between 1 and p − 1. These sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of . The randomness of these sequences has been examined by diehard tests. All of them are of form 8k + 3 or 8k + 5, because if p = 8k + 1 or 8k + 7, then 2 is a quadratic residue modulo p, so p divides , and the period of in base 2 must divide and cannot be p − 1, so they are not full reptend primes in base 2. Further, all safe primes congruent to 3 modulo 8 are full reptend primes in base 2. For example, 3, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, 1523, 1619, 1907, etc. (less than 2000). Binary full reptend prime sequences (also called maximum-length decimal sequences) have found cryptographic and error-correction coding applications. In these applications, repeating decimals to base 2 are generally used which gives rise to binary sequences. The maximum length binary sequence for (when 2 is a primitive root of p) is given by: The following is a list about the periods (in binary) to the primes congruent to 1 or 7 (mod 8): (less than 1000) None of them are binary full reptend primes. The binary period of n-th prime are 2, 4, 3, 10, 12, 8, 18, 11, 28, 5, 36, 20, 14, 23, 52, 58, 60, 66, 35, 9, 39, 82, 11, 48, 100, 51, 106, 36, 28, 7, 130, 68, 138, 148, 15, 52, 162, 83, 172, 178, 180, 95, 96, 196, 99, 210, 37, 226, 76, 29, 119, 24, 50, 16, 131, 268, 135, 92, 70, 94, 292, 102, 155, 156, 316, 30, 21, 346, 348, 88, 179, 183, 372, 378, 191, 388, 44, ... (this sequence starts at n = 2, or the prime = 3) The binary period level of n-th prime are 1, 1, 2, 1, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 1, 1, 1, 2, 8, 2, 1, 8, 2, 1, 2, 1, 3, 4, 18, 1, 2, 1, 1, 10, 3, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 6, 1, 3, 8, 2, 10, 5, 16, 2, 1, 2, 3, 4, 3, 1, 3, 2, 2, 1, 11, 16, 1, 1, 4, 2, 2, 1,