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## Twin Prime Proof Proffered

### By Eric W. Weisstein

 Author's note: After this news story was written, a serious error was found in Arenstorf's proof. In particular, Lemma 8 was found to be incorrect. As a result, the paper has been retracted and the twin prime conjecture remains fully open.

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

converges to a definite number even if the sum contains an infinite number a terms, a result known as Brun's theorem. The number B, known as Brun's constant, is difficult to compute, but is known to be approximately equal to 1.902160583104. (Amusingly, it was T. Nicely's 1995 high-precision computation of Brun's constant that first revealed a serious hardware bug in Intel's Pentium microprocessor.) Since the sum of the reciprocals of all the primes diverges (which represents a strengthening of Euclid's second theorem on the infinitude of the primes that was first proved by Euler in 1737), Brun's theorem shows that the twin primes are sparsely distributed among the primes.

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.

References

Arenstorf, R. F. "There Are Infinitely Many Prime Twins." Preprint. 26 May 2004. http://arXiv.org/abs/math.NT/0405509

Brun, V. "La serie 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ..., les dénominateurs sont nombres premiers jumeaux est convergente où finie." Bull. Sci. Math. 43, 124-128, 1919.

Guy, R. K. "Gaps between Primes. Twin Primes." §A8 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 19-23, 1994.

Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 1-70, 1923.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 5, 1979.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 30 and 219, 1993.

Tenenbaum, G. "Re: Arenstorf's paper on the Twin Prime Conjecture." NMBRTHRY@listserv.nodak.edu} mailing list. 8 Jun 2004. http://listserv.nodak.edu/scripts/wa.exe?A2=ind0406&L=nmbrthry&F=&S=&P=1119