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bio website math.ubc.ca/~bennett
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visits member for 1 year, 7 months
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Aug
2
comment How to prove $~(c - b) ^ 2 + 3cb = x^3~$ has no nonzero integer solutions?
If, say, $c=1, b=-1$ and $x=1$, we have a solution to (2).
Jun
26
comment How to find the minimum value of $|5^{4m+3}-n^2 |$
The minimum is indeed $275$ but I do not see a short, mathematics-free way to prove this….
May
9
comment Find all real $x$ ,such $8x^3-20$ and $2x^5-2$ is perfect square
Are you really offering a bounty for a problem whose solution is in the comments?
May
9
comment Infinitely many perfect squares
That doesn't affect the genus. Just multiply the equation by $(A 2^{n_0})^2$ to get a new equation of the shape $Y^2=X^3+C$ with $C=(A 2^{n_0})^2 B$.
May
7
comment Find all real $x$ ,such $8x^3-20$ and $2x^5-2$ is perfect square
Plugging f:=EllipticCurve([0,-20]); followed by IntegralPoints(f); into Magma ensures that the solution Robert found is the only one (i.e. $x=3$).
May
7
answered Infinitely many perfect squares
May
2
awarded  Nice Answer
Jan
23
awarded  Yearling
Dec
17
answered How find this equation $n!+(n+1)!+\cdots+(n+m)!=a^b$ all solutions
Nov
26
comment There are only finitely many integer solutions to $ax^n+by^n=c.$
Well, $ax^3+by^3=cz^3$ can certainly have infinitely many coprime integral solutions...
Nov
22
comment What integers can be represented by the quadratic form $4x^2 - 3y^2 - z^2$?
This case is rather easier as the form is indefinite (and can be easily shown to represent every $n \not\equiv 2 \mod{4}$ infinitely often).
Nov
21
comment What integers can be represented by the quadratic form $4x^2 - 3y^2 - z^2$?
Did you perhaps look for solutions? There are many.
Nov
20
comment Show that there are no squares included in the sequences
And $3$ is not a quadratic residue modulo $5$, for the final case.
Nov
6
comment Generalization of an inequality $0\lt e^6-{\pi}^4-{\pi}^5\lt 0.00002$
I mean $l^{-1}$ instead of $a^{-1}$ in the above comment.
Nov
6
comment Generalization of an inequality $0\lt e^6-{\pi}^4-{\pi}^5\lt 0.00002$
If we have even $| e^k - \pi^l - \pi^m| <1$ with, say, $l > m$, then, $\left| \log \pi - k/l \right| \ll a^{-1} \pi^{m-l}$ and so, if $l$ is a fair bit bigger than $m$, we would have an extraordinarily good approximation to $\log \pi$. Unfortunately, as far as I know, we cannot rule this out, but one suspects that such an approximation is unlikely to exist. This would suggest that $|l-m| \ll \log m$. For such pairs $(l,m)$, a heuristic argument suggests that we cannot force $e^k-\pi^l-\pi^m$ to be arbitrarily close to $0$.
Oct
24
comment How does one attack a divisibility problem like $(a+b)^2 \mid (2a^3+6a^2b+1)$?
While the original question can be reduced to finding the integral points on a parametrized family of elliptic curves (certainly not just one), it's extremely unclear that his approach is even slightly helpful for actually solving the problem.
Oct
24
answered Do there exist natural number solutions such that $x^m=11\cdots11$ for $m\ge 2$?
Oct
16
comment About the integer solutions of $\ y^2=x^3+9$
There might not be an easy proof. The most straightforward that I know uses Skolem's method -- for this equation, this was done in a thesis of Hemer from 1952. Your equation reduces to solving $Aa^3-Bb^3=C$ for all triples $A, B, C$ with $ABC=6$. I don't know an elementary way to do this.
Oct
15
comment About the integer solutions of $\ y^2=x^3+9$
These are the only integer solutions. Data for $y^2=x^3+k$ with $|k| < 10^4$ can be found at tnt.math.se.tmu.ac.jp/simath/MORDELL. These results were proved using lower bounds for linear forms in elliptic logarithms (though the case $k=9$ can be treated more simply, by reducing to cubic Thue equations).
Sep
17
comment Does this Diophantine cubic have solutions?
An old theorem of Delone and Nagell (discovered independently) says that, given $a$ and $b$, the equation $ax^3-by^3=1$ has at most one solution in positive integers $x$ and $y$ (and moreover tells you how to find the solution, if it exists).