PRintervallum

PRintervallum is a term used in recreational and analytic number theory to denote a class of intervals on the integers that are defined by containing a prescribed number of prime numbers. The abbreviation PR is taken from prime, and intervallum is Latin for interval. The concept is employed in discussions of prime distribution and prime gaps, and it is often used in puzzle contexts as well as in heuristic investigations.

Definition and notation

For an integer r ≥ 1, a PRintervallum of order r is an open interval (a, b) with a and b integers such that the number of primes p that satisfy a < p < b equals r. The length of the interval is b − a. A related notion is the minimal PRintervallum length L(r), defined as the smallest possible length of an interval containing exactly r primes.

Examples

The interval (1, 4) contains the primes 2 and 3, so it is a PRintervallum of order 2. Another example is (0, 5), which also contains the primes 2 and 3, giving order 2 but with a larger length. These simple instances illustrate how the same order can arise from intervals of different lengths.

Properties and considerations

The function L(r) is not fixed for all r and is studied both for its intrinsic interest and for its connection to prime gaps. Heuristic expectations based on the prime number theorem suggest that minimal lengths grow roughly on the order of r log r as r increases, reflecting the average size of prime gaps.

History and usage

The term PRintervallum emerged in online recreational and mathematical literature in the early 2010s and has since been used to frame problems about interval containment of primes, to design prime-gap puzzles, and to illustrate concepts in teaching about prime distribution. See also prime gaps, prime counting functions, and related interval problems.