# Prime gap record data fields

## Detailed characterisation of the fields representing prime gap record data

The text of this characterisation is taken from Dr. Thomas R. Nicely‘s original description presented in “First Occurrence Prime Gaps”. Dr. Nicely’s references to spatial positioning and sequence have been replaced by ad hoc terminological labels in order to distinguish the data fields.

Tom Nicely’s original rubric, copied verbatim for correctness.

The convention is followed here that the size (or measure) $G$ of a prime gap is the difference of its two bounding primes; consequently a gap $G$ contains $\left(G - 1\right)$ consecutive composite integers. Some authorities (e.g., J. W. L. Glaisher, Daniel Shanks) have taken $\left(G - 1\right)$ itself as the size of the gap, while others (e.g., Richard P. Brent) have specified the size of a gap by the parameter $r = G/2$. Furthermore, some authorities (e.g., Hans Riesel, Paul Leyland) have specified a gap by the terminating prime $p$2, rather than the initiating prime $p$1 as employed herein.

Note that the term “first occurrence prime gap” (of measure $G$) refers to that interval $p <= x <= p+G$ for which (1) $p$ and $p+G$ are primes, (2) $p+t$ is composite for each integer $t=1,...,G-1$, and (3) no smaller positive prime $p$ possesses these properties. The entity might be more accurately described as the earliest or smallest occurrence of the gap $G$, but the terminology “first occurrence prime gap” is well established in the literature. It does not imply historical precedence; a gap of equal measure, bounded by larger primes, may have been previously known.

The term “first known occurrence prime gap” refers to a gap which satisifies conditions (1) and (2) above, but for which condition (3) has not been verified; there may exist a gap (at this time undiscovered, or at least unknown to me) of the same size at some unknown location between the current upper bound of exhaustive computation and the listed gap. For prime gap measures whose first occurrence is not yet known, first known occurrences serve as upper bounds; the current upper limit of exhaustive scans serves as the lower bound.

1. gapsize

This field contains the size or measure of the gap (difference of the bounding primes).

2. ismax

The gap is a MAXIMAL gap, strictly exceeding in measure all the prime gaps preceding it (those between consecutive prime numbers smaller in magnitude). In this case, it will in addition always be a definite first occurrence and certified.

3. primecat

This field indicates the type of the gap. All of the gaps in this list are “conventional” (common, classic, standard, regular, ordinary, normal) prime gaps, indicated by the letter “C”; in other words, consecutive prime numbers differing by the measure of the gap, as defined by conditions (1) and (2) alone from the above definition of first occurrence prime gaps. This is the default; if the term “prime gap” is used without further qualification or elaboration, it refers to a conventional prime gap. Additional lists are planned, enumerating other types of prime gaps.

4. isfirst

This field indicates the first occurrence status of the prime gap.

• The character “F” signifies that the gap has definitely been established (by an exhaustive scan to or beyond that point) as a first occurrence prime gap.
• The character “N” signifies that the gap is definitely not a first occurrence (a prior occurrence is known).
• The character “?” signifies that the gap is a first known occurrence (no such gap with smaller bounding primes has been found), but that it is not presently known if it is the first occurrence (i.e., whether or not a gap of equal measure with smaller bounding primes exists).

All gaps presently in this list are first known occurrences not known (or expected) to be first occurrences; presumably, as the list evolves, entries will occasionally be replaced by newly discovered smaller instances of gaps.

5. primecert

This field indicates whether the bounding integers of the gap are certified primes (“C”) or probabilistic primes (“P”). The bounding integers of certified gaps (also titled confirmed, conclusive, deterministic, definite, definitive, or proven) have been conclusively proven prime, using trial prime divisors to the square root of the prime, or a test such as APRCL2 (Adleman-Pomerance-Rumely-Cohen-Lenstra-Lenstra).

The gap is probabilistic (also titled “Monte Carlo”) if the bounding integers have only been shown statistically prime (with a probability extremely close to one), using, for example, Miller’s test with multiple bases. For extremely large integers (hundreds or thousands of digits), probabilistic tests are orders of magnitude faster than deterministic tests, but nonetheless become time consuming in the thousands of digits.

I attempt to personally certify the smaller gaps (to perhaps 100 digits), and to verify probabilistically larger gaps (to perhaps 500 digits). For gaps with even larger initiating primes, I must rely on the discoverer’s report and the vigilance of third parties.
Declaration by originator and maintainer Dr. Thomas R. Nicely

In all cases, the interior integers of the gaps have been certified deterministically to be composite, using, for example, trial divisors, Fermat’s test, or Miller’s test.

6. discoverer

This field contains an abbreviation of the listed discoverers. A key of abbreviations to full names (and credit acknowledgements) is maintained separately.

7. year

This field reflects the most accurate value known for the actual date of discovery; if this is not known, the date of publication or the date of the preprint is shown; if this is not known, an estimate is given.

8. merit

This field states a so-called figure of merit for the gap. This indicates how much larger the gap is than the average gap (approximately ln(x), as a consequence of the Prime Number Theorem) between primes near that point; the greater the merit, the more unusual the gap. The merit is computed as $G/ln\left(p$1); variations in use (and at one time employed in these tables) include $G/ln\left(p$2) and $G/ln\left(\left(p$1 + p2)/2), where $p$1 and $p$2 are the initiating and terminating primes of the gap. For all but the first few gaps, the differences among these formulas are trivial; indeed, if the results are rounded to two decimal places (as herein), I (Dr. Thomas R. Nicely) have found no discrepancies in the resulting values for any gap exceeding $112$.

9. primedigits

This field indicates the number of decimal digits in the initiating prime.

10. primestart

This field shows the initiating prime (smaller bounding integer) of the gap.

Initiating primes longer than 200 decimal digits are subject to abbreviation, unless they can be represented by a simple formula (such as $10^999 + 7$). This unfortunate policy is implemented due primarily to a shortage of bandwidth. As the listing grows, additional restrictions of this type may become necessary, including bounds on gap sizes, figures of merit, or the size of the initiating primes.

If the complete expansion of an abbreviated prime is desired, I recommend that you contact the discoverer.