- #1
BustedBreaks
- 65
- 0
Let [tex]p_{1}, p_{2},...,p_{n}[/tex] be primes. Show that [tex]p_{1} p_{2}...p_{n}+1[/tex] is divisible by none of these primes.Let [tex]p_{1}, p_{2},...,p_{n}[/tex] be primes
Let [tex]k \in N[/tex]
Assume [tex]p_{1}p_{2}...p_{n}+1=kp_{n}[/tex]
[tex]\frac{p_{1}p_{2}...p_{n}}{p_{n}}+\frac{1}{p_{n}}=k[/tex]
[tex]p_{1}p_{2}...p_{n-1}+\frac{1}{p_{n}}=k[/tex]
This is a contradiction because the left side will not be a natural number.
My issue is that this seems to only prove [tex]p_{1} p_{2}...p_{n}+1[/tex] is not divisible by [tex]p_{n}[/tex] and not all [tex]p_{1}, p_{2},...,p_{n}[/tex].
Thanks!
Let [tex]k \in N[/tex]
Assume [tex]p_{1}p_{2}...p_{n}+1=kp_{n}[/tex]
[tex]\frac{p_{1}p_{2}...p_{n}}{p_{n}}+\frac{1}{p_{n}}=k[/tex]
[tex]p_{1}p_{2}...p_{n-1}+\frac{1}{p_{n}}=k[/tex]
This is a contradiction because the left side will not be a natural number.
My issue is that this seems to only prove [tex]p_{1} p_{2}...p_{n}+1[/tex] is not divisible by [tex]p_{n}[/tex] and not all [tex]p_{1}, p_{2},...,p_{n}[/tex].
Thanks!