The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
0answers
133 views

Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three consecutive powerful numbers. Necessary condition for this is integer solution of $$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$ What are integer solutions ...
0
votes
0answers
84 views

Find two triangles of longest side length 25? [on hold]

I'm using the quadratic Diophantine equations to solve for two integer triangles of longest side 25. It's been shown that for $a^2 + b^2 = c^2$, which goes to $x^2 + y^2 = 1$ where $x = a/c$, $y = ...
1
vote
1answer
154 views

Non-coprime solutions to x^n+y^n = z^2

Let $n$ be an odd prime. I know that the equation $x^n+y^n = z^2$ has no non-zero coprime solution in integers whenever $n \geq 5$, and that there are infinitely many solutions as soon as one drops ...
-2
votes
0answers
38 views

One Diophantine equation [migrated]

I wonder now that the following Diophantine equation: $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ have only this formula describing his decision? $a=-(k^2+2(p+s)k+p^2+ps+s^2)$ $b=2k^2+4(p+s)k+3p^2+3ps+2s^2$ ...
11
votes
3answers
289 views

(Non-)Existence of curves of low degree on affine and projective varieties

I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...
2
votes
3answers
189 views

Specific Diophantine Equation Appearing in Faa Di Bruno Formula

In a Faa Di Bruno Formula there is an equation: $m_1$+2*$m_2$+3*$m_3$+...n*$m_n$=n Is there any general solution for this equation. For example for $m_1$+$m_2$+$m_3$+...+$m_n$=n, there is a ...
1
vote
1answer
201 views

On $x^3-y^2=1728 \text{ unit}$ in number fields

Consider solution of $$x^3-y^2=1728 \text{ unit} \qquad (1)$$ in a number field. This is related to the discriminant of elliptic curve in terms of $c_4,c_6$. Via elliptic curves it might have ...
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
4answers
2k views

General integer solution for $x^2+y^2-z^2=\pm 1$

How to find general solution (in terms of parameters) for diophantine equations $x^2+y^2-z^2=1$ and $x^2+y^2-z^2=-1$? It's easy to find such solutions for $x^2+y^2-z^2=0$ or $x^2+y^2-z^2-w^2=0$ or ...
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 ...
4
votes
3answers
351 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 ...
10
votes
7answers
2k 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 ...
16
votes
6answers
3k views

Pythagorean 5-tuples

What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")? There are simple formulas describing Pythagorean n-tuples for n=3,4,6: n=3. The formula ...
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 ...
20
votes
3answers
627 views

Consecutive square values of cubic polynomials

Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square? It is known that the ...
2
votes
0answers
107 views

What is the general feeling for Hilbert's 10th problem for Q?

We know that Hilbert's 10th problem for $\mathbb{Z}$ is undecidable. I was wondering whether there is a strong opinion in the mathematical community on the decidability of Hilbert's 10th for ...
9
votes
1answer
410 views

Integer Solutions of $x+y^n = y + x^m$ for $n < m$

I found 8 of them and believe there is no more: $$2+3^2=3+2^3$$ $$2+6^2=6+2^5$$ $$6+15^2=15+6^3$$ $$3+16^2=16+3^5$$ $$3+13^3=13+3^7$$ $$2+91^2=91+2^{13}$$ $$5+280^2=280+5^7$$ $$30+4930^2=4930+30^5$$ ...
3
votes
0answers
89 views

Integers in a given box that can be represented by a polynomial

Suppose that $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ is a polynomial of degree $d$, and examine the quantity $$\displaystyle N(F;X, B) = \# \{(x_1, \cdots, x_n) \in \mathbb{Z}^n | -X ...
6
votes
4answers
230 views

Number of solutions of linear homogenous Diophantine equation inside a box

Let $a_1, ..., a_d$ be positive reals and consider the linear Diophantine equation $$ \sum_i a_in_i = 0. $$ I am interested in estimating the number of integer solutions of this equation inside a ...
3
votes
1answer
313 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 ...
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.
5
votes
1answer
393 views

Is the following consequence of the Lang conjecture known?

This came up in a discussion with a colleague of mine, who studies PDEs. He was asking for a function $f \colon \mathbb{N} \rightarrow \mathbb{N}$ such that, for all but finitely many $n$, the ...
0
votes
1answer
115 views

Solutions of the equation $X^4-DY^4=z^4$

In order to obtain infinite integer non trivial solutions of the equation $X^4-DY^4=Z^4$ (all numbers natural) we do the following. We set $X=(r_1●p+p)$, $Y=p$, $Z=(r_1●p)$, ...
-1
votes
1answer
110 views

How many integer points does my favorite ellipse goes through? [closed]

What value on P gives an ellipse with 768 lattice Points? x^2 + 3y^2 = P P= 4*7*13*19*31*37*43 gives 384 lattice points
5
votes
1answer
281 views

Subsets of all Diophantine's sets

I have asked this question on math.stackexchange already: http://math.stackexchange.com/questions/627461/subsets-of-all-diophantines-sets Function $\mathbb{N}^k \to \mathbb{N}^m$ is computable ...
9
votes
0answers
570 views

Consecutive averages of sequence (or difference quotients of partial sums) always square

I proposed the following problem for the December 2013 USA IMO TST earlier this month: Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s ...
3
votes
0answers
99 views

Curves on hypersurfaces generated by diagonal sums

This is related to an earlier question of mine ((Non-)Existence of curves of low degree on affine and projective varieties). It seems that the question is too difficult for specific surfaces, although ...
2
votes
0answers
90 views

Congruences of binary forms

Suppose $f(x,y) \in \mathbb{Z}[x,y]$ is a binary form (that is, homogeneous polynomial in two variables). Further suppose that $f$ is irreducible over $\mathbb{Z}$ and has no fixed prime divisor. Let ...
3
votes
1answer
180 views

Special Case of famous Equation

I'm interested in the following diophantine eqaution: $(5^n-1)/4=y^2$. It turns out that this is a special case of the Nagell-Ljunggren equation, where $x=5$ and $q=2$ It has been shown that for ...
0
votes
1answer
154 views

Reference request: on sums of the form $ax^m + by^n = h$

I know that equations of the form $$\displaystyle ax^d + by^d = h$$ with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
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 ...
6
votes
3answers
410 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.
25
votes
2answers
868 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 ...
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 ...
9
votes
1answer
359 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 ...
6
votes
2answers
405 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 ...
12
votes
1answer
429 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
130 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
126 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
193 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
125 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
189 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
307 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
417 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
164 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 $$x:=(j(p^2-4ps+3s^2)-(p-s)(3p^2-4ps+s^2))k^2+2(j-2(p-s))(p-s)kn+(j-p+s)n^2, $$ ...
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 ...
6
votes
2answers
583 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 - ...
23
votes
3answers
3k views

Solve in positive integers $n!=m(m+1)$

Is anybody know a solution of this problem? (Sorry, I've missed one summand in the previous post.)
1
vote
2answers
772 views

Solve in positive integers $n!=m^2$

Is anybody know a solution of this problem? (Sorry, correct question is here.)
28
votes
3answers
1k views

A binomial generalization of the FLT: Bombieri's Napkin Problem

This is an extract from Apéry's biography (which some of the people have already enjoyed in this answer). During a mathematician's dinner in Kingston, Canada, in 1979, the conversation turned ...