Questions tagged [diophantine-equations]

Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...

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94 views

Could someone check this direct proof of Fermat'sLast Theorem? [closed]

Fermat’s Conjecture: x^n +y^n =z^n with n>=3 and 1 < x < y < z (or 1 2. There is no equality between the sum (x^n+y^n) and 1^n. A. Basis step / anchoring Let’s assume z=2 with 1<=x <...
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Solutions to the Diophantine equation $a^xy+x=c$

Fix positive integers $a,c$ with $a>2$. Is it possible that the Diophantine equation $$a^xy+x=c$$ has infinitely many solutions (in positive integers $x$ and $y$)?
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119 views

The number of perfect squares which can occur in an arithmetic progression of length n

This is a similar question to https://math.stackexchange.com/questions/2023399/the-maximum-number-of-perfect-squares-that-can-be-in-an-arithmetic-progression/3693487#3693487 Let f(n) be the maximum ...
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Why is this “the first elliptic curve in nature”?

The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation $$ y^2 + y = x^3 - x^2. $$ My guess is that there is some problem ...
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Chinese remaindering to solve solvable diophantine equations

Given a diophantine equation $$f(x_1,\dots,x_z)=0$$ where $f(x_1,\dots,x_z)\in\mathbb Z[x_1,\dots,x_z]$ is of total degree $d$ and each variable degree $d_i$ where $i\in\{1,\dots,z\}$ there is no ...
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Recursively obtained hard Diophantine equation for “Baseless numbers”

An equivalent problem was originally asked on MSE as Does every number base have at least one “Baseless number”?, but did not receive any answers that would help answer the main question about "...
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400 views

Methods of sheaf theory for solving Diophantine equations

What are some examples of sheaf theory used to either provide solutions to Diophantine equations, or to state that no such solutions exist?
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Are twin primes the only solution to this equation?

Let $p,q_i, i \ge 1$ be primes, $m$ a positive integer. The equation $$ p.\prod_{i=1}^m(q_i-1)-(p-1).\prod_{i=1}^mq_i=2 $$ for $m=1$ has all twin primes $p,q_1=p+2$ as solution. Are there solutions ...
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On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means

In this post (the content of this post is now cross-posted from Mathematics Stack Exchange see below) we denote the radical of an integer $n>1$ as the product of disctinct primes dividing it $$\...
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400 views

Natural number solutions for equations of the form $\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}$

Consider the equation $$\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}.$$ Of course, there are solutions to this like $(a,b,c) = (9,8,6)$. Is there any known approximation for the ...
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On a structural decomposition of polynomials based on integral roots

Given an irreducible polynomial of structure $$f(x,y)=\sum_{\substack{i,j\in\{0,1,2\}\\i+j\leq3}}a_{i, j}x^iy^j\in\mathbb Z[x,y]$$ with $a_{2,1}a_{1,2}a_{1,1}a_{1,0}a_{0,1}a_{0,0}\neq0$ if $f(m,n)=0$ ...
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160 views

When is $\phi(a^n+b^n+c^n)=0\mod n$?

A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...
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$3$-ranks of elliptic curves and representations $p=ax^3+by^3$

Let $p$ be a prime with $p\equiv2\pmod3$ and $E_p$ the elliptic curve $y^2=x^3+9p^2$ which has a rational $3$-torsion point. Let $\alpha$ from $E_p(\mathbb Q)$ to $\mathbb Q^*/{\mathbb Q^*}^3$ be the $...
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Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples - $\mathsf{II}$

$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Define the primitive Pythagorean triple $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$. Consider the Linear Diophantine Equation $$a^{2t}u+b^{2t}...
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Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples

$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$. Is it true that there are ...
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Small linear relations between primitive Pythagorean triples $\mathsf{II}$

WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$. Now let $a^2+b^2=c^2$ be a primitive Pythagorean ...
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Cohn's eight diophantine equations

Today I was reading J.H.E. Cohn's Eight diophantine equations (1966). On p. 158 he comes across the equation $y^2 = a^3 + 3a$ for odd values of $a$ and writes that this is equivalent to $x^3 + (x+1)^3 ...
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FLT and integral points on elliptic curves

For integers $x,y,z,t,n$ define $S_n : xy(x+y)=t^n$. For $ n > 2$, Fermat's Last Theorem implies there are no integral solution on $S_n$ with $x,y$ coprime and $xy(x+y) \ne 0$ since $x,y,x+y$ are ...
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Why $n$ or $n+1$ has the form $x^4+T_y+T_z$?

For $n\in\mathbb N=\{0,1,2,\ldots\}$ let $T_n$ denote the triangular number $n(n+1)/2$. By an observation of Euler, $$\{T_y+T_z:\ y,z\in\mathbb N\}=\{y^2+z(z+1):\ y,z\in\mathbb N\}.$$ It is well known ...
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Number of integer solutions to quadratic polynomial with integer coefficients

It is known from for example Representations of Integers as Sums of Squares by Grosswald, E. that $$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{...
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Write $n^2$ as $x^2+y^2+2\times4^z$ or $x^2+y^2+5\times 4^z$

In March 2018, I formulated the following somewhat curious question. Question 1. Whether for any integer $n>1$ there is a nonnegative integer $k$ such that $n^2-2\times 4^k$ or $n^2-5\times 4^k$ ...
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$n$-variable polynomials modulo $p$

The Hasse-Weil bound implies that for any 2-variable polynomial $P(x,y)$, there exists approximately $p$ solutions in $\mathbb{F}_p$ of $P(x,y) \equiv a \pmod p$ for sufficiently large $p$, and any ...
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Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers

In April 2018, I noted that the first integer $n>1$ with $n^2\not\in\{x^2+2y^2+3\times 2^z:\ x,y,z=0,1,2,\ldots\}$ is $$5884015571=7\times17\times49445509.$$ Question. Is it true that for each ...
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Diophantine equation $10^n-a^3-b^3=c^2$

Consider the Diophantine equation: $10^n-a^3-b^3=c^2$, for $a$, $b$, $c$, and $n$ positive. Has this equation infinitely many solutions?
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Simplest diophantine equation with open solvability

What is the simplest diophantine equation for which we (collectively) don't know whether it has any solutions? I'm aware of many simple ones where we don't know (whether we know) all the solutions, ...
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204 views

Solutions in primes of the equation $\,3p^2+q^2=r^2+3$

Let's consider the Diophantine equation $\,3p^2+q^2=r^2+3$. Actually, I am interested only in the solutions represented by sets $\,(p,q,r)\,$ of prime numbers. It's easy to prove that if $\,(p,q)\,$ ...
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314 views

Does this equation have more than one integer solution?

Consider the following diophantine equation $$n = (3^x - 2^x)/(2^y - 3^x),$$ where $x$ and $y$ are positive integers and $2^y > 3^x$. Does $n$ have any other integer solutions besides the case ...
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On Kellner's result and the Erdos-Moser equation

Let $m, k$ be positive integers. Consider the Erdos-Moser expression $S_{k}(m) = 1^k + 2^k + ... + (m-1)^k$. By a result of Kellner, we know that if $m | S_{k}(m)$, then $m|B_k$, where $B_r$ denotes ...
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291 views

Which Hilbert's 10th polynomials are known to have solutions?

The Diophantine equation $$x^3 + y^3 + z^3 = 42$$ was recently solved by Booker and Sutherland: Sum of three cubes for 42 finally solved. Is there a clean partition of the form of those polynomial ...
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Request for an exact formula related to a partition in number theory

The Frobenius equation is the Diophantine equation $$ a_1 x_1+\dots+a_n x_n=b,$$ where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-...
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Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers

I am interested to know if a similar theorem that shows this answer of the post Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
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find all of rational solutions of the quartic equation?

Consider the equation $a^4+6v^2a^2-8a+v^4=0$ over the rationals. Note that the following are solutions: $(a,v)=(1,1),(0,0),(2,0)$. Are there any other rational solutions?
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On variants of the abc conjecture in terms of Lehmer means

In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$ see the reference Wikipedia Lehmer mean. The ...
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239 views

Are there any references in the literature relating to work on finding a Diophantine equation representing abc

The Davis-Putnam-Robinson-Matiyasevich theorem is: Diophantine is equivalent to listable This result has some known applications: (1) Prime-producing polynomials. (2) Diophantine statement of the ...
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226 views

Sum from combinatorics on nonnegative integer numbers

Let $n_1,n_2,\ldots,n_k\in\{0,1,2,\ldots\}$. Can you calculate the sum $$ \sum_{n_1,n_2,\ldots,n_k\geqslant0}\mathbb{1}_\left\{n_1+\frac{n_2}{2}+\ldots+\frac{n_k}{k}<1\right\}? $$ If it's helpful, ...
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395 views

Diophantine representation of the set of prime numbers of the form $n²+1$

A polynomial formula for the primes (with 26 variables) was presented by Jones, J., Sato, D., Wada, H. and Wiens, D. (1976). Diophantine representation of the set of prime numbers. American ...
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1answer
411 views

Integer points of one Mordell equation

How can I determine all integer points of the following equation $$y^2=x^3+10546$$ I tried Magma with IntegralPoints(EllipticCurve([0,10546])); but got the ...
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Research work on $ax^n-by^m=1$

I am looking for results on the equation $$ax^n-by^m=1 \tag 1 $$ where $\gcd(m,n)=1$ and $a,b,n,m$ are constants. I found literature for $ax^n-by^n=1$ (R. A. Mollin, D. T. Walker) but couldn't ...
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Normalising Beal's conjecture

Problem solved:Check counter example(MSE) Beal's conjecture Is the below mentioned equation equivalent of Beal's conjecture ? If not, is there any counter example ? If $$ \sum_{q=0}^{u}(n+qd)...
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Prove $\frac{\text{Area}_1}{c_1^2}+\frac{\text{Area}_2}{c_2^2}\neq \frac{\text{Area}_3}{c_3^2}$ for all primitive Pythagorean triples

A while ago I asked this question on MSE here. After placing a bounty it got quite a bit of attention but unfortunately it has yet to be resolved. After getting some advice from MO Meta I have decided ...
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283 views

A question about integer triples

How can we generate all integer solutions of the equation $$(qr+rp+pq)(x^2+y^2+z^2) = (p^2+q^2+r^2)(yz+zx+xy),$$ given that $p,q,r$ are integers? Clearly if any one of $(x,y,z), (x,z,y), (y,z,x), (...
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On sparse $0/1$ linear equations solvable with compressed sensing

If you have a system of $m$ linearly independent equations in $n$ variables with domain $0/1$ and we know there is at least one solution with at most $d$ variables to be $1$ then if $m$ at least a ...
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Diophantine equations that involve cubes and the volume of square frustums

This week I wondered about diophantine problems that involve the volume of certain cubes and frustums, see the Wikipedia Frustum. I wondered if each one of these problems have infinitely many ...
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286 views

Solutions of a Diophantine equation with large common divisors

There is a curious Diphantine equation showing up in my research: $$ \frac{1}{a^2-1}+\frac{1}{b^2-1}=\frac{1}{c^2-1}+\frac{1}{d^2-1}. $$ I am trying to find its integer solutions where $a$, $b$, $c$ ...
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1answer
99 views

Different solution of power Diophantine equation based on constant term

Let us define a power Diophantine equation by 2 algebraic functions $f,g$ (having different degree) and by integers $k, l >0$ where, there are finite solutions for $f(x)+k=g(y)$, but there exists $...
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326 views

Diophantine equation in Laurent polynomials

(This is a modified repost of a question from MSE; since it came out of research, I thought it might be appropriate to post it here.) Consider the equation \begin{equation*} P(x, x^{-1})^m + Q(x, x^{-...
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Quadratic factors of $l_1(x,y)^3+l_2(x,y)^3+l_3(x,y)^3-n$

Related to sum of three squares and this question. Let $l_1,l_2,l_3 \in \mathbb{Z}[x,y]$ and $n \in \mathbb{Z}$. Assume that $n$ is not a cube and not twice cube. Let $f=l_1(x,y)^3+l_2(x,y)^3+l_3(x,...
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209 views

Rational Diophantine set for the non-squares

Related to Hilbert's Tenth problem. Is there polynomial with integer coefficients $P(a,x_1,...,x_n)$ such that $P(A,X_i)=0$ has rational solutions $X_i$ iff $A$ is not the square of integer (or as ...
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1answer
110 views

On a variant of Brocard's problem using the definition of Pochhammer symbols

I've considered the following variant of Brocard's problem $$\frac{(2n-1)!}{(n-1)!}+1=m^2\tag{1}$$ for integers $n\geq 1$ and integers $m\geq 1$. I was inspired from the fact that the evaluation of ...
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Is integer circuit membership undecidable?

According to wikipedia integer circuit in its simplest form is succinct representation of multivariate polynomial with integer coefficients. Decidability if an integer is represented by the integer ...

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