# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### Some Diophantine problems for equal sums with high powers

Given rationals $R = a,b,c,d,e,f$. Define, $$F_n = a^n+b^n+c^n-(d^n+e^n+f^n)$$ If $F_\color{red}1=0$, is there a rational solution to $7F_3x^4+7F_5x^2+F_7 = 0$? Then for $k=1,2,8$, ...
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### Number of integral solutions to a polynomial

Given a polynomial of $n$th order, represented by $$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+a_{n-1}x+a_{n}=0$$ Is it possible to find the number of integral solutions/roots to ...
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### Unique Solution To The Diophantine Equation

Show that the following Diophantine equation has a unique solution in positive integers $x^n+y^n=(x+y)^m$ with $x>y, m>1,n>1$. This could be solved by a direct use of Zsigmondy's theorem. ...
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Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ... 0answers 57 views ### Solution of a equation in natural number nvolving reciprocal of prime Let p be a prime and n a natural number . Solve in \mathbb{N} the equation$$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$0answers 112 views ### How to solve x^4+y^4=n? How to solve Diophantus equation$$x^4+y^4=n $$where x,y and n are positive integers. We know that Theorem: A natural numbern n can be represented as a sum of two squares if and only if ... 0answers 110 views ### how many natural numbers on a sphere how many natural solutions are there to the following equation:$$ \sum_{i=0}^k x_i^2 = n$$where n,k \in\ \Bbb{N} i well like to get a answer for every n and k, but could do with just k=2,3. 0answers 347 views ### Diophantine with Gaussian Integer I'm trying to find the set of solutions to a specific diophantine equation over \mathbb{Z}[i]. The equation is the following:$$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$with  z_1 (resp z_2) such ... 0answers 398 views ### XX^t=A, X=?. Where X \in \{0,1\}^{n \times m} The problem: XX^t=A, \quad (X_{ij}\in{0,1}, \quad \sum_{j=1}^m x_{ij}=2), \quad X=? Details: n,m \in N A \in \{0,1,2\}^{n \times n} X \in \{0,1\}^{n \times m} A is a ... 0answers 281 views ### Counting Solutions of Diophantine Inequalities I understand that Diophantine Analysis is an enormous field! Without first determining the solution set, suppose I'd like to calculate the number of non-negative integer solutions (x,y,z) of ... 0answers 212 views ### Upper bound for the quality of an abc-triple A triple of positive integers (a,b,c) is an abc-triple if a and b are coprime and c = a + b. Define the quality or power of an abc-triple as P(a,b,c) = \frac{\log c}{\log ... 0answers 16 views ### If a\not\equiv 0\mod{p} then there are p-1 solutions (ordered pairs) to x^2-y^2\equiv a\mod{p} Let p be an odd prime, and let a\in\mathbb{Z}_p such that a\not\equiv 0. I need to show that there are p-1 ordered pairs (x,y) such that x^2-y^2\equiv a \mod{p}. As I see it, the ... 0answers 37 views ### Diophantine System Solution Could you please help with finding of general solution of diophantine system for rational a, b, c, d (a^2+b^2)(c^2+d^2)=A^2 (a^2-b^2)(c^2-d^2)=B^2 for some rational A and B. This is related ... 0answers 69 views ### Quartic Diophantine equation  2 x^4 - 2 x^2 = 3 (y^2 - 1) About the quartic Diophantine equation:$$ 2 x^4 - 2 x^2 = 3 (y^2 - 1)$$On oeis.org/A180445 it says that all positive solutions (x,y) are:$$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$... 0answers 52 views ### On the integer solutions to u^2+163v^2=w^3 and others It seems the solution of,$$u^2+dv^2 = w^3\tag1$$involves the class number h(d). Assume \gcd(u,v)=1. Q: For which \color{red}{prime}\; d is the complete solution of (1) in the integers ... 0answers 57 views ### Without solving it, is there an elementary way to show that X^3+Y^3=Z^3 has a finite number of primitive [and non-trivial] integer solutions? Considering the cubic case of Fermat’s Last Theorem, I make the following claim: Proposition: The Diophantine equation$$ X^3 + Y^3 = Z^3 \tag{$\star$} $$has a finite number of primitive [and ... 0answers 21 views ### Reformulating diophantine inequalities Assume we are given inequalities x \not\equiv a_i\text { (mod }b_i) for i=1,\ldots,n where 1 \leq a_i \leq b_i, x \in \mathbb{Z}. Can we somehow reformulate the problem as x \not\equiv ... 0answers 14 views ### Largest Erdős–Diophantine graphs A Diophantine graph is a set of vertices in the plane with integer coordinates, all at integer distances from eachother. An Erdős–Diophantine graph is a maximal Diophantine graph, so that it cannot ... 0answers 135 views ### Prove that (a-b)^n\mid (a^n-b^n) \iff n=1 under given conditions Suppose that a,b,(a-b) are pairwise co-prime (i.e. a\perp b\perp (a-b)\perp a), and that \frac{a}{2}<b<a, where a and b are both positive integers greater than 2. Let n be odd. ... 0answers 82 views ### An interesting equation in natural numbers Let n be a fixed natural number. How to solve the following equation in natural numbers:$$ \frac{1}{x_1} + \frac{2}{x_2} + \cdots + \frac{n}{x_n} = 1 $$(I can find many soltions but I am looking ... 0answers 56 views ### Help solving the quadratic equation ax^2-4bx+4bc-\frac{d^2}{a}=0 I have been struggling to solve this quadratic equation in the variable x with integral coefficients:$$ax^2-4bx+4bc-\frac{d^2}{a}=0$$a\neq 0 of course.How do I ensure that x is an integer? ... 0answers 71 views ### Which positive integers satisfies a^{b^2} = b^a How one can find all integers satisfying a\geq 1,b\geq 1,a^{b^2} = b^a? I think that the solutions are  (a,b)=(1, 1), (16, 2),(27, 3). 0answers 46 views ### Show that x^4+py^4+p^2z^4=p^3w^4 has no solutions, where p is any prime. I am trying to show that the equation: $$x^4+py^4+p^2z^4=p^3w^4$$ has no solutions. Assuming there is a nonzero solution (x_0,y_0,z_0,w_0), with w_0 minimal, then it ... 0answers 23 views ### Constellations of three powers How can I prove that for all i, j \in \mathbb{N} there are only a finite number of solutions to x^a + i + j = y^b + j = z^c with a,b,c,x,y,z \in \mathbb{N} and a,b,c \ge 2? This is a weaker ... 0answers 77 views ### How to prove every term of this sequence is not a natural number Sorry for the repost and for my "bad" English. I made a lot of errors in the previous one, so here's my actual question: Let's take a look at this sequence: (1) [a_1,a_2,a_3,a_4,...,a_x] where ... 0answers 23 views ### L-existential and L-diophantine Could you explain to me the last sentence: "Whenever we want to stress dependence on the language, we will use the self-explanatory terms and L-existential and L-diophantine" ? What does ... 0answers 49 views ### Finding solutions to a symmetric divisibility condition x\mid p(y),\;y\mid p(x) In general, are there strategies for finding all integers x and y such that x \mid p(y) and y \mid p(x) for some polynomial p with integer coefficients? For example, could we find all ... 0answers 43 views ### Diophantine eqution with odd prime HOW to find all possible set of solutions of an equation type y^p \pm 2 = x^2, where p is any odd prime High regards to one and all 0answers 79 views ### Finding all solutions: a^2 + b^2 = c^2 + d^2 I want to find all solutions to the problem of two squares equaling two other squares.$$a^2 + b^2 = c^2 + d^2 \qquad b \le N$$Clearly, without loss of generality, I can assume that$$gcd(a,b,c,d) = ...
I have been reading about Runge's theorem on diophantine approximation Theorem. Let $\xi$ be an algebraic real number of degree $d\geq 3$. For every $\epsilon >0$ there is a number $\gamma >0$ ...