High Quality Content by WIKIPEDIA articles! A safe prime is a prime number of the form 2p + 1, where p is also a prime. (Conversely, the prime p is a Sophie Germain prime.) The first few safe primes are 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907. (sequence A005385 in OEIS) With the exception of 7, a safe prime q is of the form 6k – 1 or, equivalently, q 5 (mod 6) — as is p > 3 (c.f. Sophie Germain prime, second paragraph). Similarly, with the exception of 5, a safe prime q is of the form 4k – 1 or, equivalently, q 3 (mod 4) — trivially true since (q – 1) / 2 must evaluate to an odd natural number. Combining both forms using lcm(6,4) we determine that a safe prime q > 7 also must be of the form 12k–1 or, equivalently, q 11 (mod 12). These primes are called "safe" because of their relationship to strong primes.

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