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15h
comment $x^4+5y^4=z^2$ doesn't have an integer solution.
@SiXUlm I still do not know whether there are only finitely many different families of solutions. I believe the relevant topic is called "elliptic curves" whicfh sometimes can show for Diophantine equations of degree 3 or 4 that there can only be a finite number of such families. Unfortunately I'm not expert enough about this topic to say more for this example.
16h
awarded  Enlightened
16h
awarded  Nice Answer
19h
comment Suppose $a<b<c<d$ and $p(x)=(x-a)(x-b)(x-c)(x-d)$. Show that $\int_a^b \frac{dx}{\sqrt{|p(x)|}} = \int_c^d \frac{dx}{\sqrt{|p(x)|}}$
It also seems odd that the parameter $d$ is not in either integral, even if the problem is to show they're equal or something.
1d
comment For any rng $R$, can we attach a unity?
It's OK if the starting ring $R$ happens to contain a unity, say $e$.It won't conflict via the embedding with the element $(1,0_R)$ since the image of $e$ is $(0,e)$ and the latter is not an identity for the full ring on $\mathbb{Z} \times R.$.
1d
revised Sum of Series as $1,(2),1,(2,2),1,(2,2,2),1,(2,2,2,2),1…$
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1d
revised Sum of Series as $1,(2),1,(2,2),1,(2,2,2),1,(2,2,2,2),1…$
added 294 characters in body
1d
answered Sum of Series as $1,(2),1,(2,2),1,(2,2,2),1,(2,2,2,2),1…$
May
25
comment What will be the equation of side $BC$.
If side AB goes with equation $x+y=5$ then it seems the point labeled B should be the one your diagram has labeled C. (and vice-versa for the point labeled C). I don't know if this affects the calculation...
May
24
revised Find all solutions in N of the following Diophantine equation
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May
24
comment Find all solutions in N of the following Diophantine equation
Good answer, and it proves we have expressed all the integer solutions. +1.
May
24
comment Find all solutions in N of the following Diophantine equation
@user241340 I just put some more material at the end about the case of solutions with $x,y,z$ positive integers, to your starting equation before you introduced new variables $A,B.$ What it is consists of a parametrization in terms of two integers $m,n$ sort of like the usual one for Pythagorean triples. However I don't know at this point if all solutions are in the family I found here.
May
24
revised Find all solutions in N of the following Diophantine equation
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May
24
revised Find all solutions in N of the following Diophantine equation
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May
23
comment Find all solutions in N of the following Diophantine equation
@user241340 It isn't always the best move for solving integer equations to first find all rational solutions and then from that obtain integer solutions as special cases, or at least infinite families of integer solutions. (I haven't yet looked at this issue for your equation...) For some equations, there may be a clever thing to notice from the given equation which allows one to proceed and be sure not to miss any integer solutions. But this is more difficult than just getting the rational solutions, as I mentioned.
May
23
answered Find all solutions in N of the following Diophantine equation
May
22
comment Product of two sets with density zero has density zero?
@DanielFischer Thanks for this comment, which fills in where I only gave a hand-waving heuristic. Is there a reference for the natural density result you mention, which would be then be good to include in the answer?
May
21
comment Product of two sets with density zero has density zero?
My attempts to re-code the squarefree integers into one or the other of the above $A,B$ involved moving blocks of them to the right by uncontrollable amounts, which means the natural density could easily change from being positive to winding up zero.
May
21
revised Product of two sets with density zero has density zero?
added 714 characters in body
May
21
awarded  number-theory