# The Gaps Between Primes

### By Chris Caldwell

### Contents:

- Introduction and definition of g(
*n*) - lim inf g(
*n*) = 1(?) and lim sup g(*n*) = infinity - A Table and Graph of Record Gaps
- Bounds on g(
*p*) - What about g(
*p*)/log(*p*), g(*p*)/(log*p*)^{2}?

### 1. Introduction and definition of g(*n*)

It is frequently asked how large the gap between consecutive primes may
get. Before we answer this, let us first carefully define gap (there are
two different standard definitions). For every prime *p* **let g( p)
be the number of composites between p and the next prime**. So
letting

*p*be the

_{n}*n*th prime we have:

*p _{n}*

_{+1}=

*p*+ g(

_{n}*p*) + 1.

_{n}That is, **g(p_{n}) is the (size of) gap** between

*p*and

_{n}*p*

_{n}_{+1}.

By the prime number theorem we know there
are approximately *n*/log(*n*) (natural log) primes
less than *n*, so the "average gap" between primes less than *n*
is log(*n*). But how wide of range can these gaps have? We will discuss
several aspects of this question below.

### 2. lim inf g(*n*) = 1(?) and lim sup g(*n*)
= infinity

First note that g(*p*) = 1 for twin primes *p*, *p*+2. So
from the twin prime conjecture we have the conjecture (almost certainly
true) that g(*p*) = 1 infinitely often (or equivalently lim inf g(*n*)
= 1).

Second note that g(*p*) can be arbitrarily large. To see this let
*n* be *any* integer greater than one and consider the following
sequence of consecutive integers:

*n*!+2, *n*!+3, *n*!+4, *n*!+5, ..., *n*!+*n*

Notice that 2 divides the first, 3 divides the second, ..., *n* divides
the *n*-1st, showing all of these numbers are composite! So
if *p*
is the largest prime less than *n*!+2 we have g(*p*) > *n*-1. Obviously
there should be smaller numbers which produce the same gaps. For
example, there is a gap of 777 composites after the prime 42842283925351--this
is the least prime which produces a gap of 777 and it is far smaller than
778!+2 (which has 1914 digits). (Rather than use *n*!, one
can also use the smaller
*n* primorial: *n*#).

In the last paragraph we showed that lim sup g(*n*) = infinity,
but we expect much more since the "average gap" is about log(*n*). In 1931 Westzynthius [Westzynthius31]
proved that

lim sup g(*n*)/log *p _{n}* = infinity

which means that for every *ß* > 0 there are infinitely many
primes *p* with g(*p*) > *ß* log *p*. Before
we say more we should look at some numerical evidence.

### 3. A Table and Graph of Record Gaps

In the following table we list the **maximal gaps** through 381. These
are the first occurrences of gaps of *at least of this length*. For
example, there is a gap of 879 composites after the prime 277900416100927. This
is the first occurrence of a gap of this length, but still is not a
maximal gap since 905 composites follow the smaller prime 218209405436543
[Nicely99].

Gap | After | Gap | After | Gap | After | Gap | After |
---|---|---|---|---|---|---|---|

0 | 2 | 33 | 1327 | 117 | 1349533 | 247 | 191912783 |

1 | 3 | 35 | 9551 | 131 | 1357201 | 249 | 387096133 |

3 | 7 | 43 | 15683 | 147 | 2010733 | 281 | 436273009 |

5 | 23 | 51 | 19609 | 153 | 4652353 | 287 | 1294268491 |

7 | 89 | 71 | 31397 | 179 | 17051707 | 291 | 1453168141 |

13 | 113 | 85 | 155921 | 209 | 20831323 | 319 | 2300942549 |

17 | 523 | 95 | 360653 | 219 | 47326693 | 335 | 3842610773 |

19 | 887 | 111 | 370261 | 221 | 122164747 | 353 | 4302407359 |

21 | 1129 | 113 | 492113 | 233 | 189695659 | 381 | 10726904659 |

For each non-negative integer *g* let ** p(g)**
be the smallest prime that is followed by at least

*g*composites. This table tells us

*p*(148) =

*p*(149) = ... =

*p*(153) = 4652353. See [Nicely99] and [NN99] for information on searches which extended this table to maximal gaps through 1132. We give a more complete list of maximal gaps on a separate page. See also Jens Kruse Andersen's page of top twenty gaps.

For
the values in the table above we have graphed ln *p*(*g*) (the
natural log) versus *g* in the graph on the right. Perhaps you can
begin to see why Shanks conjectured in 1964 that

ln *p*(*g*) ~ sqrt(*g*),

and Weintraub estimated in 1991 that

ln *p*(*g*) ~ sqrt(1.165746*g*).

### 4. Bounds on g(*p*)

It is possible to put an upper limit on g(*p*) given

*p*. By the prime number theoremwe can show that for every real number

*e*> 0 and there is some integer

*m*

_{0}such that there is always a prime

*p*satisfying

*m* < *p* < (1 + *e*)*m* (for every
*m* > *m*_{0})

This shows that g(*p*) < *ep* for all *p* > max(*m*_{0},1+1/*e*). Or more succinctly, g(*p _{n}*) <

*ep*for

_{n}*n*>

*n*

_{0}. Here are several specific pairs

*e*,

*n*

_{0}quoted from [Ribenboim95p252-253]:

- g(
*p*) < (1/5)_{n}*p*for_{n}*n*> 9 (Nagura 1952) - g(
*p*) < (1/13)_{n}*p*for_{n}*n*> 118 (Rohrbach & Weis 1964) - g(
*p*) < (1/16597)_{n}*p*for_{n}*n*> 2010760 (Schoenfeld 1976)

In 1937 Ingham refined pioneering work of Hoheisel to show that g(*p*)
is bounded by a constant times *p*^{5/8+eps} (for every eps>0). Many folk have improved on the 5/8, the most recent record that I know of
is 0.535 due to R. Baker and G. Harman [BH96] (but surely this has been improved on by now).

### 5. What about g(*p*)/log
*p*, g(*p*)/(log *p*)^{2} ?

Again, the prime number theorem proves that the average value of g(*p*)/log
*p* is one, but what do we know of the sequence {g(*p*)/log*p*}?
Ricci [Ricci56] showed that set
of limit points of this set has positive Lebesgue measure, but so far the
only proven limit point is infinity (mentioned above) [see EE85, p22].

Various upper bounds for lim inf g(*p*)/log*p* have been found,
including 0.248 [Maier85] (of course,
both the twin prime conjecture and the prime k-tuple conjectures require
that the limit inferior be zero). In a related conjecture Cramér
[Cramér36]
conjectured that

lim sup g(*p*)/(log *p*)^{2} = 1.

Granville
altered Cramer's conjecture, suggesting that it underestimates the size
of the gaps. Granville conjectures that for any constant *c* less
than Euler's gamma:

g(*p*) __>__ 2 e^{-c} log^{2} *p*

infinitely often. Here the constant comes from analogy with Merten's theorem.

Can this be proven? Not yet, but Cramér did show that **if** the Riemann Hypothesis holds, then
we would have the far weaker result:

g(*p*) < *k* *p*^{1/2} log *p.*

There is a lot more to say concerning conjectures and theorems on both the
size of these gaps and how often a given gap occurs... Why not just spend
an evening with section 4.2 ("The *n*th Prime and Gaps") of [Ribenboim95].