Difference between revisions of "2008 OIM Problems/Problem 4"
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== Solution == | == Solution == | ||
| − | {{ | + | Clearly <math>21|2008!</math>, so <math>21|x^{2008}</math> as well. Then <math>21|x</math>, so let <math>x=21a</math> for some positive <math>a</math>. Our equation would become: |
| + | <cmath>21^{2008}a^{2008}+2008!=21^y</cmath> | ||
| + | But notice that | ||
| + | <cmath>21^y=21^{2008}a^{2008}+2008!>21^{2008}a^{2008}\ge 21^{2008}</cmath> | ||
| + | implying that <math>y>2008</math>. This will be used later. | ||
| + | |||
| + | Next, we note that by [[Legendre's Formula]], there are <math>331</math> powers of <math>7</math> and <math>1000</math> powers of <math>3</math> in the prime representation of <math>2008!</math>. Then we can let <math>2008!=3^{1000}7^{331}b</math> for some positive integer <math>b</math> that is not divisible by <math>3</math> or <math>7</math>. Substituting: | ||
| + | <cmath>21^{2008}a^{2008}+3^{1000}7^{331}b=21^y</cmath> | ||
| + | <cmath>\Rightarrow 21^{2008}a^{2008}+3^{669}21^{331}b=21^y</cmath> | ||
| + | <cmath>\Rightarrow 21^{1677}a^{2008}+3^{669}b=21^{y-331}</cmath> | ||
| + | <cmath>\Rightarrow 3^{669}b=21^{y-331}-21^{1677}a^{2008}</cmath> | ||
| + | Since <math>y>2008>331</math>, the right-hand side is divisible by <math>7</math>. However, the left-hand side is not, which is not possible; thus there are no positive integer solutions to the equation. | ||
| + | |||
| + | ~ [https://artofproblemsolving.com/wiki/index.php/User:Eevee9406 eevee9406] | ||
== See also == | == See also == | ||
[[OIM Problems and Solutions]] | [[OIM Problems and Solutions]] | ||
Latest revision as of 22:20, 25 March 2025
Problem
Prove that there are no positive integers 
and 
such that
![\[x^{2008}+2008!=21^y\]](http://latex.artofproblemsolving.com/2/0/2/2026207e891339a1b45c23c12da28a179da6f8e2.png)
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
Clearly 
, so 
as well. Then 
, so let 
for some positive 
. Our equation would become:
![\[21^{2008}a^{2008}+2008!=21^y\]](http://latex.artofproblemsolving.com/1/a/a/1aa993786354b094816bf7ffc66f2515d0df262a.png)
But notice that
![\[21^y=21^{2008}a^{2008}+2008!>21^{2008}a^{2008}\ge 21^{2008}\]](http://latex.artofproblemsolving.com/8/9/f/89f763773e4ddaeffafd8f39c1294ea8720a9ed0.png)
implying that 
. This will be used later.
Next, we note that by Legendre's Formula, there are 
powers of 
and 
powers of 
in the prime representation of 
. Then we can let 
for some positive integer 
that is not divisible by or . Substituting:
![\[21^{2008}a^{2008}+3^{1000}7^{331}b=21^y\]](http://latex.artofproblemsolving.com/5/1/6/516109a4de2451957f0101b13af1d9387b57c187.png)
![\[\Rightarrow 21^{2008}a^{2008}+3^{669}21^{331}b=21^y\]](http://latex.artofproblemsolving.com/d/f/1/df1343201a0456046c62fcdd376651b1b14df7d8.png)
![\[\Rightarrow 21^{1677}a^{2008}+3^{669}b=21^{y-331}\]](http://latex.artofproblemsolving.com/f/2/1/f2146483c552e1c37ef1fdb8f8f923eda2f1682e.png)
![\[\Rightarrow 3^{669}b=21^{y-331}-21^{1677}a^{2008}\]](http://latex.artofproblemsolving.com/7/5/5/75591857c8734a7155ce0affa2f172d8185f9be5.png)
Since 
, the right-hand side is divisible by . However, the left-hand side is not, which is not possible; thus there are no positive integer solutions to the equation.
