Find all primes p such that 2^p + p^2 is also prime.
Prove there are no
others.
Let f(p)=2^p+p^2
Then, f(2)=2^2+2^2=8-> NOT a prime number.
f(3)= 2^3+3^2=17-> A PRIME NUMBER --------(i)
Now, let us consider p>3. We know that, all primes greater than 3 are odd.
Since all odd powers of 2 reduces to 2 in module 3 system, we must have:
2^p=2(mod 3) for p>3 .......(ii)
Now, for p>3, any multiple of 3 is composite and thus NOT a prime number.
Thus, p=1 or 2(mod 3), whenever, p>3
=> p^2=1 or 4 (mod 3)=1 (mod 3)..........(iii)
Accordingly, from (ii) and (iii):
f(p)=(2+1) (mod 3),
= 0 (mod 3), whenever p>3
This is a contradiction, as f(p) is then clearly a composite number, being a multiple of 3.
Consequently, the only possible prime number value of p for which 2^p+p^2 is a prime number MUST be 3.
Edited on January 8, 2022, 8:56 pm
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