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.

integers x champions
for x integers x champions
for x

3 - 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 10¹²) 6

181 - 378 2

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 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^.10³⁵, where 30 becomes the champion. Moreover, they estimate that 30 is replaced as a jumping champion by 210 around x=10⁴²⁵. 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. Erdös and E. 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.

integers x	champions for x	integers x	champions for x
3 - 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 10¹²)	6
181 - 378	2