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1200
Prime Expression — Editorial
number theoryprimesmodular arithmetic
If pp is an odd prime other than 3,3, then modulo 33 we have 2p≡22^p \equiv 2 (since pp is odd) and p2≡1,p^2 \equiv 1, so 2p+p2≡0(mod3).2^p + p^2 \equiv 0 \pmod{3}. Since 2p+p2>3,2^p + p^2 > 3, it must be composite. Checking p=2p = 2: 4+4=8,4 + 4 = 8, not prime. Checking p=3p = 3: 8+9=17,8 + 9 = 17, which is prime. The only qualifying prime is p=3,p = 3, so the sum is 3.3.