp^2 +1=q^5$

The Question- Find all triples $(m,p,q)$ where $m$ is a positive integer and $p , > q$ are primes. ****$2^mp^2 +1=q^5$**** After trying my best to solve this problem, and progressing a ...
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### Solving a Diophantine equation $3a^3+3b^3=a^3+c^3$

A friend of mine gave me the following problem Find all integers solutions to $$3a^3+3b^3=a^3+c^3$$ Of course $(0,0,0)$ is a solution, and I think that there are no others, but I can’t prove it. I ...
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### Solving the Diophantine equation $7x+4y=100$

I have attempted to solve this Diophantine equation: But the solution I got is different from the one given above, my general solution is: $x = -100 + 4t$ & $y = 200 - 7t$ , is the solution ...
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### If $x, y, z$ are three distinct positive integers, where $x + y + z = 13$ and $xy, xz, yz$ form an arithmetic…

If $x, y, z$ are three distinct positive integers, where $x + y + z = 13$ and $xy, xz, yz$ form an increasing arithmetic sequence, what is the value of $(x + y)^z$ ? I've been trying to solve this by ...
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### Solving the diophantine $y^3=2x^2$

I am trying to find the general solutions to the diophantine $y^3=2x^2$. Usually I can solve these using coprime/prime factorisation methods but I have tried and cannot seem to get the correct general ...
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### Solvability of a Diophantine equation in mod 8 class

How does one solve an equation of the form $$3^a - b^3 \equiv 1 \ ({\rm mod}\ 8)?$$
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### Pirates and Bag of Coins [duplicate]

A group of pirates with 17 members steal a bag of golden coins. When they share their loot evenly, it left with 3 coins. When they argue who should get the remaining of the coins, one of them is ...
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### Find $1 \le a < b \le n$ such that $a\cdot b + a + b = n\cdot (n+1)/2$

Find $1 \le a < b \le n$ such that $$a\cdot b + a + b = \frac{n\cdot (n+1)}2$$ Is there a more efficient way than picking $a$ or $b$ and trying all values between $1$ and $n$ ?
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### Integer Solutions to a Two-Sheeted Hyperboloid

During some free time I had, I was wondering how to find the integer solutions $(x,y,z)$ to this generalized equation: $$z^2=axy+bx+cy+d$$ I am specifically looking for ways that do no involve ...
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### Difference between diophantine equations $ax + by = c$ and $ax - by = c$

I had just solved this problem of Sierpinski Prove that the sequence $2^n - 3, \; n = 2,3,4, \dots$ contains infinitely many terms divisible by $5$ and $13$ but no terms divisible by $5\cdot 13$...
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### How do you generate for a given solution for a linear diophantine equation more solutions

How can I generate for a given solution of a linear diophantine equation all solutions? For example let $21x+12y+9z=9$. I found one solution to be $(-3+3t,6-6t,t),t\in\mathbb Z$. How can I generate ...
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### General Solution of Linear Diophantine Equation with 2 and more than 2 variable

We know the theorem of linear diophantine equation from Bezout's identity that the solution is in ordered pair form: $\left(x + m \dfrac{b}{\text{gcd}(a,b)}\,,\,y-m \dfrac{a}{\text{gcd}(a,b)}\right)$ ...
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### Solving a non linear diophantine equation $px+y^{p-1}=2017$ [duplicate]

I had been tasked in an exam to solve this equation: $px+y^{p-1}=2017$ where $p$ is a positive prime number, and $x$ and $y$ are natural numbers. I was able to prove that if $p≥5$ then $p=7$, so now I ...
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