Jumping Champions
Ian Stewart leaps over the gaps between prime numbers

M

athematics is full of surprises. Who would haveimagined, for instance, thatsomething as straightforward as the natural numbers (1, 2, 3, 4,...)
could give birth to anything so bafflingas the prime numbers (2, 3, 5, 7, 11, . . . )?
The pattern of natural numbers is obvious: no matter which one you pick, it’seasy to determine what the next one is.
You can’t say that for the primes. And yet
it’s a simple step from natural numbers
to primes. Just take the natural numbersthat have no proper divisors.
We know a lot about the primes, including some powerful formulas that provide
good approximations when exact answers
aren’t forthcoming. The Prime Number
Theorem states that the number of primes
less than x is approximately x / log x,
where log denotes the natural logarithm.
So, for instance, we know that there areroughly 4.3 × 1097 primes with less than
100 digits— but the exact number is a total mystery.
Recently Andrew Odlyzko of AT&TLabs, Michael Rubinstein of the University of Texas and Marek Wolf of the University of Wroclaw in Poland turned their attention to the gaps between successive
primes. In an article in Experimental Mathematics (Vol. 8, No. 2, 1999), they addressed the following problem: What
number is the most common gap between successive primes less than x? This
question was posed in the late 1970s by
Harry Nelson of Lawrence Livermore National Laboratory. Later on, John Horton
Conway of Princeton University coined
the phrase “jumping champions” to describe these numbers.
The primes up to 50 are 2, 3, 5, 7, 11,13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.
The sequence of gaps— the differences between each prime and the next— is 1, 2,2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2 and 4. The
number 1 appears only once because allprimes except for 2 are odd. The rest ofthe gaps are even numbers. In this sequence, 2 occurs six times, 4 occurs fivetimes, and 6 occurs twice. So when x = 50,
the most common gap is 2, and this num-106

ber is therefore the jumping champion.
Sometimes several gaps are equallycommon. For instance, when x = 5 thegaps are 1 and 2, and each occurs once.
For higher x, the sole jumping championis 2 until we reach x = 101, when 2 and 4
are tied for the honor [see illustration below]. After that, the jumping champion iseither 2 or 4, or both, until x = 179, when2, 4 and 6 are involved in a three-way tie.
At that point the challenge from 4 and 6
dies away, and 2 reigns supreme until x =379, where 2 is tied with 6. Above x = 389
the jumping champion is mostly 6, occasionally tied with 2 or 4, or both. But
when x ranges from 491 to 541, thejumping champion reverts to 4. From x =947 onward the sole jumping champion
is 6, and a computer search shows that
this continues up to at least x = 1012.
It seems reasonable to conclude thatapart from some initial competition from1, 2 and 4, the only long-term jumping
champion is 6. But even a pattern that

MOST COMMON GAPS between successive prime numbers are 2, 4 and 6 for the sequences running up to 1,000. But no one knows the jumping champions for very long sequences of primes.

Scientific American December 2000

Mathematical Recreations

ALL ILLUSTRATIONS BY BRYAN CHRISTIE

Mathematical Recreations

NUMBER THEORY_PRIMES

persists up to numbers in the trillions
may well change as the numbers get stillbigger. And that’s where the surprisecomes in. Odlyzko and his colleagues provide a persuasive argument that somewhere near x = 1.7427 × 10 35 the jumping champion changes from 6 to 30.
They also suggest that it changes again,
to 210, near x = 10 425.
Except for 4, the conjectured jumping
champions fit into an elegant pattern,which becomes obvious if we factor
them into primes:
2=2
6=2×3
30 = 2 × 3 × 5
210 = 2 × 3 × 5 × 7
Each number is obtained by multiplying successive primes together. Thesenumbers are called primorials— like factorials, but using primes— and the next
few are 2,310, 30,030 and 510,510. In
their article, Odlyzko and his co-authors
propose the Jumping Champions Conjecture: the jumping champions are precisely the primorials, together with 4.
Here’s a brief explanation of their analysis. Anyone who looks at the sequence ofprimes notices that every so often two
consecutive odd numbers are prime: 5and 7, 11 and 13, 17 and 19. The Twin
Prime Conjecture states that there are infinitely many such pairs. It is based on
the idea that primes occur “at random”
among the odd numbers, with a probability based on the Prime Number Theorem. Of course, this sounds like nonsense— a number is either prime or not;there isn’t any probability involved— but
it is reasonable nonsense for this kind ofproblem. According to a calculation ofwww.sciam.com

probabilities, there is no chance that thelist of twin primes is finite.
What about three consecutive oddnumbers being prime? There is only one
example: 3, 5, 7. Given any three consecutive odd numbers, one must be a multiple of 3, and that number is therefore notprime unless it happens to equal 3. Yet
the patterns p, p + 2, p + 6 and p, p + 4, p +6 cannot be ruled out by such arguments, and they seem to be quite common. For example, the first type of pattern occurs for 11, 13, 17 and again for41, 43, 47. The second type of pattern occurs for 7, 11,13 and again for 37, 41, 43.
About 80 years ago English mathematicians Godfrey Harold Hardy and John
Edensor Littlewood analyzed patterns of
this kind involving larger numbers ofprimes. Using the same kind of probabilistic calculation that I described for thetwin primes, they deduced a precise formula for the number of sequences ofprimes with a given pattern of gaps. The
formula is complicated, so I won’t show it
here; see the article in Experimental Mathematics and the references therein.
From the Hardy-Littlewood work, Od-READER_FEEDBACKIn a recent column on logical paradoxes [“Paradox Lost,” June], I argued thatthe Surprise Test paradox rests on an inconsistent interpretation of the word
“surprise” and isn’t really a paradox at all.
Several readers drew my attention to an article entitled “Surprise Maximization”
in American Mathematical Monthly (Vol. 107, No. 6, June–July 2000). The authors
define a measure of surprise and ask what strategy the teacher should follow tomaximize the students’ surprise. They conclude that in choosing the day of theweek for the test, the teacher should use a probability distribution that remains
roughly constant through the early part of the week but increases rapidly in thelast few days. Under this strategy, Friday would be chosen most often.
— I.S.
Scientific American December 2000

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Mathematical Recreations

LOGARITHMIC PLOT shows how the number of gaps between successive primes less than xvaries with the size of the gap (2d ). The plot suggests that 210 may be a jumping champion.

lyzko and his colleagues extracted a formula for N (x, d ), which is the number ofgaps between consecutive primes when
the gap is of size 2d and the primes are lessthan x. (We use 2d rather than d because
the size of the gap has to be even.) The formula is expected to be valid only when 2dis large and x is much larger. The illustration at the left shows how log N (x, d )varies with 2d for 13 values of x ranging
from 2 20 to 2 44 (in this graph, log denotes a base 10 logarithm). Each plot line
is approximately straight but has lots ofbumps. A particularly prominent bumpoccurs at 2d = 210, the conjectured jumping champion for very large x. (It wouldlook even more prominent if the logarithmic graphing didn’t flatten it out.)
This kind of information suggests that
the N (x, d ) formula is not too wide off
the mark.
Now, if 2d is going to be a jumping
champion, the value of N (x, d ) has to bepretty big. The best way to achieve this is
if 2d has many distinct prime factors.
Also, 2d should be as small as possible
subject to this condition, so the most plausible choices for 2d are the primorials. The
known jumping champion 4 is presumably an exception. It occurs at a size where
the N (x, d) formula isn’t a good approximation anyway. The formula also lets us
work out roughly when a given primorial
takes over from the previous one as thenew jumping champion.
What’s left for recreational mathematicians to do? Prove the Jumping Champions Conjecture, of course— or disprove it.
If you can’t do either, try searching forother interesting properties of the gaps
between primes. For example, what is theleast common gap (that actually occurs)
between consecutive primes less than x?
And which gap occurs closest to the average number of times? As far as I know,these questions are wide open, even for
relatively small values of x.