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:
“No government power can be abused long. Mankind will not bear it.... There is a remedy in human nature against tyranny, that will keep us safe under every form of government.”
—Samuel Johnson (1709–1784)
“The prime lesson the social sciences can learn from the natural sciences is just this: that it is necessary to press on to find the positive conditions under which desired events take place, and that these can be just as scientifically investigated as can instances of negative correlation. This problem is beyond relativity.”
—Ruth Benedict (1887–1948)