Two Elusive Prime Number Problems Solved

After centuries of flummoxing number crunchers, two mathematical puzzles about prime numbers were cracked this year.

Prime numbers — those divisible only by 1 and the number itself, like 5, 11 or 37 — are like the atoms of mathematics: All numbers are formed by multiplying these building blocks together. 

But what happens when you add a number to a prime number? When will the sum be prime? Or, conversely, when is a number a sum of primes? Mathematicians have been working to answer these fundamental questions for centuries, and on the same day in May, two mathematicians finally found tantalizing partial answers to both of them.

To imagine the answer to the first question, start by adding the number 2 to a prime. When the sum is also prime, the pair is called a “twin prime,” like 5 and 7. As numbers get bigger, primes become more rare; you might then expect the spacing between them to grow consistently larger, too, so that very large twin primes would never occur.

Yet the famous but unproven “twin prime conjecture” states there are an infinite number of primes that differ by 2 — no matter how high you count, you will never run out of twin primes. A related, more general conjecture suggests there are also infinitely many pairs of primes that differ by 4, or 6, or any even number at all. 

But conjecture is all it was until May 13, when a nearly unknown mathematician, Yitang Zhang of the University of New Hampshire, made a serious dent in the twin primes conjecture. During a talk at Harvard, he presented a proof of the related, general conjecture that as prime numbers increase toward infinity, the spaces between them — counterintuitively — do not always do the same: No matter how big prime numbers get, you’ll always find pairs of them that differ by, at the very most, 70 million. 

Admittedly, 70 million is a lot bigger than 2, so the twin primes conjecture remains unsolved. But Zhang established for the first time a necessary (and supremely difficult) first step — that the spread between successive primes does not increase toward infinity.

On the same day Zhang emerged from obscurity to reveal his stunning proof, Harald Helfgott of the Ecole Normale Superieure in Paris cracked another famously elusive problem involving prime numbers — a variation on the Goldbach conjecture, which claims that every even number greater than 2 is the sum of two primes. (For example: 16 = 5 + 11.) 

Instead, Helfgott posted a proof of the “odd Goldbach conjecture,” which states that every odd number above 5 is the sum of three primes. (19 = 3 + 5 + 11.) It’s a big step in the right direction because the full Goldbach conjecture implies the odd version: Just take your odd number (say, 19), subtract the prime number 3 (now you have 16), and apply the Goldbach conjecture to the resulting even number. (16 = 5 + 11.)

While Helfgott’s proof does not solve the full conjecture, which is considered much harder, it shines a light on the intricate dance prime numbers engage in. Now the full conjecture, along with Zhang’s almost-but-not-quite-proven twin primes conjecture, remain a tantalizing plum for future mathematicians

Voyager 1 Goes Interstellar

More than three decades after it left our planet, Voyager 1 entered a realm where no Earthborn spacecraft has gone before.

It took more than 35 years and a journey over 15 billion miles, but: “Voyager 1 is the first human-made object to make it into interstellar space — we’re actually out there,” says Don Gurnett, lead author of a September Science paper announcingthe feat. 

The probe first gained fame in the 1970s and ’80s with visits to the solar system’s outer planets; it’s been racing toward this next milestone ever since. In recent years, various scientists had prematurely trumpeted Voyager 1’s crossing into interstellar space, the area dominated by gas ejected from other stars. This time, however, NASA’s scientists are sure, thanks to three key pieces of evidence — two of which were published earlier in 2013. 

First, astronomers announced that Voyager 1 had recorded a steep drop in the “solar wind,” a stream of charged particles emanating from the sun. At the same time, the spacecraft also detected a corresponding uptick in galactic cosmic rays, ultrafast particles that come from outside the solar system. 

This waning of the solar wind amid growing gusts from interstellar space suggested Voyager 1 had crossed the edge of the heliosphere, the bubble of charged particles blown by our sun that surrounds the solar system. At the bubble’s edge, the expansion of the sun’s hot, ionized gas, or plasma, is halted by the pressure of cooler, denser plasma in the space between the stars.

But that wasn’t enough to prove that Voyager 1 had sailed through the heliosphere; knowing for sure required determining the density of plasma bathing the spacecraft. Alas, Voyager 1’s plasma sensor failed back in 1980, near Saturn. Fortunately, it still has a working plasma wave instrument, which measures the frequency of plasma vibrations (as opposed to the density). All the instrument needed was something to set the surrounding sea of plasma in motion.

A lucky explosion on the sun fit the bill: When this blast of charged, magnetic particles reached Voyager 1 in April, the instrument detected these vibrations and revealed the plasma to be more than 40 times denser than previously measured in the heliosphere. Combined with previous data, this was consistent with an escape into interstellar space at the same time as the measured drop in the solar wind.

“It all really fits,” says Gurnett. “That’s why we’re so confident this is the answer.”

Voyager 1 has enough power in its nuclear generator to send dispatches until the mid-2020s. Beyond that, its momentum will carry this most distant and devoted scout silently toward the stars, a testament to humanity’s will to explore.