Full reptend prime

Full reptend prime

In number theory, a full reptend prime or long prime in base "b" is a prime number "p" such that the formula

:frac{b^{p - 1} - 1}{p}

(where "p" does not divide "b") gives a cyclic number. Therefore the digital expansion of 1/p in base "b" repeats the digits of the corresponding cyclic number infinitely. Base 10 may be assumed if no base is specified.

The first few values of "p" for which this formula produces cyclic numbers in decimal are OEIS|id=A001913

: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 …

For example, the case "b" = 10, "p" = 7 gives the cyclic number 142857, thus, 7 is a full reptend prime. Furthermore, 1 divided by 7 written out in base 10 is 0.142857142857142857142857...

Not all values of "p" will yield a cyclic number using this formula; for example "p" = 13 gives 076923076923. These failed cases will always contain a repetition of digits (possibly several).

The known pattern to this sequence comes from algebraic number theory, specifically, 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.

The term "long prime" was used by John Conway and Richard Guy in their "Book of Numbers". Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers."

The corresponding cyclic number to prime "p" will possess "p" - 1 digits if and only if "p" is a full reptend prime.

Patterns of occurrence full reptend primes

Advanced modular arithmetic can show that any prime of the following forms:


#40"k"+1
#40"k"+3
#40"k"+9
#40"k"+13
#40"k"+27
#40"k"+31
#40"k"+37
#40"k"+39
can "never" be a full reptend prime in base-10. The first primes of these forms, with their periods, are:

However, studies show that "two-thirds" of primes of the form 40"k"+"n", where "n" ≠ {1,3,9,13,27,31,37,39} are full reptend primes. For some sequences, the preponderance of full reptend primes is much greater. For instance, 285 of the 295 primes of form 120"k"+23 below 100000 are full reptend primes, with 20903 being the first that is not full reptend.

References

*MathWorld|urlname=ArtinsConstant|title=Artin's Constant
*MathWorld|urlname=FullReptendPrime|title=Full Reptend Prime
*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.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Mersenne prime — Named after Marin Mersenne Publication year 1536[1] Author of publication Regius, H. Number of known terms 47 Conjectured number of terms Infinite …   Wikipedia

  • List of prime numbers — This is an incomplete list, which may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. By Euclid s theorem, there are an infinite number of prime numbers. Subsets of the… …   Wikipedia

  • Cuban prime — A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y. The first of these equations is: and the first few cuban primes from this equation are (sequence A002407 in OEIS): 7 …   Wikipedia

  • List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

  • 233 (number) — 233 is the natural number between 232 and 234. It is also a prime number.In mathematics233 is an irregular prime, a full reptend prime, a cousin prime, a Chen prime, and a sexy prime. It is an Eisenstein prime of the form 3n 1 with no imaginary… …   Wikipedia

  • 131 (number) — Number|number = 131 cardinal = one hundred [and] thirty one ordinal = st ordinal text = one hundred [and] thirty first numeral = factorization = prime prime = 32nd divisor = unicode = greek prefix = latin prefix = misc =131 is the natural number… …   Wikipedia

  • 167 (number) — 167 is the natural number following 166 and preceding 168.It is a prime number, a safe prime and an Eisenstein prime with no imaginary part and a real part of the form 3n 1. Since the next odd number, 169, is a square of a prime, 167 is a Chen… …   Wikipedia

  • 149 (number) — 149 is the natural number between 148 and 150. It is also a prime number.In mathematics149 is the 35th prime number, and with the next prime number, 151, is a twin prime, thus 149 is a Chen prime. 149 is a strong prime in the sense that it is… …   Wikipedia

  • 20000 (number) — Number number = 20000 range = 10000 100000 cardinal = 20000 ordinal = th ordinal text = twenty thousandth factorization = 2^5 cdot 5^4 bin = 100111000100000 oct = 47040 hex = 4E2020,000 (twenty thousand) is the number that comes after 19,999 and… …   Wikipedia

  • Transposable integer — A summary of this article appears in Repeating decimal. The digits of some specific integers permute or shift cyclically when they are multiplied by a number n. Examples are: 142857 × 3 = 428571 (shifts cyclically one place left) 142857 × 5 =… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”