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6
votes
3answers
425 views

On the equation $a^n + b^n = c^2$

I am interested in the possible natural solutions of the equation $a^n + b^n = c^2$ where $n \geq 4$ is fixed. I am not sure if it is well-known or not, so any suggestion would be helpful.
10
votes
1answer
387 views

Can the sum of two non-zero coprime fifth powers be powerful?

I am wondering if the sum of two non-zero coprime fifth powers can be powerful. There are no small solutions. Q1 Can the sum of two non-zero coprime fifth powers be powerful? Got a partial ...
25
votes
2answers
925 views

Does Fermat's last theorem hold in the ordinals?

My question is whether there are no nontrivial solutions in the ordinals of the equations arising in Fermat's last theorem $$x^n+y^n=z^n$$ where $n\gt 2$, and where we use the natural ordinal ...
6
votes
2answers
433 views

The Theory of Transfinite Diophantine Equations [closed]

The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
13
votes
1answer
453 views

State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations $$a^5 + b^5 = c^5 + d^5$$ and $$a^6 + b^6 = c^6 + d^6$$ have no known nontrivial solutions, but $$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$ and ...
2
votes
1answer
138 views

Integral values of rational map

This question is related to this post on Math.MO. A theorem of B.Segre tells us that if there is one rational point on a non-singular cubic surface $X$ over $\mathbb{Q}$, then the surface is ...
-4
votes
1answer
141 views

$p=4x^2+27y^2$,with $p$ a prime [closed]

p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ , $p=1,9\pmod{20}$.
4
votes
0answers
213 views

Counting Special Rational Points on Cubic Surfaces

A paper of Heath-Brown gives an heuristic argument for the density of rational points on two cubic surfaces: $x^3+y^3+z^3=kw^3,k=2,3$, say, the number of rational points of height less than $N$ on ...
2
votes
1answer
133 views

Paired Quadratic diophantine equations

For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$$$0<|ta|^2,|tb|^2,|tc|^2,|td|^2<|x|,|y|$$ If so, how to ...
3
votes
0answers
202 views

Ternary form related to identity for abc conjecture

Consider the identity: $$ \begin{aligned} f_1 &= 4 (4 x + z) \cdot z^{3} \\ f &= x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4} + 4 x^{3} z + 12 x^{2} y z + 12 x y^{2} z + 4 y^{3} z + ...
4
votes
2answers
328 views

On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$

Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the ...
7
votes
2answers
493 views

The equation $x^m-1=y^n+y^{n-1}+…+1$ in prime powers $x,y$

Does the equation $x^m-1=y^n+y^{n-1}+...+1$ have only finitely many solutions $(x,y,m,n)$ where $x,y$ are prime powers with $y>2$ and $m,n$ are integers with $m,n>1$? This question arose in the ...
0
votes
1answer
220 views

Solutions of system of diophantine equations

The system of diophantine equations $$\{x^2-y^2+z^2-u^2+q^2-t^2=0,\,xy+zt-uq=0 \}$$ is given. Do the formulas $$x:=(j(p^2-4ps+3s^2)-(p-s)(3p^2-4ps+s^2))k^2+2(j-2(p-s))(p-s)kn+(j-p+s)n^2, $$ ...
7
votes
3answers
804 views

How many integer points does my favorite ellipse go through?

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ x^2+y^2 - ...
11
votes
2answers
674 views

On Generalizations of Fermat's Conjecture

We know the following facts: (1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$. (2) For all $3\leq n$ the equation ...
3
votes
1answer
341 views

Proving conditions on $(r+s)^2 \mid (4r^4+1)$, related to Pell oblongs

While working on another problem (Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$), I came across a question which seems to be of [semi-] independent interest. Conjecture. If ...
1
vote
0answers
194 views

Integral points on affine rational curves over $\mathbb{Q}$

Given a rational curve $C:(f_1(t),f_2(t))$, where $f_i(t),i=1,2$ are rational functions with rational coefficients. Question: Is there any criterion(proved or conjectural) for the existence of ...
0
votes
0answers
103 views

Rational solutions of equations of the form $y^2 x = f(x)$

Let $k$ be any number field, and suppose we want to study the $k$-rational points on $$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 3. In other words, $y^2 x = f(x)$ is a ...
1
vote
0answers
75 views

Is there any track for proving $D=NP$, besides showing that $D$ has polynomial-bounded universal quantifiers?

Background By the MRDP theorem, every for every recursively enumerable set $S$, there exists a Diophantine polynomial $p$ such that $$x \in S \iff \exists y_1, \dots, y_n \in \mathbb{N} \text{ such ...
9
votes
3answers
2k views

Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$

I'm working on solving the quartic Diophantine equation in the title. Calculations in maxima imply that the only integer solutions are \begin{equation} (r,s) \in \{(-3, -2), (-2, 3), (-1, 0), (0, ...
3
votes
1answer
267 views

Is there an easy proof of this equation related to simultaneous Pell equations?

Working with the famous Baker-Davenport system of simultaneous Pell equations \begin{align} 3x^2-2 &= y^2, & 8x^2-7 &= z^2, \qquad(\star) \end{align} I am left, after a series of ...
5
votes
1answer
125 views

Relative size of Egyptian fraction denominators

Suppose we have a finite Egyptian fraction decomposition of a rational: $$\frac{n}{m} = \sum_{i=1}^k \frac{1}{x_i}$$ such that (i) $x_i>0$, (ii) $x_i \neq x_j$ for $i \neq j$, and (iii) $\gcd(m, ...
10
votes
7answers
3k views

Is there an algorithm to solve quadratic Diophantine equations?

I was asked two questions related to Diophantine equations. Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all ...
6
votes
2answers
385 views

On integers as sums of three integer cubes revisited

It is easy to find binary quadratic form parameterizations $F(x,y)$ to, $$a^3+b^3+c^3+d^3 = 0\tag{1}$$ (See the identity (5) described in this MSE post.) To solve, $$x_1^3+x_2^3+x_3^3 = 1\tag{2}$$ ...
7
votes
3answers
509 views

Diophantine equation - $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$)

How can I find the general solution of $a^4+b^4=c^4+d^4$ ($a,b,c,d > 0$)? And how did Euler find the solution $158^4+59^4=133^4+134^4$?
0
votes
0answers
135 views

On unique solutions to linear diophantine equations

Let $\sum_{i=1}^k a_ix_i = N$ be the equation with $a_i \in [2^t,2^{t+1}]$ being distinct primes. If we seek unique solutions $x_i\in R_i = (0,a_i)\cap \Bbb Z$, then in general it is not possible. ...
5
votes
2answers
1k views

Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

This question has been asked previously on math.SE without receiving any answers. http://math.stackexchange.com/questions/479740/solving-xkx1kx2k-cdotsxk-1k-xkk-for-k-in-mathbb-n Letting $k$ be a ...
1
vote
0answers
208 views

Can six square numbers be simultaneously represented in a single sum of consecutive odd numbers? [closed]

I had some free time from my work to do a little exploration regarding the existence (or non existence) of perfect cuboids. A solution is represented by the set of Diophantine equations: $a^2 + b^2 = ...
3
votes
0answers
125 views

Effective Lang-Weil bounds for del Pezzo surfaces

Let $X$ be variety in $\mathbb{P}^N$ over $\mathbb{F}_q$ of dimension $n$ and degree $d$. By the Lang-Weil bounds $ |\# X(\mathbb{F}_q) - q^n| \le (d-1)(d-2)q^{n-1/2} + Cq^{n-1}$for a constant $C$ ...
2
votes
1answer
240 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
792 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
492 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
243 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
260 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
942 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
395 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 ...
0
votes
1answer
543 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
231 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
161 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 ...
9
votes
3answers
1k 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
607 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
644 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
282 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 ...
4
votes
4answers
444 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
973 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
745 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
254 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
317 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
968 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
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 - ...