MathWorld Headline News

Twin Prime Proof Proffered

By Eric W. Weisstein

June 9, 2004--A recent preprint by Vanderbilt University mathematician R. F. Arenstorf appears to come close to settling the long-standing question of the infinitude of twin primes. Twin primes are pairs of prime numbers such that the larger member of the pair is exactly 2 greater than the smaller, i.e., primes p and q such that q - p = 2. Explicitly, the first few twin primes are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), and (41, 43).

The properties and the distribution of twin primes (so named by P. Stäckel, 1892-1919) are active areas of mathematical research. While the distribution of twin primes has remained elusive, mathematician V. Brun proved in 1919 that the sum of the reciprocals of the members of each twin prime pair

The twin prime conjecture states that there are an infinite number of twin primes. While Hardy and Wright (1979) note that "the evidence, when examined in detail, appears to justify the conjecture," and Shanks (1993) states even more strongly, "the evidence is overwhelming." Hardy and Wright also note that the proof or disproof of conjectures of this type "is at present beyond the resources of mathematics."

In fact, no proof of the twin primes conjecture had been constructed despite the efforts of dozens of mathematicians over almost a century. In contrast, a recent preprint has apparently succeeded in showing the existence of prime arithmetic progressions of any length k, a related and also long-outstanding problem (MathWorld headline news story, April 12, 2004).

In a May 26 preprint, however, R. F. Arenstorf published a proposed proof of the twin prime conjecture in a stronger form due to Hardy and Littlewood (1923). The proof uses methods from classical analytic number theory, including the properties of the Riemann zeta function, ideas from the proof of the prime number theorem, and a so-called Tauberian theorem by Wiener and Ikehara from 1931, the last of which leads almost immediately to Arenstorf's main result.

While Arenstorf's approach looks promising, an error in one particular step of the proof (specifically, Lemma 8 on page 35; a lemma is short theorem used in proving a larger theorem) has recently been pointed out by French mathematician G. Tenenbaum of the Institut Élie Cartan in Nancy (Tenenbaum 2004). While mathematicians remain hopeful that any holes in the proof can be corrected, Tenenbaum opines that this particular error may have serious consequences for the integrity of the overall proof. Additional analysis by other mathematicians over the coming weeks and months will establish whether, like the originally flawed proof of Fermat's last theorem, the twin prime result can also be corrected, thus finally settling this long-open problem, or if it requires additional insight and tools before it can finally be cracked.