# Tagged Questions

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

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### diophantine linear equation with condition on the sum of coefficients [closed]

I am trying to solve the following diophantine equation x*a + y*b == c where a,b,c are integers and the absolute value of their sums is, for instance, 4. Then I ...
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### how to find sum of variable c1+c2+c3 for expression combinations

how to find sum of variable c1 + c2 + c3 for expression combinations: ...
By Mathematica, I try to obtain solutions of Diophantine equations such as: $$F_{n_1}F_{n_2}\cdots F_{n_k}+1=F_t$$ where the sequence $\{F_n\}$ is the Fibonacci sequence,$n_1 < n_2 < \cdots &... 0answers 85 views ### Find out whether solutions exist rather than full-blown solution search I have some quadratics, and I am trying to find out whether there exist solutions in the integers. The following tells me that the first does, wheraeas the second doesn't: ... 4answers 241 views ### How to merge permutations obtained from Solve on multiple variables? When I compute the following expression to find integer solutions of the equation (x^2 + y^2 + z^2 == 14^2) ... 2answers 192 views ### Diophantine equation: getting a list of solutions for each coeficient I'm trying to solve the following diophantine equation,$$x^3+dy^3=1$$ such that $$x,y \in \mathbb{Z} \land y \neq 0$$ and I want to form a list of solutions for each $$d\in[1,1000]; d\in\mathbb{N}$$ ... 2answers 118 views ### Optimization problems over the Integers Should be an easy question. How can I define a function which gets only integers in a form that I can use to find max and min for instance? Example: Find the Max of the expression ... 0answers 172 views ### Solving Thue equations A Thue equation is a 2-variable homogeneous integer polynomial of degree at least 3. It's well-known that such equations have only finitely many solutions over the integers. I'm trying to solve some ... 1answer 78 views ### equidistant solutions in Sequence Sums That Are Squares In my demonstration "Sequence Sums That Are Squares" I demonstrate two "lines" of solutions (of infinite length) and ask whether someone can find one more line. I am looking forward to your solution. ... 2answers 119 views ### Problem on Consecutive Sequence Sums that are Squares Here is a seemingly simple Problem: I have two natural numbers n and m, n < m such that S1:= n +...+ m is a square and also S2:= n +...+ m + (m+1) is a square. Problem a) : Find n and m. You ... 4answers 601 views ### find the number of integral solutions a+b+c+d+e+f = 18 [duplicate] Find the number of integral solutions of a + b + c + d + e + f = 18 where a, b, c, d, e, f are elements of the range ... 1answer 289 views ### Efficient way to solve equal sums$x_1^k+x_2^k+\dots+x_5^k=y_1^k+y_2^k+\dots+y_5^k$with Mathematica? I need to solve the system of equations, call it$S_1$, in the integers $$x_1x_2x_3x_4x_5 = y_1y_2y_3y_4y_5$$ $$x_1^k+x_2^k+\dots+x_5^k=y_1^k+y_2^k+\dots+y_5^k,\;\; k= 2,4,6$$ I used a very ... 1answer 776 views ### How to solve this equation with integers as a solution? I want to solve the equation $$x^y + y = y^x + x$$ with$x$,$y$are integer numbers. I tried Solve[x^y + y == y^x + x, {x, y}, Integers] How to solve the ... 3answers 196 views ### How do I generate a set of n-tuples containing integral solutions to a linear equation provided certain constraints? Let$m,k,p$be fixed positive integers. I want to create a table of k-tuples$(x_1,x_2,\ldots,x_k)$comprised of solutions in positive integers to the equation below: $$x_1+x_2+\cdots+x_k=m\quad\... 2answers 657 views ### Extracting Reduce results I'm solving a Diophantine equation inside of a function using Reduce but I'm having trouble extracting the necessary parts of the answer. For example, if my input ... 0answers 338 views ### Computing Ehrhart's polynomial for a convex polytope Is there a Mathematica implementation for computing the Ehrhart polynomial of a convex polytope which is specified either by its vertices or by a set of inequalities? I am interested in knowing this ... 6answers 3k views ### How to find lattice points on a line segment? How do I find points on the line segment joining {-4, 11} and {16, -1} whose coordinates are positive integers? 2answers 749 views ### Solving/Reducing equations in \mathbb{Z}/p\mathbb{Z} I was trying to find all the numbers n for which 2^n=n\mod 10^k using Mathematica. My first try: Reduce[2^n == n, n, Modulus -> 100] However, I receive ... 2answers 452 views ### Is there a way to use functions like Prime[n] within Solve[]? I'm trying to see if a number can be written as the sum of two prime numbers. Ideally, I would like to use Solve[ Prime[n] + Prime[m] == 100, {n, m}] But that ... 1answer 1k views ### How to solve this equation with positive integers as a solutions? This is a problem of United Kingdom Mathematical Olympiad. Find all triples (x,y,z) of positive integers such that$$\biggl(1+\dfrac{1}{x}\biggr)\cdot \biggl(1+\dfrac{1}{y}\biggr)\cdot \biggl(1+\... 1answer 776 views ### Using 'Reduce' to solve a set of inequalities, specified by a list I have a two lists$LHS$and$RHS$, both of size$n\$. I want to solve a system of inequalities of the form: $$LHS[1] \leq RHS[1]$$ $$LHS[2] \leq RHS[2]$$ $$LHS[3] \leq RHS[3]$$ $$\vdots$$ $$LHS[n]... 3answers 2k views ### Finding the number of solutions to a diophantine equation I want to count total number of the natural solutions (different from 0) of the equation 2x + 3y + z = 100, but don't know how. How can I calculate it using Mathematica? I tried: ... 3answers 1k views ### How can I solve the equation with integers as a solution? I want to solve the equation$$(x-1)^2 + (y-1)^2 + (z-1)^2 = 49$$where x, y, z are integer and x \neq 1, y \neq 1, x \neq 1. How do I tell Mathematica to do that? 4answers 585 views ### Defining a Unique Domain for Solving Diophantine Equations I am working on a research problem in discrete geometry to do with sphere packings, and believe it or not, I have been able to reduce it to finding the solutions to the Diophantine equation,$$n = \...
I tried to find a non-trivial integer solution to the equation $$2012^2=a^2+b^2+c^2+d^2+e^2$$ with Mathematica but the computation takes minutes; I might be doing something wrong. ...