# Tag Info

## Hot answers tagged diophantine-equations

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### Estimating the size of solutions of a diophantine equation

This problem turned out to be much more interesting than I originally thought. Let me give my solution, which seems to be slightly different from (but essentially the same as) the solution in the ...
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### Is equation $xy(x+y)=7z^2+1$ solvable in integers?

There is no solution. It is clear that at least one of $x$ and $y$ is positive and that neither is divisible by 7. We can assume that $a := x > 0$. The equation implies that there are integers $X$, ...
• 11.2k
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### Universality of $y^4-x^3$ mod $p$

The curve $C:y^4-x^3z=az^4$ is nonsingular over $\mathbb F_p$ for $p\ge5$ and $a\ne0$. It has genus $3$. So Weil's theorem says that $$\bigl| \#C(\mathbb F_p) - p - 1 \bigr| \le 6\sqrt{p}.$$ There ...
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### Permutations $\pi\in S_n$ with $\sum_{k=1}^n\frac1{k+\pi(k)}=1$

Claim: $a_n>0$ for all $n\geq 6\quad (*)$. Proof: We use induction to prove $(*)$. We have $a_6,a_7,a_8,a_9,a_{10},a_{11}>0$. Assume $(*)$ holds for all the integers $\in [6,n-1]$. We want ...
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### The "stubborn" solutions to sums of three cubes

This lim sup indeed goes to $\infty$. We can prove this using exactly the strategy Lucia suggested. We will count the number of $x,y,z$ in a box with $x^3+y^3+z^3$ not a cubic residue modulo $p$ for a ...
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### Can you solve the listed smallest open Diophantine equations?

The equation $$x^3 + y^3 + z^3 + xyz = 5$$ is solvable in integers. For example, take $$x=-3028982, \quad y=-3786648, \quad z=3480565.$$ Verification is straightforward, but I would like to add ...
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### Can $y^2-4$ be a divisor of $x^3-x^2-2 x+1$?

No. The roots of $x^3 - x^2 - 2x + 1$ are $-(\zeta + \zeta^{-1})$ where $\zeta$ is a 7th root of unity; this soon implies [see below] that any prime factor is either $7$ or $\pm 1 \bmod 7$, and thus ...
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### Diophantine equation $3^n-1=2x^2$

This problem happens to have appeared on the Polish Mathematical Olympiad camp in 2015. Here is the official solution of the problem: (I use $m$ in place of $x$ because this is how the problem was ...
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### Why is this "the first elliptic curve in nature"?

I actually only wrote the part that says that this curve is a model for $X_1(11)$, not the first part, which I think was written by John Cremona. It is standard to order elliptic curves by conductor (...
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### Rational inscribed realization of the regular dodecahedron

An example Yes, here is a list of rational coordinates lying on the unit sphere, the convex hull of which is combinatorially equivalent to a regular dodecahedron. This polyhedron is invariant under ...
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### How many cubes are the sum of three positive cubes?

There are bivariate coprime polynomial parametrizations: https://sites.google.com/site/tpiezas/010: $$(a^4-2ab^3)^3 + (a^3 b+b^4)^3 + (2a^3 b-b^4)^3 = (a^4+a b^3)^3.$$ Added If you drop the positivity ...
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### Are there infinitely many positive integer solutions to $(3+3k+l)^2=m\,(k\,l-k^3-1)$?

It does have infinitely many positive solutions. Here is just one such series. Consider the following recurrence sequence: $$u_0=1,\ u_1=2,\ u_{n+1} = 23 u_n - u_{n-1} - 4\qquad (n\geq 1).$$ Let $t,k$ ...
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### Does the equation $(xy+1)(xy+x+2)=n^2$ have a positive integer solution?

It looks that Vieta jumping helps. For fixed positive integer $y$ choose a minimal positive integer $x$ for which $(xy+1)(xy+x+2)$ is a perfect square. Denote $4(xy+1)(xy+x+2)=4n^2=(2xy+x+3-z)^2$ ...
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