# jumping champion

Here are the first few primes:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 43.The differences between these primes are:

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, and 2.For these primes 2 occurs most often as a gap between primes, so we call it a jumping champion.

An integer *n* is **jumping champion** if
*n* is the most frequently occurring difference
between consecutive primes < *x* for some
*x*. The example above shows 2 is a jumping champion
for *x*=43 (in fact for any *x* with
7 ≤ *x* < 131). John Horton Conway coined
the term jumping champion in 1993. Harry Nelson may have
first suggested the concept (without the term) in 1977-8.
(Jumping champions have also called **high
jumpers**.)

Sometimes there are more than one jumping champion for
a given *x* (because a couple gaps show up an equal
number of times). For example, when *x*=5 we have
the two champions 1 and 2. When *x*=179 we have the
three champions 2, 4 and 6.

The only champions we see in this table are 1, 2, 4, and 6. It is conjectured that if we extend this table far enough we will get other champions including first 30, then 210, and then 2310. In fact it is conjectured that the only jumping champions are 1, 4 and the primorials 2, 6, 30, 210, 2310… To prove this conjecture will probably first require the proof of the

integers xchampions

forxintegers xchampions

forx3 - 4 1 379 - 388 2, 6 5 - 6 1, 2 389 - 420 6 7 - 100 2 421 - 432 2, 6 101 - 102 2, 4 433 - 438 2 103 - 106 2 439 - 448 2, 6 107 - 108 2, 4 449 - 462 6 109 - 112 2 463 - 466 2, 6 113 - 130 2, 4 467 - 490 2, 4, 6 131 - 138 4 491 - 546 4 139 - 150 2, 4 547 - 562 4, 6 151 - 166 2 563 - 940 6 167 - 178 2, 4 941 - 946 4, 6 179 - 180 2, 4, 6 947 - (at least 10 ^{12})6 181 - 378 2

*k*-tuple conjecture, so it could be quite awhile.

We also see in the table that the jumping champions
for a given *x* seem to be growing as *x*
does. (In the table: 1 last occurs at 6, 2 at 490,
and 4 at 946). It is conjecture that the jumping
champions tend to infinity. Odlyzko, Rubinstein and Wolf used a heuristic argument to estimate that 6 stays the sole jumping champion from 947 to about 1.7427^{.}10^{35}, where 30 becomes
the champion. Moreover, they estimate that 30 is
replaced as a jumping champion by 210 around
*x*=10^{425}. Erdös and Straus
have shown that this second conjecture follows
from a form of the *k*-tuple conjecture.

**See Also:** GilbreathsConjecture

**Related pages** (outside of this work)

- The gaps between primes (an article on prime gaps)
- When does 30 take over from 6?

**References:**

- Brent74
R. P. Brent, "The distribution of small gaps between succesive primes,"Math. Comp.,28(1974) 315--324.MR 48:8356- Brent75
R. P. Brent, "Irregularities in the distribution of primes and twin primes,"Math. Comp.,29(1975) 43--56.MR 51:5522- ES80
P. ErdösandE. G. Straus, "Remarks on the differences between consecutive primes,"Elem. Math.,35(1980) 115--118.MR 84a:10052- Guy94
R. K. Guy,Unsolved problems in number theory, Springer-Verlag, 1994. New York, NY, ISBN 0-387-94289-0.MR 96e:11002[An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]- Nelson78
H. Nelson, "Problem 654,"J. Recreational Math.,11(1978-79) 231.