Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the
subsequence of
prime numbers
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 ways ...
that occupy prime-numbered positions within the sequence of all prime numbers.
The subsequence begins
:3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ... .
That is, if ''p''(''n'') denotes the ''n''th prime number, the numbers in this sequence are those of the form ''p''(''p''(''n'')).
used a computer-aided proof (based on calculations involving the
subset sum problem) to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers. Their proof relies on a result resembling
Bertrand's postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3, there always exists at least one prime number p with
:n < p < 2n - 2.
A less restrictive formulation is: for every , there is always ...
, stating that (after the larger gap between super-primes 5 and 11) each super-prime number is less than twice its predecessor in the sequence.
show that there are
:
super-primes up to ''x''.
This can be used to show that the set of all super-primes is
small
Small may refer to:
Science and technology
* SMALL, an ALGOL-like programming language
* Small (anatomy), the lumbar region of the back
* ''Small'' (journal), a nano-science publication
* <small>, an HTML element that defines smaller text ...
.
One can also define "higher-order" primeness much the same way and obtain analogous sequences of primes .
A variation on this theme is the sequence of prime numbers with
palindromic prime
In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such ...
indices, beginning with
:3, 5, 11, 17, 31, 547, 739, 877, 1087, 1153, 2081, 2381, ... .
References
*
*.
*.
*.
External links
A Russian programming contest problem related to the work of Dressler and Parker
Classes of prime numbers
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