Full reptend prime

Base 10

Base 10 may be assumed if no base is specified, in which case the expansion of the number is called a repeating decimal. In base 10, if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., 9 appears in the reptend the same number of times as each other digit. (For such primes in base 10, see .) In fact, in base b, if a full reptend prime ends in the digit 1, then each digit 0, 1, ..., b − 1 appears in the repetend the same number of times as each other digit, but no such prime exists when b = 12, since every full reptend prime in base 12 ends in the digit 5 or 7 in the same base. Generally, no such prime exists when b is congruent to 0 or 1 modulo 4.

The values of p for which this formula produces cyclic numbers in decimal are: : 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983, 1019, 1021, 1033, 1051...

This sequence is the set of primes p such that 10 is a primitive root modulo p. Artin's conjecture on primitive roots is that this sequence contains 37.395...% of the primes.

Binary full reptend primes

In base 2, the full reptend primes are: (less than 1000) :3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797, 821, 827, 829, 853, 859, 877, 883, 907, 941, 947, ...

For these primes, 2 is a primitive root modulo p, so 2n modulo p can be any natural number between 1 and p − 1.

: a(i) = 2^i \bmod p \bmod 2. These sequences of period p − 1 have an autocorrelation function that has a negative peak of −1 for shift of (p-1)/2. The randomness of these sequences has been examined by diehard tests.

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 1/p (when 2 is a primitive root of p) is given by Kak.

References

• Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.
• Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers"; in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240–246.

References

1. Dickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co.
2. Bellamy, J. "Randomness of D sequences via diehard testing". 2013. {{arXiv. 1312.3618.
3. Kak, Subhash, Chatterjee, A. "On decimal sequences". IEEE Transactions on Information Theory, vol. IT-27, pp. 647–652, September 1981.
4. Kak, Subhash, "Encryption and error-correction using d-sequences". IEEE Trans. On Computers, vol. C-34, pp. 803–809, 1985.