The Coolest Exponential Diophantine Equation from Dutch Math Olympiad

In this video, we are solving an interesting exponential diophantine equation from Dutch Math Olympiad. We are looking for positive integer solutions to this interesting exponential diophantine equation. Dutch Math Olympiad is full of interesting yet challenging problems that I am planning to cover many of them. How to factor the function out, and what is the mathematical logic and thought process to apply to solve this exponential diophantine equation? Make sure to watch the video until the end. More to come! Stay tuned!

#exponentialequations #diophantineequation #matholympiad

[domedebali]

Fantastic video and explanation. Your logic and thought process sounds so great

[marinoceccotti]

I had to slow down the video, but indeed, a good one !

[EnnioPiovesan]

If we write the equation as:
2^n=(m^2-1)/3=(m-1)(m+1)/3
we have two cases
A) m-1=3*(2^a) and m+1=2^b with a+b=n
B) m-1=2^a and m+1=3*(2^b) again with a+b=n

A) m+1=(m-1)+2 --> 2^b=3*(2^a)+2 --> 2^b=2(3*(2^(a-1))+1), but 3*(2^(a-1))+1 can be a power of 2 only if a-1=0 so a=1. As a consequence:

2^b=2*(3+1)=2^3 --> b=3
m-1=3*2 --> m=7
n=a+b=1+3--> n=4

B) m+1=m-1+2--> 2^a+2=3*(2^b) --> 2(2^(a-1)+1)=3*(2^b). This can only happen when 2^(a-1)+1=3 (and b=1). That is 2^(a-1)=2 --> a=2.
So m-1=2^2 --> m=5
n=a+b=1+2 --> n=3.

[mathnerd]

oh I thought there are 3 answers including (2, 0), but it was about positive integer, not non-negative integer. nice professor!

[cristianoronaldoofficial]

Sir I have a doubt:
'x^4=16'.Find all 4 solutions.

[vamshichinna]

(2, 0) is also solution it is correct (or ) wrong please clarify my doubt sir

[echandler]

Why is 6 the only possible even multiple of 3 for one factor? If 3 was multiplied by a power of 2 that is greater than 2, then the other number (3*2^k +/- 2) would be an odd multiple of 2, hence not a power of 2.