Mathematicians have taken a step forward in understanding patterns within the primes, numbers divisible only by 1 and themselves. According to the new work, the population of prime numbers contains an infinite collection of arithmetic progressions—number sequences in which each term differs from the preceding one by the same fixed amount.
For example, in the sequence 3, 5, 7, each prime number is 2 more than the preceding one. Another example of such a sequence is 5, 11, 17, 23, 29, in which successive primes differ by 6.
For centuries, mathematicians have wondered how many arithmetic progressions such as these exist among the set of prime numbers and how long the progressions can get. In 1939, the Dutch mathematician Johannes van der Corput proved that there are infinitely many progressions with three terms. Whether longer progressions are infinitely plentiful or limited in number and size had remained a matter of conjecture.
The longest known progressions have just 22 terms and lie in remote stretches of the number line. For instance, one 22-term progression starts at 11,410,337,850,553 and the difference between successive terms is 4,609,098,694,200.
Now, a pair of mathematicians offers a proof that in one fell swoop demonstrates that there are infinitely many prime progressions of every finite length. Ben Green of the University of British Columbia in Vancouver and Terence Tao of the University of California, Los Angeles report their findings in a preprint that they posted on the Internet on April 8.
It may be months before mathematicians have finished checking the proof. Nevertheless, Green and Tao’s report has sparked excitement in the math community.
Proving anything about progressions with more than three terms had seemed beyond reach, says Andrew Granville, a mathematician at the University of Montreal. “If they’ve succeeded in breaking that barrier, it’s an extraordinary achievement.”
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In their proof, Green and Tao considered how the primes relate to a larger set of numbers that the pair calls almost-primes—numbers that are a product of at most 10 primes. Although prime numbers are scarce among the whole numbers, they are more plentiful in the narrower setting of the almost-primes.
Green and Tao showed that the sequence of almost-primes is pseudorandom; roughly speaking, the almost-primes are “nicely spread out all over the place,” Green says. The mathematicians then deduced that the prime numbers are arranged within the spread of almost-primes with enough regularity to ensure that the overall sequence of primes does indeed contain arithmetic progressions of every length.
Unfortunately, the new insight about prime numbers won’t give mathematicians much of a handle on where to look for long progressions. All it does is guarantee that for any length k, there’s a prime progression of that length that starts at a number smaller than 2 raised to the power 2 raised to the power 2 raised to the power 2, repeated k times. But for even small values of k, that upper limit quickly becomes astronomical.
“The bounds our argument gives are ridiculously bad,” Green acknowledges. They are essentially useless, even for mathematicians with access to the world’s most powerful computers, he says.
Green and Tao are now trying to pin down the location of prime arithmetic progressions more precisely. “It’s going to force us to understand the primes better,” Green says.