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3
votes
1answer
200 views

Diophantine equation with primitive nth root of unity

Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - ...
2
votes
0answers
134 views

Reference for original paper (but translated to English) of Matiyasevich's proof of Fibonacci relation being Diophantine?

Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems that is accessible to ...
12
votes
2answers
287 views

A sequence based on Catalan–Mihăilescu problem

It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The conjecture was ...
3
votes
2answers
699 views

What is known about a^2 + b^2 = c^2 + d^2

Could you state or direct me to results regarding the Diophantine equation $a^2+b^2=c^2+d^2$ over integers? Specifically, I am looking for a complete parametrization. In the case that a complete ...
17
votes
1answer
822 views

Solutions to $\binom{n}{5} = 2 \binom{m}{5}$

In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says: On National Public Radio, the Weekend Edition program posed the following probability problem: Given a certain number of ...
6
votes
3answers
560 views

References on techniques for solving equations with discontinuous functions such as floor and ceiling?

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good ...
4
votes
0answers
262 views

$a^5+b^5=c^5+d^5$ and polynomial identities

No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known. (1) has infinitely many solutions in an extension of $\mathbb{Z}$ (root of $9-15x+37x^2 $ ) resulting from genus 0 curve ...
5
votes
1answer
292 views

What analytic tools can provide a lower bound for this Diophantine equation?

The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken. I felt that the following ...
8
votes
1answer
291 views

Binary expansion of squares

I came across the following simple question: what odd integer squares have exactly 3 ones in their binary expansion? After looking at it for a while I convinced myself that the only solutions to $r^2 ...
12
votes
2answers
804 views

sum of three cubes and parametric solutions

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples ...
2
votes
2answers
230 views

Catalan-type equations for prime powers

Do there exist nonzero integers $a,b,c$ for which the equation $$aX + bY = cZ$$ has infinitely many solutions with $X,Y,Z$ distinct prime powers? For example, if there are infinitely many Sophie ...
7
votes
0answers
221 views

When adding a constant makes a multivariate polynomial reducible?

Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of smaller degrees? ...
5
votes
2answers
247 views

Semimagic Squares and Partitions

Say, we have a semimagic square $X$, that is, an $n\times n$ square matrix with entries from natural numbers, such that each row and column of it sums up to the same natural number $s$. Let $M$ be a ...
0
votes
1answer
434 views

Like Diophantine equation

Dear all, I have posted this question on m.s.e. Unfortunately, no one responded to answer. I hope, this site and members of this site will answer my questions. The equation $x^n - ny^x-nxy$ = $0$ ...
4
votes
0answers
223 views

Expressions of $tanh$ type whose continued fractions have two shifts per period

This is a follow-up of another thread about quasi periodic continued fractions, a.k.a. Hurwitz fractions, with some linear shifts. I seem to have found the pattern of a subclass of them, as given ...
5
votes
6answers
526 views

Representations with Triangular Numbers

A well known theorem of Gauss says that any natural number $n$ may be written as the sum of three triangular numbers - $$ n={a_{1} \choose 2}+{a_{2} \choose 2}+{a_{3} \choose 2} $$ The following ...
10
votes
1answer
258 views

Sets of integers represented by degree zero rational functions

Suppose $f(x_1,x_2,\dots)=\frac{P}{Q}$, where $P,Q$ are polynomials in several variables with integer coefficients that have the same degree. Let's denote by $S(f)$ the set of integers $n$ for which ...
6
votes
4answers
884 views

Does every polynomial diophantine equation have solutions modulo p?

Obviously, this is not exactly true; what I am really asking is whether any diophantine polynomial equation with integer coefficients (let's call them DPEICs) who's solution does not admit ...
3
votes
0answers
237 views

quasi periodic continued fractions and powers of e, tanh, tan

It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N $, when written as regular continued fractions (R.C.F.), yield what can be called a ...
4
votes
2answers
423 views

Are there Heronian triangles that can be decomposed into three smaller ones?

Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...
3
votes
1answer
210 views

Any non-conforming numbers?

Consider the function $x^m \pm y^n \pm z^p$, where $x, y, z, m, n, p$ are integers such that $m, n, p \geq 2$. The question is, are all numbers expressable using this function? Are there any ...
3
votes
0answers
89 views

Weak classes of diophantine functions

From a well-known work(s) by Putnam, Davis, Robinson and Matiyasevich, we know that every partially recursive function is diophantine. Now it seems a natural question to ask: can we say something ...
0
votes
0answers
324 views

Diophantine equation solutions

I am not able to make headway on solving the diophantine equation $x^m - y^n = 6.$ Are there any solutions to this? What about $x^m - y^n = 14,$ and $= 30$ (both $m$ and $n$ are at least $2$).
2
votes
2answers
195 views

Maximum size of powers with a given difference

Pillai's conjecture -- that the gap between (nontrivial) powers is unbounded below -- is still open (it would be a consequence of the $abc$ conjecture, were that proven). But I wonder what the right ...
41
votes
3answers
2k views

What is the geometry of an undecidable diophantine equation?

As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine ...
4
votes
2answers
644 views

The diophantine equation X^2 - Y^2 - Z^2 = +- 1

Hi everybody. I'd like to know if the diophantine equation (1) $$X^2 - Y^2 - Z^2 = \pm 1$$ has been studied and if the set of its solutions $(X,Y,Z)$ is known. I appreciate any reference. Thank you ...
6
votes
1answer
352 views

Solving equations in a subset of rational numbers

Let $S$ be a set of all positive rational numbers $x$ such that $2x^2 - 1$ is a square, excluding $x=1$. I am interested in computing as many as possible solutions in $S$ to either the following ...
3
votes
1answer
1k views

Integer solutions of x^n + y^n = z^{n-1}

This is related to another question I am interested in the non-trivial integer solutions of $$ x^n + y^n = z^{n-1} $$ for $n \ge 4$. A solution is trivial if $xyz=0$ or $x = \pm y$. There are ...
6
votes
2answers
411 views

Diophantine Equation with Polynomial Coefficients

DISCLAIMER: I'm primarily a graph theorist and am fairly inept when it comes to classical number theory. Recently I have been looking at the possibility (or impossibility) of embedding various graphs ...
0
votes
1answer
330 views

Erdős-Straus with 4 terms

The Erdős-Straus conjecture states that any fraction of the form $\frac{4}{n}$ can be decomposed as an Egyptian fraction with just 3 terms. In related research, I've recently come across conditions on ...
6
votes
1answer
276 views

Examples of finiteness of rational points for hypersurfaces in $\mathbb P^3_{\mathbb Q}$ of degree $>4$.

Given an homogeneous polynomial $F(X,Y,Z,T)\in \mathbb Q[X,Y,Z,T]$ of degree $>4$, the surface it defines is well-known to be of general type. Suppose, moreover, that this surface doesn't contain ...
5
votes
4answers
441 views

seeking an integer parameterization for A^2+B^2=C^2+D^2+1

I'm looking for a complete [integer] parameterization of all integer solutions to the Diophantine equation $A^2+B^2=C^2+D^2+1$, analogous to the classical parameterization of the Pythagorean ...
5
votes
2answers
1k views

Rational solutions to x^3 + y^3 + z^3 - 3xyz = 1

I can show that there infinitely many solutions to this equation. Is it possible that the set of rational solutions is dense?
8
votes
1answer
379 views

what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...
2
votes
2answers
775 views

is there a solution to system of linear Diophantine equations?

I have a matrix A \in Z^{n \by m}, where m > n and a vector b \in Z^n. Then, under what conditions does an integer solution exist to the equation Ax = b. Is there a way to bound the norm of the ...
0
votes
0answers
91 views

how do you bound exponent of x^2+1=y^p

for p a prime exponent using linear forms in logs? So far I have (x-i)(x+i)=y^p which are coprime and hence x+i=(a+ib)^p , now how do I get a linear form in logs so that I can find an upper bound on ...
1
vote
0answers
383 views

Diophantine equation over Z[i]

I'm trying to generate the set of solutions of a specific diophantine equation over 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 that ...
1
vote
0answers
167 views

Efficient counting of Egyptian fractions with bounded denominators

I was amazed to discover that sequence http://oeis.org/A020473 in the OEIS has almost four hundred terms computed. I wonder how one can get that far? E.g., how one can compute A020473(100)? P.S. ...
11
votes
1answer
348 views

Symmetric functions on three parameters being perfect squares

Is it possible for $x+y+z, xy+yz+zx$, and $xyz$ to be perfect squares at the same time for positive integer values of $x,y,z$?
2
votes
0answers
625 views

0,1 solution to system of linear integer equations.

I have the following problem: $A x = b$ where $A, b$ - $m \times n$-maxtrix and $m$-vector of nonnegative intgers (respectivelly). $x \in \{0,1\}^n $ - vector of binary variables, which need to be ...
10
votes
0answers
315 views

Growth of $n=n(k)$ for which there's a non-trivial solution to $x_1^k+\cdots+x_n^k=y^k$?

Walter Hayman just asked me the following question. What, if anything, is known about the growth of the function $n(k)$, where $k\geq1$ is an integer, and $n=n(k)\geq2$ is the smallest integer for ...
1
vote
1answer
218 views

A problem on cubic Diophatine equations

What is the best algorithm to find all the integer points (X,Y) on this curve $X^3+aX-bY^3=m,a,b,m\in\mathbb{Z}$(a>0,b>0,b is not a cubic number)?
9
votes
3answers
772 views

a family of Pellian equations

I have a question concering the family of Pellian equations $$x^2 - (k^2+1)y^2 = k^2. \qquad (*)$$ For an integer $k\geq 2$, the equation (*) has at least three classes of solutions in ...
2
votes
1answer
363 views

Infinite solutions of a diophantine equation [closed]

Given the Diophantine equation$$ax^2+bxy+cy^2+dx+ey+f=0$$ if the coefficients $(a,b,c,d,e,f)$ are chosen among all the prime numbers, we have infinite equations. Is it possible to prove that the ...
2
votes
0answers
212 views

Hurwitz integers and $F_4$

The Hurwitz integers are $$ \mathcal H= \{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}. $$ I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
8
votes
1answer
510 views

how many consecutive integers $x$ can make $ax^2+bx+c$ square ?

The following problem was raised in a Mathlinks thread: If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ? The ...
12
votes
5answers
2k views

Special arithmetic progressions involving perfect squares

Some time ago the following rather easy problem appeared in an online publication called "Problems in Elementary NT" by Hojoo Lee: Prove that there are infinitely many positive integers $a$, $b$, $c$ ...
1
vote
1answer
287 views

A good introduction to S unit equations

I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper. ...
31
votes
1answer
1k views

On a remark of Tait on FLT for the exponent 3

This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below: In the ...
5
votes
5answers
965 views

Impossible Heronian Triangles (Ratio of 2 Sides)

There is no Heronian triangle (or simply consider triangles on an integer lattice which also have integer side lengths) for which one side is half the length of another side. What other "side-side ...