A throwback number theory problem

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[jkid]

Better than being a throwback to the Balkans in the 90s

[aln]

So satisfying to watch an olympiad problem being solved👍🏻

[goodplacetostop]

11:01 Spoiler alert, the original statement says to…




Prove that the equation y^2 = x^5 - 4 has no integer solutions.

So yeah, no solution was the answer

[wojteksocha]

If p=4k+3 and p divides x^2 + y^2, then p divides both x and y. That means that 4 divides both sides and 8 does not, but if 4 divides y^5, then 32 divides y^5

[Bodyknock]

One small thing that makes it slightly easier at 8:00 is, instead of doing the chart from n=0 to 10, do it from n= -5 to 5. You get the same results but it's immediately obvious why there's symmetry about 0 and why you only need to actually calculate n=0 through 5.

[virajagr]

6:11 ah yes, 10 = 1 -11, the good old Michael is back

[HershO.]

This was a nice revision of number theory.

[jesusalej]

According to FLT if y^5 = 1(mod 3), then x^2+1 must be 1(mod 3), that means x^2 = 0(mod 3). From there come the solutions, x^2+1 cannot be 2(mod 3), because is a contradiction.

[dandjr]

So he checked mod 3 and mod 11. Does that prove there are no solutions? Or do you also have to check mod 17, mod 31, .. mod 1187, ...

[mehmetozdemir]

Why are mod3 and mod11 enough to say there is no solution?

[Anonymous-zphb]

Ah quadratic residues. Of course. Didn't get very far with this problem. Nice solution.

[АнтонФ-цй]

Hi. 10!/6! = 7!
Any other a, b, c from N (Z) exsists?

[leif]

Surely you can solve without fermats little theorem or modular arithmetic so why use them at all?

[lawrencecataylo]

So if the value of y^5 = x^2 + 4 = 1(mod 3) and y^10 = (x^2 + 4)^2 (mod 11), how do we get the modulo value, based on Fermat's little theorem? That is the modulo 3 and 11? is it like the main remainder of the given equation of y? thank you.

[johns.]

How about for x^2=y^5+4 ? I already see a solution!

[frentz]

You put a *hint* on the chalkboard *with* the statement of the problem. Come on .... at least put the problem up there for five seconds, for people who want to try it!

[quite_unknown_]

Mod 11 is sufficiënt on itself, no need to check mod 3