wHITTy | 413
Moreover, every prime number can be obtained in this way, by making a suitable choice of
whole numbers a, b, c, . . . , z:
()
()()()()
()()
()()()()()
()()()
()
()+−++−
−+++ ++
−++++−
−+++−− +++−
−+−− +−
−+ −++−+
−+−−−+−+ −++−
−+−−++−−−−
−+ −−++−−−−
−+ −+ −−−kwzhjq
gk gk hjhz
kknf
npqzeeea o
ay xrya u
auua ndyxcu
al maiklnlvy
plan banann m
qyap sapapp x
zpla ptap ppm2{1
[2 1–]
[16 1211 ]
2[ 21 1]
[( –1)1][16 (–1) 1]
{([()] –1)[ 41 ]
(–1) 1}[1 1]
[1(2 22 2) ]
[1(2 22 2) ]
[( 21 )]}.2
2(^3222)
(^23222)
22 22 24222
22 2 2 22
22 22 2 2
22
22
22
It is not too hard to see that this polynomial can take a positive value only if all the squared
terms are 0, so to find prime-number values we must solve fourteen Diophantine equations.
Each of these fourteen equations has solutions that grow rapidly in size and seem to defy
analysis; to combine the solutions to all of the equations seems to be a formidable task—indeed,
to the best of my knowledge, even the prime number 2 has never been exhibited explicitly as a
value of the polynomial. Nevertheless, it is still a fact that every prime number can be produced
in this way! It strikes one as the kind of problem that Turing would have enjoyed tackling; it is
conceivably a problem that could never have existed without him.