Prime Numbers

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.