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Prime Numbers

Prime Numbers Prime Numbers

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8.4 Diophantine analysis 417 9262 3 + 15312283 2 = 113 7 , 17 7 + 76271 3 = 21063928 2 , 43 8 + 96222 3 = 30042907 2 . (The latter five examples were found by F. Beukers and D. Zagier.) There is a cash prize (the Beal Prize) for a proof of the conjecture of Tijdeman and Zagier that (8.2) has no solutions at all when p, q, r ≥ 3; see [Bruin 2003] and [Mauldin 2000]. It is known [Darmon and Granville 1995] that for p, q, r fixed with 1/p+1/q+1/r ≤ 1, the equation (8.2) has at most finitely many coprime solutions x, y, z. We also know that in some cases for p, q, r the only solutions are those that appear in our small table. In particular, all of the triples with exponents {2, 3, 7}, {2, 3, 8}, {2, 3, 9}, and{2, 4, 5} are in the above list. In addition, there are many other triples of exponents for which it has been proved that there are no nontrivial solutions. These results are due to many people, including Bennett, Beukers, Bruin, Darmon, Ellenberg, Kraus, Merel, Poonen, Schaefer, Skinner, Stoll, Taylor, and Wiles. For some recent papers from which others may be tracked down, see [Bruin 2003] and [Beukers 2004]. The Fermat–Catalan conjecture is a special case of the notorious ABC conjecture of Masser. Let γ(n) denote the largest squarefree divisor of n. The ABC conjecture asserts that for each fixed ɛ>0 there are at most finitely many coprime positive integer triples a, b, c with a + b = c, γ(abc)

418 Chapter 8 THE UBIQUITY OF PRIME NUMBERS Armed with this definition of a Diophantine set, formal mathematicians led by Putnam, Davis, Robinson, and Matijasevič established the striking result that the set of prime numbers is Diophantine. That is, they showed that there exists a polynomial P —with integer coefficients in some number of variables—such that as its variables range over the nonnegative integers, the set of positive values of P is precisely the set of primes. One such polynomial given explicitly by Jones, Sato, Wada, and Wiens in 1976 (see [Ribenboim 1996]) is (k +2) 1 − (wz + h + j − q) 2 − ((gk +2g + k + 1)(h + j)+h − z) 2 − (2n + p + q + z − e) 2 − 16(k +1) 3 (k + 2)(n +1) 2 +1− f 2 2 − e 3 (e + 2)(a +1) 2 +1− o 2 2 − a 2 y 2 − y 2 +1− x 2 2 − 16r 2 y 4 (a 2 − 1) + 1 − u 2 2 − ((a + u 4 − u 2 a) 2 − 1)(n +4dy) 2 +1− (x + cu) 2 2 − (n + l + v − y) 2 − a 2 l 2 − l 2 +1− m 2 2 − (ai + k +1− l − i) 2 − (p + l(a − n − 1) + b(2an +2a − n 2 − 2n − 2) − m) 2 − (q + y(a − p − 1) + s(2ap +2a − p 2 − 2p − 2) − x) 2 − (z + pl(a − p)+t(2ap − p 2 − 1) − pm) 2 . This polynomial has degree 25, and it conveniently has 26 variables, so that the letters of the English alphabet can each be used! An amusing consequence of such a prime-producing polynomial is that any prime p can be presented with a proof of primality that uses only O(1) arithmetic operations. Namely, supply the 26 values of the variables used in the above polynomial that gives the value p. However, the number of bit operations for this verification can be enormous. Hilbert’s “tenth problem” was eventually solved—with the answer being that there can be no algorithm as sought—with the final step being Matijasevič’s proof that every listable set is Diophantine. But along the way, for more than a half century, the set of primes was at center stage in the drama [Matijasevič 1971], [Davis 1973]. Diophantine analysis, though amounting to the historical underpinning of all of number theory, is still today a fascinating, dynamic topic among mathematicians and recreationalists. One way to glimpse the generality of the field is to make use of network resources such as [Weisstein 2005]. A recommended book on Diophantine equations from a computational perspective is [Smart 1998]. 8.5 Quantum computation It seems appropriate to have in this applications chapter a brief discussion of what may become a dominant computational paradigm for the 21st century.

8.4 Diophantine analysis 417<br />

9262 3 + 15312283 2 = 113 7 ,<br />

17 7 + 76271 3 = 21063928 2 ,<br />

43 8 + 96222 3 = 30042907 2 .<br />

(The latter five examples were found by F. Beukers and D. Zagier.) There is<br />

a cash prize (the Beal Prize) for a proof of the conjecture of Tijdeman and<br />

Zagier that (8.2) has no solutions at all when p, q, r ≥ 3; see [Bruin 2003] and<br />

[Mauldin 2000]. It is known [Darmon and Granville 1995] that for p, q, r fixed<br />

with 1/p+1/q+1/r ≤ 1, the equation (8.2) has at most finitely many coprime<br />

solutions x, y, z. We also know that in some cases for p, q, r the only solutions<br />

are those that appear in our small table. In particular, all of the triples with<br />

exponents {2, 3, 7}, {2, 3, 8}, {2, 3, 9}, and{2, 4, 5} are in the above list. In<br />

addition, there are many other triples of exponents for which it has been<br />

proved that there are no nontrivial solutions. These results are due to many<br />

people, including Bennett, Beukers, Bruin, Darmon, Ellenberg, Kraus, Merel,<br />

Poonen, Schaefer, Skinner, Stoll, Taylor, and Wiles. For some recent papers<br />

from which others may be tracked down, see [Bruin 2003] and [Beukers 2004].<br />

The Fermat–Catalan conjecture is a special case of the notorious ABC<br />

conjecture of Masser. Let γ(n) denote the largest squarefree divisor of n. The<br />

ABC conjecture asserts that for each fixed ɛ>0 there are at most finitely<br />

many coprime positive integer triples a, b, c with<br />

a + b = c, γ(abc)

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