Prime numbers

For discussing Olympiad Level Number Theory problems
Katy729
Posts: 47
Joined: Sat May 06, 2017 2:30 am

Prime numbers

For how many natural numbers \$n\$, both \$n\$ and \$(n -6)^2 + 1\$ are prime?

Abdullah Al Tanzim
Posts: 20
Joined: Tue Apr 11, 2017 12:03 am

Re: Prime numbers

use parity.
Everybody is a genius.... But if you judge a fish by its ability to climb a tree, it will spend its whole life believing that it is stupid - Albert Einstein

aritra barua
Posts: 57
Joined: Sun Dec 11, 2016 2:01 pm

Re: Prime numbers

\$n\$=\$2,5,7\$.

ankon dey
Posts: 3
Joined: Wed Nov 22, 2017 2:16 pm

Re: Prime numbers

a prime is odd (ignore 2!!) .......odd-even=odd..... \$odd^2\$ =odd.....odd+1=even...but it cannot be prime(ignore \$2\$)......

so remaining is \$2\$...setting \$n=2\$ we get, \$(n-6)^2 + 1\$ =\$17\$....
setting \$n=5 or 7\$ we get the term as 2...it is a prime

so....solution....these type of primes are 2,5 ,7

Tasnood
Posts: 73
Joined: Tue Jan 06, 2015 1:46 pm

Re: Prime numbers

Basically the solution may be this kind of:
\$n\$ is a natural prime number. So, it can be either an odd number or the only even number,\$2\$
For, \$n=2\$, we get: \$(n-6)^2+1=(2-6)^2+1=(-4)^2+1=17\$; which is a prime.

But for, \$n\$=odd, \$n-6\$=odd, \$(n-6)^2\$=odd, but \$(n-6)^2+1\$=even
The only even prime=\$2\$
So, \$(n-6)^2+1=2\$
\$\Rightarrow (n-6)^2=1\$
\$\Rightarrow n-6=+1/-1\$
\$\Rightarrow n-6=+1 \Rightarrow n=7\$
or, \$\Rightarrow n-6=-1 \Rightarrow n=5\$
So, the answers are:\$2,5,7\$