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Find an integer in the range $100leq n leq 1997$ such that $frac2^n+2n$ is an integer.

Showing $x^8equiv 16 pmodp$ is solvable for all primes $p$Congruences for Fermat QuotientsCan the relation of $a^p equiv apmod 2p$ if (p prime and p>2) be added to Fermat's little theorem?Prove that $n^2 leq fracc^n+c^-n-2c+c^-1-2$.Prove that if $n$ is sufficently large, there is a prime gap $G(p_k, p_k+1)$ with $p_k leq n$ and $p_k+1 - p_k > frac12ln(n)$Investigating the density of special integersSuppose that $n$ is a composite, squarefree integer such that for every prime divisor $p$ of $n$…Fermat's little theorem and reciprocals modulo $p$Prove that $1835^1910 + 1986^2061$ is divisible by 7.Proving there exists a prime $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

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$begingroup$

I am having trouble with this question. It comes from the 1997 APMO:

Find an integer in the range $100leq n leq1997$ such that $frac2^n+2n$ is an integer.

When I first attempted this problem, I thought, maybe FLT (Fermat's Little Theorem) might apply to this. However, I soon found out that $n$ must be even, and the only even prime is 2, so FLT is useless. I really don't know what to do. Can someone please help?

number-theory contest-math

share|cite|improve this question

asked Mar 10 at 20:46

A RA R

715

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why do you think that $n$ must be even?
  $endgroup$
  – Misha Lavrov
  Mar 10 at 20:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Actually, I just found out that $n$ can be odd, since odd times even = even. Silly me! However, FLT still does not work. I tried to factor out 2 from the numerator and use FLT on the term that is inside, but I still can't seem to make it work.
  $endgroup$
  – A R
  Mar 10 at 20:56
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  SPOILER ALERT: mks.mff.cuni.cz/kalva/apmo/asoln/asol972.html
  $endgroup$
  – Dr. Mathva
  Mar 10 at 21:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Dr.Mathva So, then, a question whose answer is readily found using the brute force of mathematical software, but which nobody could reasonably be expected to find under exam conditions.
  $endgroup$
  – Rosie F
  2 days ago
  

  
  
  
  

add a comment | 

0

$begingroup$

I am having trouble with this question. It comes from the 1997 APMO:

Find an integer in the range $100leq n leq1997$ such that $frac2^n+2n$ is an integer.

When I first attempted this problem, I thought, maybe FLT (Fermat's Little Theorem) might apply to this. However, I soon found out that $n$ must be even, and the only even prime is 2, so FLT is useless. I really don't know what to do. Can someone please help?

number-theory contest-math

share|cite|improve this question

asked Mar 10 at 20:46

A RA R

715

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why do you think that $n$ must be even?
  $endgroup$
  – Misha Lavrov
  Mar 10 at 20:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Actually, I just found out that $n$ can be odd, since odd times even = even. Silly me! However, FLT still does not work. I tried to factor out 2 from the numerator and use FLT on the term that is inside, but I still can't seem to make it work.
  $endgroup$
  – A R
  Mar 10 at 20:56
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  SPOILER ALERT: mks.mff.cuni.cz/kalva/apmo/asoln/asol972.html
  $endgroup$
  – Dr. Mathva
  Mar 10 at 21:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Dr.Mathva So, then, a question whose answer is readily found using the brute force of mathematical software, but which nobody could reasonably be expected to find under exam conditions.
  $endgroup$
  – Rosie F
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

I am having trouble with this question. It comes from the 1997 APMO:

Find an integer in the range $100leq n leq1997$ such that $frac2^n+2n$ is an integer.

When I first attempted this problem, I thought, maybe FLT (Fermat's Little Theorem) might apply to this. However, I soon found out that $n$ must be even, and the only even prime is 2, so FLT is useless. I really don't know what to do. Can someone please help?

number-theory contest-math

share|cite|improve this question

asked Mar 10 at 20:46

A RA R

715

$endgroup$

share|cite|improve this question

asked Mar 10 at 20:46

A RA R

715

share|cite|improve this question

asked Mar 10 at 20:46

A RA R

715

asked Mar 10 at 20:46

A RA R

715

asked Mar 10 at 20:46

A RA R

715