# Questions tagged [diophantine-equations]

Questions on the use of Mathematica to find integer/rational solutions to equations.

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### Solving Diophantine Equation with Only Specific Variable Values Allowed

I am trying to Solve an equation like this: $$\sum _{i=1}^{\text{iMax}} (-i+\text{iMax}+1) n(i)=546$$ Where all the $n(i)$ are can only take on the values 2,3,6,7, or 8. I really have no idea how to ...
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### Finding coprime solutions to Diophantine equations

I'm hoping to use Mathematica to find solutions to Diophantine equations. Below is a toy example of something I would like to try. Consider the case of $$x^2+y^2 = 17$$ which has the solution $x=1,y=4$...
170 views

### Extremum of the graph of a function has integer coordinates

Graph of the fuction $y=\dfrac{(x-26)(x+9)}{(x+14)(x+19)}$ has maximum point and minimum point are (-16,-49) and (-4,-1) whose ...
1 vote
85 views

### Solving a-two-variable equation in primes

How solve the following equation in Mathematica (preferably in one line) for pairs of $(x,y)$ such that $x$ and $y$ are primes? $x^3-y^4=1$
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### How can I find all integer numbers so that mydistance is an integer number?

I am tring to selec two points A, B on the sphere (x-2)^2 + (y-4)^2 + (z-6)^2 ==9^2 so that EuclideanDistance[pA,pB] is an ...
878 views

### How can I select four points on a sphere to make a regular tetrahedron so that its coordinates are integer numbers?

I want to select four points lie on the sphere (x-1)^2 + (y-3)^2 + (z-5)^2 = (5* Sqrt)^2 so that its coordinates are integer numbers to make a regular ...
382 views

### How can I get all squares on this sphere so that its coordinates are integer numbers?

I have the sphere (x-2)^2 + (y-4)^2 + (z-6)^2 = 15^2. I want to select all squares on this sphere so that its coordiantes are twelve different integer numbers like ...
218 views

### How can I find four points on a circle to make a square so that its coordinates are integer numbers?

I have a circle and I want to select four points to make a square so that its coordiantes are eight different integer numbers like this ...
1 vote
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### Parametric integer solution of $aX^2+bY^2-cZ^2=0$

Let $a,b,c\in \mathbb{Z^+}$ and unknowns $X,Y,Z\in \mathbb{Z}$. How can I get the parametric solution $(X,Y,Z)$ of equation $$aX^2+bY^2-cZ^2=0$$ in Mathematica? In pari/gp, this solution can be ...
221 views

### How can I reduce the time to compile this solve?

I want to find five integer numbers $a, b, c, d, e$ so that $a+ b+ c + d + e = 9 k$ and none of two number in which are equal. I tried. ...
942 views

### Optimize search for rational numbers on unit circle?

Crossposted: https://community.wolfram.com/groups/-/m/t/2763509 I'd like to classify rational numbers on a unit circle by the following property: number of digits in the denominator. It relates to an ...
258 views

### Number of solutions in natural numbers $a+b+c+d=13$

I would appreciate it if somebody could help me with the following problem: Q: To find the number of ordered pairs of natural numbers in the following equation $$a+b+c+d=13$$ we want to add the ...
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### One more solution of the Mordell equation

Please, help to find one more solution of the Mordell equation $y^2=x^3+n,\quad n\in\mathbb{Z}$ $$y^2=x^3-307$$ Using Solve I was able to go up to ...
477 views

### Calculating the integral points of an elliptic curve

I asked this question on Math stachexchange. The question I have is: Can I use Mathematica to find the (integral points on the following elliptic curve) or can I find when the number $\text{n}$ is a ...
1 vote
Well, I'm trying to investigate the density of solutions to Diophantine equations. What is a general method to describe that density? I've two functions: $$\varphi\left(\text{a},\text{b},\text{c}\... 2 votes 2 answers 157 views ### Finding a program that can find integer solutions for large values of a variable I've the following code: ... 2 votes 0 answers 74 views ### Never terminating when solving linear Diophantine systems Say I have: (1): Set of Diophantine systems where each defines a (possibly unbounded) polyhedron. This is solvable with existing algorithms: (*): The (1) can be solved in SageMath with ... -1 votes 1 answer 76 views ### Quadratic equation with integral coefficients [closed] Let a,b,c  be Natural Numbers, such that roots of the equation ax^2+bx+c=0 are distinct and both lie in the interval (0,1) (1,2) (2,3) (Brackets signify open interval, roots are IN BETWEEN  ... 4 votes 1 answer 900 views ### Why does Solve lock up when trying to solve the quadratic equation with large integers? Why does Solve lock up when trying to solve the equation ... 2 votes 4 answers 306 views ### FrobeniusSolve with solutions only being 0 or 1 being acceptable I want an EFFICIENT way of solving the Frobenius Equation a1 n1 + a2 n2 + a3 n3 + ... + aK nK = U where n1, n2, ..., nK are restricted to be either 0 or 1. I ... 3 votes 4 answers 248 views ### How do I declare a countably infinite list of variables as being integers? I want to do this, Solve[ { n1 + n2 + n3 + .... = 10, n1 + 2 n2 + 3 n3 + .... = 100}, {n1,n2,n3,....}, Assumptions -> {n1,n2,n3,.... are Integers >=0} ] ... 2 votes 2 answers 142 views ### On the solvability of the Diophantine equation a(x^2-y^2)+2bxy=1 Take two integers a,b with \mathrm{GCD}(a,b)=1 and a odd. Consider the Diophantine equation$$ a(x^2-y^2)+2bxy=1,\qquad x,y\in\mathbb Z $$Is there any efficient way to determine whether such ... 0 votes 1 answer 137 views ### Unexpected omission by Wolfram Alpha [closed] When Alpha is submitted the equation a(a^2-1)=2b^2, it unexpectedly forgets the integer solution a=1,b=0. What could explain this ? http://www.wolframalpha.com/input/?i=a(a%5E2-1)%3D2b%5E2 4 votes 2 answers 792 views ### How can I reduce the time to run this code? I am trying to find the integers  a, b, c, d \in [-15, -1] \cup [1, 15]  so that the equation  \left| x^2 + a x + b \right| = c x + d  has four distinct integeral solutions different from  0 . ... 5 votes 1 answer 326 views ### Pythagorean quadruples I am very interested in the problem that deals with the Pythagorean quadrruples, which are listed in rosettacode. Unfortunately, I have not been able to find any way to compute them with Mathematica. ... 3 votes 1 answer 108 views ### Guess Diophantine equation from its solutions Take for example the equation$$ n^2=x^2+y^2+1 $$where x,y are integers. This equation admits solutions only for some values of n, to wit,$$ n=1,3,9,17,19,33,35,51,73,81,99,\dots $$Can we ... 1 vote 1 answer 430 views ### How do we show reduction steps? This produces 3 equivalent Diophantine equations. How can we show the reduction steps? I need to do this for a paper. ... 4 votes 3 answers 644 views ### Finding Smallest Positive Solution to Diophantine Equation Mathematica can solve v^2- d u^2=4 quickly, even for nonsmall d: d = 400004; Reduce[v^2 - d u^2 == 4, Integers] d = 400012; Reduce[v^2 - d u^2 == 4, Integers] ... 4 votes 3 answers 737 views ### Solving a system of diophantine equations from a mathematical competition There is a problem from the 66th Putnam Mathematical Competition, B2 Find all positive integers n,k_1,k_2,...,k_n such that$$k_1+\cdots+k_n=5n-4$$and$$\frac{1}{k_1}+\cdots+\frac{1}{...
Solve this equation using Mathematica: $2^x + 113^y = z^2$, $\operatorname{gcd}(x, y) = 1$.