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).
Read more about Safe Prime: Applications, Further Properties, Records
Famous quotes containing the words safe and/or prime:
“My paternal grandmother would not light a fire on the Sabbath and piled all Sunday’s washing-up in a bucket, to be dealt with on Monday morning, because the Sabbath was a day of rest—a practice that made my paternal grandfather, the village atheist, as mad as fire. Nevertheless, he willed five quid to the minister, just to be on the safe side.”
—Angela Carter (1940–1992)
“... unless the actor is able to discourse most eloquently without opening his lips, he lacks the prime essential of a finished artist.”
—Julia Marlowe (1870–1950)