# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...