Unknown

Notes on Explorations 403

f(k-i) E f(k+i) (mod q) f or each integer i, it can be argued that u 5 k.
We have

4k(k-1)+1=q2< f(u)<f(k)=k(k-l)+p

whence 3(k - 1)2 < p, so that u 5 m + 1. But this contradicts the
hypothesis.
A polynomial of several variables whose positive values are prime is given
in J.P. Jones, D. Sato, H. Wada & D. Wiens, Diophantine representations
of the set of prime numbers, Amer. Math. Monthly 83 (1976), 449-464;
and Problem P. 291, Canadian Math. Bull. 24 (1981), 505.

E.14. With respect to the assertions in (b), we have the following. If x 2 y,
then

I(y - x)y - x21 = x2 + xy - y2 = x(x + y) - y2 >_ 2y2 - y2 = y2,

so that (*) implies x = y = 1. If y - x > x, then

4y2 - 4xy - 4x2 = y2 + 2y(y - 2x) + (y2 - 4x2) 2 y” 2 4,

so that (*) implies y = 2, x = 1. In the induction step, if F,, < x < y <
F ,,+I, then y - x < F,,-1 and

(Y - X:>Y - x2 I (FeI - l)Fn+l - (Fn + l)2

= (Fe-lFn+l - F,” - 1) - Fn+l - 2F,,^5 -3,

which contradicts (*). See James P. Jones, Diaphantine representation of
the Fibonacci numbers, Fibonacci Quart. 13 (1975), 84-88; and Problem
3, Int. Math. Olympiad 1981, Math. Mug. 55 (1982), 55.
The issue raised in this exploration is related to the tenth problem
posed by David Hilbert in his famous keynote address to the International
Congress of Mathematicians in 1900. He sought to give a prospectus of
the main topics requiring the attention of mathematical researchers during
the coming century. To provide a focus, he posed thirty-seven problems,
and these have tended to become benchmarks for progress in mathematics.
For a biographical account, read Constance Reid, David Hilbert (Springer,
1970).
The tenth problem is: Specify a procedure which in a finite number of
steps enables one to determine whether or not a given diophantine equa-
tion with an arbitrary number of indeterminates and with rational integer
coefficients has a solution in rational integers. While it was later shown
that no such general procedure exists, research into the question has led to
significant developments in the foundations of mathematics. One direction
has involved diophantine sets, i.e. sets S of natural numbers for which there
is a polynomial f (y, x) over Z for which y belongs to S if and only if there
are numbers x for which f(y, x) = 0. If we define

!l(YT 4 = (Y + 1x1 - MY, xN2) - 1,