Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers. In other words, if prime numbers are matched with ordinal numbers, starting with prime number 2 matched with ordinal number 1, then the primes matched with prime ordinal numbers are the super-primes. The subsequence begins That is, if p(n) denotes the nth 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, 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. 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 indices, beginning with