Tagged Questions

Questions on finding integer/rational solutions of equations.

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The exponent on Thue's theorem

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 $\gamma >0$ ...
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Given the following 4 equations, can you find 4 whole number answers using whole number variable inputs? $x,y,z$ where $x>y>z$ $Eq 1 = (x^2-2xy+y^2-2xz+z^2)^{\frac{1}{2}}$ $Eq 2 = (x^2-2xy+y^... 0answers 60 views The Nagell-Ljunngren Equation I have been trying to find the papers of Nagell and Ljunngren, which deal with the equation $$\frac{x^n - 1}{x - 1} = y^2$$ and solve it completely. Many papers cite these papers, but I haven't found ... 0answers 355 views An argument for “Brocard's problem has finite solution” Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem $$x^{2}-1=... 0answers 62 views Exponential diophantine: 2^x-7^y=z^2 Find all integers x,y,z such that 2^x-7^y=z^2. For example: 2^3-7^1=1^2 2^5-7^1=5^2 2^7-7^1=11^2 (But note that \sqrt{2^9-7^1}\not\in \mathbb{Z}.) The problem with this particular ... 0answers 77 views About Runge's method I have been reading about some Diophantine equations (like Runge's theorem and Cassel's theorem) and in the text says that these theorems are solved using Runge's method, but it doesn't say what ... 0answers 94 views Transforming the cubic Pell-type equation for the tribonacci numbers The Lucas and Fibonacci numbers solve the Pell equation,$$L_n^2-5F_n^2=4(-1)^n\tag1$$The tribonacci numbers z = T_n are positive integer solutions to the cubic Pell-type equation,$$27 x^3 - 36 ... 0answers 85 views The number of solution of a Diophantine equation If we fixe$n\in \mathbb{N}$. I was wondring if there is an estimation of the number of the integer solutions of the equation : $$x_1^2+x_2^2+\cdots+x_n^2=n^3$$ where$x_i>0$for all$i=1,\cdots,n$... 0answers 91 views How to solve this equation$x^5 +4^y=2013^z$in positive integers? I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that$x$must be an odd number and further$x=1 \, mod\, 4$. Any hint is appreciated. 0answers 165 views Primes as sum of squares. If$p_{i}$and$p_{j}$are two primes of the form$4k+1$, with$p_{j} > p_{i}$, show that if$p_{j} \neq$sum of two squares$p_{i}$is also not equal to sum of two squares. It is well ... 0answers 80 views An elliptic curve for the multigrade$\sum^8 a_n^k = \sum^8 b_n^k$for$k=1,2,3,4,5,9\$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...