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:
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“Sometimes it takes years to really grasp what has happened to your life. What do you do after you are world-famous and nineteen or twenty and you have sat with prime ministers, kings and queens, the Pope? What do you do after that? Do you go back home and take a job? What do you do to keep your sanity? You come back to the real world.”
—Wilma Rudolph (1940–1994)