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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often as you rebuild me like blocks,
and promise to come visit
when I’m finally adjusted on safe land,
and am livable, joist to joist
with storm windows and screens ...”
—Anne Sexton (1928–1974)
“What was lost in the European cataclysm was not only the Jewish past—the whole life of a civilization—but also a major share of the Jewish future.... [ellipsis in source] It was not only the intellect of a people in its prime that was excised, but the treasure of a people in its potential.”
—Cynthia Ozick (b. 1928)