2005 IMO Problems/Problem 4

Revision as of 22:02, 25 November 2023 by Kscv (talk | contribs) (Solution)

Problem

Determine all positive integers relatively prime to all the terms of the infinite sequence \[a_n=2^n+3^n+6^n -1,\ n\geq 1.\]

Solution

For all primes $p>3$, by FLT, $n^{p-1} \equiv 1\pmod p$ if $n$ and $p$ are relatively prime. This means that $n^{p-3} \equiv \frac{1}{n^2} \pmod p$. Plugging $n = p-3$ back into the equation, we see that the value mod $p$ is simply $\frac{2}{9} + \frac{3}{4} + \frac{1}{36} - 1 = 0$. Thus, the expression is divisible by $p$. Because the expression is clearly never divisible by $2$ or $3$, our answer is all numbers of the form $2^a3^b$.

See Also

2005 IMO (Problems) • Resources
Preceded by
Problem 3
1 2 3 4 5 6 Followed by
Problem 5
All IMO Problems and Solutions