outline of a guess.. any easy solution not requiring exhaustion and guessing? if p | 2^32+1 2^32 = -1 mod p 2^(p-1) = 1 mod p –> guess p-1 = 64k 65 not prime, 129 not prime, 193 prime 257 prime, 321 not prime, 385 not prime 449 prime, 513 not prime, 577 prime 641 prime, 705 not prime, 769 prime 833 not prime, 897 not prime, 961 not prime 193, 257, 449, 577, 641, 769 (here all equal signs are under modulo) 193–> 2^8=2^6*3+2^6=2^6-1; 2^16=2^12-2^7+1=2^10-2^4-2^7+1=2^8-2^2-2^4-2^7+1=-4-16-64=-84; 2^32+1=84^2+1=16*441+1=16*55+1=881=109 257–> 2^8^4+1=2 449–> 2^9=2^6*7+2^6=2^6-1; 2^12=2^9-2^3=2^6-2^3-1; 2^15=2^12-2^6=-2^3-1; 2^32+1=9^2*4+1=325 641!–> 2^7*5+1; 5^4=625=641-16=-16; 1=(2^7*5)^4=2^28*(-16)=-2^32