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2
votes
1answer
203 views

Are all sums of subsets of roots of unity unique?

For a prime $p$, let $S$ be the set of all $p$-roots of unity in the complex plane. Now, consider the sum, $W(R)$ of the members of a set $R$ which is a proper subset of $S$. I suspect that $R \ne R'$ ...
8
votes
2answers
709 views

Which integers can be expressed as a sum of three cubes in infinitely many ways?

For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1) $$ If $n$ is a cube or twice a cube, identities exist. Elkies suggests no other polynomial identities are known. ...
2
votes
3answers
457 views

Algorithm to count number of positive integer solutions of $x^2(8x-3)=y^2z$?

Given the Diophantine equation $$ x^2(8x-3)=y^2z, $$ is there a way to efficiently count the number of solutions that satisfy $x+y+z\leq n$, where $n$ is a fixed given integer? Also, for any fixed ...
2
votes
3answers
223 views

Bounds on solutions to Diophantine equations of the form p(x)=q(y)

Thanks to responses in a previous question I asked, I was able to show that if $p$ and $q$ are two distinct polynomials with equal degree greater than 2 and coefficients in $\mathbb{N}$ that the ...
2
votes
2answers
246 views

Four-Square Theorem for Negative Coefficient

What integers are not in the range of $a^2+b^2+c^2-x^2$ (for all integer combinations of a, b, c, and x)? This form is similar to that of Lagrange's Four-Square Theorem, for which the answer would be ...
27
votes
6answers
888 views

Patterns among integer-distance points

Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal. ...
8
votes
1answer
328 views

Fundamental units of imaginary quartic fields

Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the ...
1
vote
1answer
390 views

Is surface $x^2+z^2=2\cdot y^2$ something of a Möbius strip?

This question is naive. My association with Möbius strip comes from not being able to smoothly extract positive solutions of the diophantine equation $$x^2+z^2=2\cdot y^2$$ I got a parametrization ...
2
votes
0answers
218 views

Algorithm for solutions to quadratic forms over number fields

Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)? I am especially interested in the quaternary case. There exist some ...
5
votes
0answers
152 views

When does the Lloyd polynomial have only integral roots?

For a $t$-error correcting code of length $n$ over the finite field $\mathbb{F}_q$, the Lloyd polynomial is given by $$ L_t(n,x):=\sum_{j=0}^t(-1)^j\binom{x-1}{j}\binom{n-x}{t-j}(q-1)^{t-j}. $$ A ...
8
votes
3answers
953 views

The modular arithmetic contradiction trick for Diophantine equations

It is a slick, and seemingly ad-hoc, technique often used to prove that a Diophantine equation has no solutions. The equation $f(x_1,\ldots, x_k)=0$, with variables $x_i\in\mathbb{Z}$ and some ...
0
votes
0answers
495 views

Diophantine: a^n + b^n + c^n = d^n and a^n + b^n = c^n + d^n

Let us consider the equation $a^n+b^n=c^n$ for positive integers $a,b,c$ and $n\ge 2$. The $n=2$ case has a well-known and beautiful parametrization known as Pythagorean triples. Fermat's Last Theorem ...
17
votes
1answer
613 views

Is there an online encyclopedia of Diophantine equations (OEDE)?

Hello all! I'm just wondering if there is an online encyclopedia of Diophantine equations (OEDE), analogous to the OEIS for sequences. While trying to solve one Diophantine equation, I reduced the ...
0
votes
2answers
208 views

Classification of these Binary Quadratic Forms

What are necessary and sufficient conditions on a binary quadratic form $ax^2+bxy+cy^2$, with integer coefficients and solution set in integers, to be equivalent to $x^2-y^2$, and separately to ...
3
votes
2answers
273 views

Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares? Be it 1,2,3,...n or k,k+1,k+2,...n.
1
vote
1answer
805 views

The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial. In particular, it's the case ...
6
votes
4answers
703 views

Can repunits be perfect cubes?

Is it true that the equation $10^{n}-9m^{3}=1$ has only one positive integer solution, namely $n=m=1$? I can't find the answer. This has an equivalent description that the repunits $R_n = 11\dots1$ ...
2
votes
2answers
246 views

Help with this system of Diophantine equations

A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it. This is the actual question which I'm trying to solve:- For ...
1
vote
3answers
309 views

Help with this Diophantine equation

Note: This question was posted in error, and should be closed as no longer relevant. The correct question is posted at Help with this system of Diophantine equations (End of note) For a research ...
10
votes
1answer
878 views

Effect of abc conjecture on Fermat's Last Theorem

A website ( http://www.math.unicaen.fr/~nitaj/abc.html#Consequences ) says that the $abc$ conjecture implies that there are only finitely many solutions to the equation $x^n+y^n=z^n$ with ...
3
votes
1answer
192 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 - ...
1
vote
0answers
128 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
281 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
664 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
763 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
500 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 ...
3
votes
0answers
245 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
283 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
279 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 ...
11
votes
2answers
753 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
219 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
215 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
245 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
428 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
215 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
499 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
252 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
835 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
214 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
375 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
206 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
86 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
320 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
193 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
1k 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
579 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
330 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
383 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
303 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 ...