In mathematics, a covering set for a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s refers to a set of

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

s such that ''every'' term in the sequence is

divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

by ''at least one'' member of the set. The term "covering set" is used only in conjunction with sequences possessing

exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...

Sierpinski and Riesel numbers

The use of the term "covering set" is related to Sierpinski and

Riesel number In mathematics, a Riesel number is an odd natural number ''k'' for which k\times2^n-1 is composite for all natural numbers ''n'' . In other words, when ''k'' is a Riesel number, all members of the following set are composite: :\left\. If the f ...

s. These are odd

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s for which the formula (Sierpinski number) or (Riesel number) produces no prime numbers. Since 1960 it has been known that there exists an

infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group) Infinite ( ko, 인피니트; stylized as INFINITE) is a South Ko ...

number of both Sierpinski and Riesel numbers (as solutions to families of congruences based upon the set but, because there are an infinitude of numbers of the form or for any , one can only prove to be a Sierpinski or Riesel number through showing that ''every'' term in the sequence or is divisible by one of the prime numbers of a covering set. These covering sets form from prime numbers that in base 2 have short periods. To achieve a complete covering set,

Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and t ...

showed that a sequence can repeat no more frequently than every 24 numbers. A repeat every 24 numbers give the covering set , while a repeat every 36 terms can give several covering sets: ; ; and . Riesel numbers have the same covering sets as Sierpinski numbers.

Other covering sets

Covering sets (thus Sierpinski numbers and Riesel numbers) also exists for bases other than 2. Covering sets are also used to prove the existence of composite generalized

Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

s with first two terms coprime ( primefree sequence), such as the sequence starting with 20615674205555510 and 3794765361567513. The concept of a covering set can easily be generalised to other sequences which turn out to be much simpler. In the following examples + is used as it is in

regular expression A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a search pattern in text. Usually such patterns are used by string-searching algorithms for "find" ...

s to mean 1 or more. For example, 91⁺3 means the set An example are the following eight sequences: * (29·10ⁿ − 191) / 9 or 32⁺01 * (37·10ⁿ + 359) / 9 or 41⁺51 * (46·10ⁿ + 629) / 9 or 51⁺81 * (59·10ⁿ − 293) / 9 or 65⁺23 * (82·10ⁿ + 17) / 9 or 91⁺3 * (85·10ⁿ + 41) / 9 or 94⁺9 * (86·10ⁿ + 31) / 9 or 95⁺9 * (89·10ⁿ + 593) / 9 or 98⁺23 In each case, every term is divisible by one of the primes . These primes can be said to form a covering set exactly analogous to Sierpinski and Riesel numbers.List of near-repdigit-related (probable) prime numbers, sorted by difficulty
/ref> The covering set is found for several similar sequences, including: * (38·10ⁿ − 137) / 9 or 42⁺07 * (4·10ⁿ − 337) / 9 or 4⁺07 * (73·10ⁿ + 359) / 9 or 81⁺51 * 9175·10ⁿ + 1 or 91750⁺1 * 10176·10ⁿ − 1 or 101759⁺ * (334·10ⁿ − 1)/9 or 371⁺ * (12211·10ⁿ − 1)/3 or 40703⁺ * (8026·10ⁿ − 7)/9 or 8917⁺ Also for bases other than 10: * 521·12ⁿ + 1 or 3750⁺1 in

duodecimal The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...

* (1288·12ⁿ − 1)/11 or 991⁺ in

* (4517·12ⁿ − 7)/11 or 2X27⁺ in

* 376·12ⁿ − 1 or 273E⁺ in

The covering set of them is An even simpler case can be found in the sequence: * (76·10ⁿ − 67) / 99 (''n must be odd'') or (76)⁺7 equence: 7, 767, 76767, 7676767, 767676767 etc. Here, it can be shown that if: * w is of form (n = ): (76)⁺7 is divisible by 7 * w is of form (n = ): (76)⁺7 is divisible by 13 * w is of form (n = ): (76)⁺7 is divisible by 3 Thus we have a covering set with only three primes .Smoothly Undulating Palindromic Primes
/ref> This is only possible because the sequence gives integer terms ''only for odd n''. A covering set also occurs in the sequence: * (343·10ⁿ − 1) / 9 or 381⁺. Here, it can be shown that: * If n = , then is divisible by 3. * If n = , then is divisible by 37. * If n = , then is algebraic factored as . Since can be written as 23⁺, for the sequence 381⁺, we have a covering set of – a covering set with ''infinitely many'' terms. The status for (343×10^n−1)/9 is like that for 3511808×63^n+1: * If n = , then is divisible by 37. * If n = , then is divisible by 109. * If n = , then is algebraic factored as Thus we have a covering of or – a covering set with ''infinitely many'' terms. A more simple example is 4×9^n−1, it is equal to (2×3^n−1) × (2×3^n+1), thus its covering sets are and , more generally, if k and b are both r-th powers for an odd r>1, then k×b^n+1 cannot be prime, and if k and b are both r-th powers for an r>1 then k×b^n−1 cannot be prime. Another example is 1369×30^n−1, its covering is

Sierpinski and Riesel numbers

Other covering sets

See also

Notes

References

External links