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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Was Fermat's Last Theorem known for infinitely many primes before Wiles?
Before Andrew Wiles's 1997 proof of Fermat's Last Theorem, in 1985, Étienne Fouvry et al. proved that the first case of FLT holds for infinitely many primes $p$.
Is there any infinite class of primes ...
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Is this proof that $A^3+B^3=C^n$ has no primitive solutions correct? [closed]
I am an independent researcher. This arose in the context of studying the Beal conjecture.
Setup: Factor $A^3+B^3=(A+B)(A^2-AB+B^2)=C^n$. For coprime $A,B$: $\gcd(A+B, A^2-AB+B^2)$ divides $3$. This ...
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Finding all integer solutions to a family of elliptic curves depending on a parameter $n$
Consider this equation
\begin{equation}
y^2 = x^3 + (36n + 27)^2 \cdot x^2 + (15552 n^3 + 34992 n^2 + 26244 n + 6561) \cdot x + (46656 n^4 + 139968 n^3 + 157464 n^2 + 78713 n + 14748)
\end{equation}
...
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On smooth pronic numbers
By Størmer's theorem, for every fixed $k$, there are only finitely many $n$ such that the set of prime divisors of $n(n+1)$ is a subset of the first $k$ primes. The OEIS sequence A141399 tracks those $...
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Does this specific polynomial identity generate infinitely many triples satisfying $c > \text{rad}(abc)$?
Starting from a math problem involving a square and a unit circle, I used some elementary algebraic transformations and discovered that for any arbitrary $u, v$, if we define $a, b, c$ as follows:
$$a ...
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Is $y^2z+z^2y = x^3+x^2+3x-1$ solvable in integers?
As it is clear from the title, the question is whether there exist integers $x,y,z$ such that
$$
y^2z+z^2y = x^3+x^2+3x-1.
$$
My two-years-old notes claim that I checked it up to $|x|\leq 450,000,000$ ...
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Numbers that are not represented as $3ijk - (ij + ik + jk)$ leading to prime numbers
Let $f(n)$ be an integer function such that $$ f(n) = \sum\limits_{i=1}^{n} \sum\limits_{j=1}^{i} \sum\limits_{k=1}^{j} [(3ijk - (ij + ik + jk)) = n]. $$
Here square bracket denotes Iverson bracket.
I ...
user587997
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Integer solution to $(3x-1)y^2 + x z^2 = x^3-2$
Do there exist integers $x,y,z$ satisfying
$$
(3x-1)y^2 + x z^2 = x^3-2 \quad ?
$$
Hilbert's 10th Problem is unsolvable in general, but is still open for cubic equations: it is unknown whether there ...
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Empirical observations on the sum of reciprocal minimum exponents in the prime factorization of coprime triples
Let $h(n)$ be the function that returns the minimum exponent in the prime factorization of $n$. We adopt the convention $h(1) = \infty$, so $1/h(1) = 0$.
According to the abc conjecture, there should ...
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Parameterization for diophantine $a^4+28a^3b+70a^2b^2+28ab^3+b^4=c^4+28c^3d+70c^2d^2+28cd^3+d^4$
How to find new parametric forms for the OP diophantine equation
$$A^4+28A^3B+70A^2B^2+28AB^3+B^4=C^4+28C^3D+70C^2D^2+28CD^3+D^4$$
The smallest parametric solution can be found in this collection by ...
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Heuristic plausibility of a lower bound for $S_a+S_b+S_c$ in coprime triples
Let $a,b,c$ be coprime positive integers with $a+b=c$. For $n=\prod p_i^{e_i}$, define $S_n=\frac{1}{\omega(n)}\sum_{i=1}^{\omega(n)}\frac{1}{e_i}.$ where $\omega(n)$ is the number of distinct prime ...
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Looking for non-trivial parametric solutions to $(r^2+s^2)(r^2s^2+1)=t^2$
I am trying to find parametric families of rational solutions $(r, s, t) \in \mathbb{Q}^3$ for the Diophantine equation:
$$
(r^2 + s^2)(r^2 s^2 + 1) = t^2
$$
I consider the following solutions to be ...
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Involutions and idoneal numbers
Fermat two squares theorem has a well known proof due to Heath Brown that uses involutions, that is, linear operators that are equal to their own inverses. This allows so called one sentence proofs, ...
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Sum of two prime powers equals a third
Are there infinitely many primitive solutions to the equation $x+y=z$ where $x,y,z$ are all prime powers? This follows from the infinitude of prime gaps of size not exceeding 4, but I wonder If other ...
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$ax^2 - by^2 = c$ and prime divisors of $a,b,x,y$
Consider the Pell-type equation $ax^2 - by^2 = c$. When $c = \pm 1$, the following Stormer-type result turns out to hold (see e.g. Theorem 3.2):
Theorem. Let $a,b$ be coprime positive integers whose ...
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