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2d

comment 
Collatz property implying infinite “fall below” trajectories, is it known?
OK, so what you are asserting is that for every $m$ there exists $n$ such that for every $a$ $T^k(a\cdot2^n+m)<a\cdot2^n+m$ for some $k$, where $T$ is the Collatz iteration, right? 
2d

comment 
Collatz property implying infinite “fall below” trajectories, is it known?
I think you are claiming that for every $m$ there exists $n$ such that for every $a$ Collatz is true for $a\cdot2^n+m$. Is that right? 
2d

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The Diophantine equation $x^p  4y^p = z^2$
@Dietrich, there are more where those came from, I think eight examples with $x<1000$. 
Aug 20 
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The Diophantine equation $x^p  4y^p = z^2$
Note that my earlier comment referred to an earlier version of the question. 
Aug 20 
comment 
The Diophantine equation $x^p  4y^p = z^2$
$78^34\times29^3=614^2$. $93^34\times53^3=457^2$. 
Aug 19 
comment 
A conjectured formula for Apéry numbers
@Will, my late colleague, Alf van der Poorten, is also credited as an author of that book. 
Aug 19 
comment 
Maximal score for the 2048 game
See also mathoverflow.net/questions/160703/… 
Aug 19 
comment 
Expected value when rolling multiple ksided dice and keeping the highest score and 1s cancelling higest remaining values
MO is for questions of mathematical research. I'm not convinced that this is a research question. 
Aug 18 
comment 
Integer valued polynomial through some points with rational coordinates
Please include a link to the m.se question, and include a link there to this question. 
Aug 17 
comment 
Linear dependency of real numbers with integer coefficients adding up to zero
There are "integerrelation finding algorithms" around, you might want to do a search for those. 
Aug 14 
comment 
Smallest prime in an arithmetic progression
Xylouris has 5.18 in Xylouris, Triantafyllos, On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet Lfunctions, Acta Arith. 150 (2011), no. 1, 65–91, MR2825574 (2012m:11129). 
Aug 14 
comment 
When is a cubic polynomial a cube?
You posted on m.se and MO without linking each to the other. That's rude. 
Aug 12 
comment 
Maximum possible number of similar three colored triangles
I suppose for the $2\times2\times2$ you insist on planarity, else the vertices of a regular octahedron will do. 
Aug 12 
comment 
Nontransitive dice
See also math.stackexchange.com/questions/57338/… 
Aug 12 
comment 
A Conjecture About Directed Graphs that are the Union of Two Trees
Where is the example of Chudnovsky & Seymour? 
Aug 12 
comment 
Divisor sums over values of binary forms of primes
Possibly simpler questions would be $$\sum\tau(x^2+y^2)$$ over all $x,y$ up to $n$ (so, not restricting $x,y$ to primes), or $$\sum\tau(p+1)$$ summing over primes $p<n$. Are asymptotics known for these? 
Aug 10 
awarded  Nice Answer 
Aug 8 
comment 
Characterize the set of roots of cubics with certain properties
You may also want to look at mathoverflow.net/questions/21267/… 
Aug 8 
comment 
Characterize the set of roots of cubics with certain properties
Pity, that. It's a good book to have, if you are interested in this sort of question. Are there any libraries where you live? 
Aug 8 
comment 
Characterize the set of roots of cubics with certain properties
Have you gone through Chapter 7.3 of Alaca and Williams yet? 