Intereting Posts

When does variété mean manifold?
Roulette betting system probability
Sum of power functions over a simplex
Distribution of hitting time of line by Brownian motion
Union of the conjugates of a proper subgroup
$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.
Hahn-Banach theorem: 2 versions
Differential equation related to Golomb's sequence
How to solve this determinant
Show discontinuity of $\frac{xy}{x^2+y^2}$
Limit power series at boundary
Under what conditions can I interchange the order of limits for a function of two variable?
Singular Value Decomposition of Rank 1 matrix
Ramanujan-type identity $\sum_{n=1}^{\infty}\frac{n^3}{e^{2^{-k}n\pi}-1}=\sum_{n=0}^{k}16^{n-1}$
The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements

Does the equation $$x^3 = 7y^3 + 6 y^2+2 y\tag{1}$$ have any positive integer solutions? This is equivalent to a conjecture about OEIS sequence A245624.

Maple tells me this is a curve of genus $1$, and its Weierstrass form is $s^3 + t^2 + 20 = 0$, with $$ \eqalign{ s = \dfrac{-2(7 y^2 + 6 y + 2)}{x^2},& \

t = \dfrac{-2(3 x^3 + 14 y^2 + 12 y + 4)}{x^3}\cr

x = \dfrac{-2s(t-6)}{s^3+56},&\ y = \dfrac{4t-24}{s^3+56}}$$

So I can find rational points on both curves, but I haven’t been able to find integer points on (1) other than the trivial $(0,0)$.

- Are there integer solutions to $9^x - 8^y = 1$?
- $x,y$ are integers satisfying $2x^2-1=y^{15}$, show that $5 \mid x$
- Find all solutions of ${\frac {1} {x} } + {\frac {1} {y} } +{\frac {1} {z}}=1$, where $x$, $y$ and $z$ are positive integers
- Solve the Diophantine equation $ 3x^2 - 2y^2 =1 $
- Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers?
- Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

- Is there always a telescopic series associated with a rational number?
- Why 1728 in $j$-invariant?
- For which primes $p$ does $px^2-2y^2=1$ have a solution?
- (USAJMO)Find the integer solutions:$ab^5+3=x^3,a^5b+3=y^3$
- Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?
- Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$
- Distribution of the sum reciprocal of primes $\le 1$
- Number of solutions of $x_1+2x_2+\cdots+kx_k=n$?
- Any Implications of Fermat's Last Theorem?
- Pythagorean quadruples

$$ \gcd(y, 7y^2 + 6 y + 2) = 1,2 $$

The first case is odd $y,$ so that $7y^2 + 6y+2$ is odd and $\gcd(y, 7y^2 + 6 y + 2) = 1.$ Both $y$ and $7y^2 + 6 y + 2$ must be cubes. Take $y = n^3.$ We want $7n^6 + 6 n^3 + 2$ to be a cube. Cubes are $1,0,-1 \pmod 9.$ If $n \equiv 0 \pmod 3,$ then $7n^6 + 6 n^3 + 2 \equiv 2 \pmod 9$ and is not a cube. If $n^3 \equiv 1 \pmod 9,$ then $7n^6 + 6 n^3 + 2 \equiv 6 \pmod 9$ and is not a cube. If $n^3 \equiv -1 \pmod 9,$ then $7n^6 + 6 n^3 + 2 \equiv 3 \pmod 9$ and is not a cube.

Next

$\gcd(y, 7y^2 + 6 y + 2) = 2.$ Both $y= 4n^3$ and $7y^2 + 6 y + 2 = 2 w^3$ .

give me a minute, it is not guaranteed to be easy just because the other case was. Hmmm, it is possible both mod 9 and mod 7, which reflects the solution with my $n=0, w=0.$ Sigh. Just taking $y=4u,$ there may be a tractable way to deal with $56u^2 + 12 u + 1 = w^3.$

Monday: computer suggests the only integer point on $56u^2 + 12 u + 1 = w^3$ is $(0,1),$ which would finish the problem if confirmed. CONFIRMED: see Integral solutions to $56u^2 + 12 u + 1 = w^3$

For what it’s worth, I took a generator $P$ of the Mordell-Weil group of $y^2=x^3-20$ and computed the preimage of $k\cdot P$ for $|k| \leq 300$ back to the original curve, and the only instance where the point was integral was for $k=0$.

- At a party $n$ people toss their hats into a pile in a closet.$\dots$
- Proof that there can be no planar graph with 5 faces, such that any two of them share an edge.
- Prove that formula in monadic second order logic exists – for each node path is finite
- Show that if $n$ is not divisible by $2$ or by $3$, then $n^2-1$ is divisible by $24$.
- Proving the relation $\det(I + xy^T ) = 1 + x^Ty$
- Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares
- What is a proof of this limit of this nested radical?
- Infinite Product is converges
- Contradiction! Any Symbol for?
- Standard Deviation Annualized
- Show that for a $2\times 2$ matrix $A^2=0$
- Show that $B$ is invertible if $B=A^2-2A+2I$ and $A^3=2I$
- If $(x_n)$ is a Cauchy sequence, then it has a subsequence such that $\|x_{n_k} – x_{n_{k-1}}\| < 1/2^k$
- Suppose $|G| = 105$. Show that it is abelian if it has a normal $3$-Sylow subgroup.
- How to endow topology on a finite dimensional topological vector space?