Are There Any Integer Solutions?

We determine if there are any integer solutions to a^m + 1 = 2^n (for a, m, n greater than 1). This problem is a special case of Catalan's conjecture.
Factor theorem:
en.wikipedia.org/wiki/Factor_...
00:00 Intro & constraints
00:50 Considering odd & even numbers
02:50 Factorising
04:30 More odd & even numbers
05:54 Ruling out more solutions
06:46 Conclusion

[@mrigayu]

Whenever I try to complete proofs like these, I seem to always forget a minor case or two. But you've explained it really methodically and it's helpful to see it done this way. Thanks for yet another elegant explanation!

[@dbmalesani]

Very neat solution. This is also an "easy" case of Catalan's conjecture / Mihăilescu's theorem, that there are no non-trivial solutions to the equation xᵃ - yᵇ = 1 besides 3² - 2³ = 1.

[@gtweak7]

CompSci grad here. I highly enjoy your small neat proofs like this one - thank you Dr Barker.

[@virgodem]

6:34 a^m = a can also mean m = 1. Yes, that is also ruled out. But it was a forgotten case.

[@DrBarker]

Good point - well spotted!

[@rogerkearns8094]

After watching this, a strange philosophical question suddenly arose in my mind: did we already know all of the solutions even before we learned that there were none?

[@want-diversecontent3887]

I think I've heard similar cases before where we already knew the answer but didn't know if it was the actual answer until later, but I can't pinpoint any exact examples.

[@rogerkearns8094]

@@want-diversecontent3887 OK, thank you.

[@sebastianmanterfield3132]

We didn't know the solution to the question "What are all the integer solutions?", but even though we were unaware of it, we knew every solution of the equation.

[@rogerkearns8094]

@@sebastianmanterfield3132 Yes, that's what I thought. It seems a bit weird, though, somehow. Cheers :)

[@lenskihe]

We knew all the solutions, but we didn't know we knew all the solutions. (Kind of similar to the Riemann Hypothesis 😳)

[@jamiepianist]

I hope your channel blows up soon, your videos are amazing

[@DrBarker]

Thank you!

[@seb6859]

Very interesting video

[@MichaelRothwell1]

Nice problem and very neat solution!

[@Sceptique_Smike7]

Hello! Now I think I should read about the original proof of the Catalan conjecture. In a sense, this topic reminds me a bit of log-concavity even though I don't know much about it. Good explanation! 👍

[@General12th]

Hi Dr. Barker! Very cool!

[@johnporter7915]

I love your channel. Can you do some imo algebra based problems? I’d love to try some. Thanks.

[@GaetanoDiCaprio]

Excellent!

[@GaetanoDiCaprio]

You might be interested in making a video on the following problem: find all positive integer x such that x^3+x^2+x+1 is a perfect square. I've made a video on a similar problem (x^4+x^3+x^2+x+1 perfect square) but I haven't been able to solve this variant. Thank you.

[@robertsandy3794]

Hi, don't understand why you looked at the case when a = -1. Your initial conditions were that a, m, n > 1 Thanks

[@stewartzayat7526]

He didn't look at the case when a=-1, he simply used the fact that -1 is a root of the polynomial, so you can factor it like he did

[@lifeisfakenews]

2:30 shouldnt it be plus 1 in the expansion? (2k+1)(2k+1) -> 4k^2 + 2k + 2k + 1

[@astherphoenix9648]

on that line, theres' an additional plus 1 because he took a^m, which is equal to what you have above + 1

[@jaafars.mahdawi6911]

Interesting quest. Neat persecution. As always. Keep it up, bro. (Oh and i enjoy the accent, too!)

[@johns.8246]

Yo, I need help proving there are no positive integer solutions x, y, z satisfying x^3 - y^3 = 2^z .

[@xchomphk.9788]

This violates the Catalan conjecture, QED

[@caspermadlener4191]

1. It doesn't. 2. The Collatz conjecture is unproven. 3. You mean the Catalan conjecture. 4. You aren't allowed to use the Catalan conjecture if you don't know the proof (although the proof is pretty long, it isn't really complicated)

[@xchomphk.9788]

@@caspermadlener4191 oh shit i meant catalan

[@xchomphk.9788]

@@caspermadlener4191 also I meant this as a joke ik its dumb to just slap an overly complicated theorem onto a problem and say QED chillax

[@caspermadlener4191]

@@xchomphk.9788 Some people used Catalan on the Dutch selectiom for the IMO. They did get part of the points for it.

[@xchomphk.9788]

@@caspermadlener4191 wait you aren't allowed to use the catalan conjecture for Math Olympiad problems even though its already proven? why would you need to reprove something that already exists

[@ritamauriya-er4fh]

A=m=n=2