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I've tested some of them with PARI/GP. There is no probable prime with the Hamming distance 1. There are several (probably 3--400, haven't exhaustively listed) probable primes with the Hamming distance 2 (p ^ (1<<30) ^ (1<<14) is one example).


About 1 in 700 1024-bit numbers is prime. There are 1023512 Hamming-distance 2 numbers for a 1024-bit number, about 750 of them will be prime.

If you found no Hamming distance 1 primes (expected: 0.68) and 3-400 Hamming distance 2 primes (expected: 750), this is one really* bad "prime"...




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