I Solved An Exponential Diophantine Equation

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PLAYLISTS 🎵 :

[rajeshbuya]

WOW WOW WOW WOW!!! ❤❤❤❤

Number Theory, Modulo Arithmetic - HOUDINI OF MATHEMAGIC!!!

While the Pythogarean triplets gave away 1 solution by just a look at the problem, but then proving this was the ONLY Integer solution was sheer magic!!!

Absolutely LOVED IT ❤❤❤

[mahmoudalbahar]

Thank you for your great videos, but 1 is congruent to -1 mod 2 so it's true for any integers z and y.

[mcwulf]

Very good, and thoroughly explained 👍😁.

I wonder if Pythagoras could have proven that?

[null.dev.]

Cute. By the way, maybe this is a nitpick or maybe I am wrong but since you mention the a>b clauses, shouldn't you mention the n > 0 when doing mod 3 at around @7:20 mark? if n = 0, 3^0 mod 3 = 1 != 0, so n has to be greater than 0. Of course, later you solve it for being precisely that but I still think you should have added it since the mod isn't lossless... or am I just been too long away from doing maff? :)

[mircoceccarelli]

3^x + 4^y = 5^z

( x, y, z ) = { ( 0, 1, 1 ), ( 2, 2, 2 ) }

😊👍👋

[goldfing]

The most obvious solution is x = y = z = 2, from Pythagoras.

[chaosredefined]

Slight problem... -1 = 1 mod 2. The first time you use this, it's fine, because what you did is also valid mod 4. From there... 3^2n = 9^n, so we have 9^k + 4^y = 5^z. Next, if y > 1, then 4^y is divisible by 8. So, taking mod 8 we get 1^k + 0 = 5^z. This only holds if z is even, and we're good to continue... unless y = 1. So, you still need to handle the case of 4 = 5^z - 9^k.

[michaelpenklis]

I thought it couldn't be too easy

[허공]

we can also know that x and z are even using mod 4 and mod 3.

[irrelevant-mr]

Is'nt minus 1 is congruent to plus 1 modulo 2?

[etzbert]

The next time use more variables. Just kidding. Nice explanation 👌

[williamBryan-ke]

there is a very simple fact in math. the number of equations to solve MUST match ( or be more than ) the number of variables. Here 1 vs 3 - not possible to solve. Another- you cannot say that using mod and then doing math on mod results will work. Here is good example 5 + 5 = 10 - now take mod 3. you get 2 + 2 = 1 - not quite true is it? So, anymone want to tell me I am wrong?