1.1. Quadratic Forms as Polynomials 3

Example 1.5. Euler (1707-1783) considered representation of primes by

quadratic forms of the form

x2

±

Ny2

over Z. A typical result: If p is prime

and satisfies p = 1 (mod 20), then p —

x2

+

5y2.

Example 1.6. Lagrange (1736-1813) showed that every nonnegative integer

n can be expressed as a sum of at most four squares; that is,

Ti

^ X-\

~~\~ Xn

I

Xo

I XA.

Lagrange also substantially developed the theory of binary (two-variable)

quadratic forms over the integers.

Example 1.7. Let p and q be odd primes, with p q, and consider repre-

sentations of the product pq by the quadratic form Q(x, y) =

x2

—

y2.

Over

the field Q of rational numbers there are infinitely many representations

pq — Q. In fact if (3 is any rational number, then for all A G Q* we have

On the other hand, a straightforward arithmetic argument (Try it!) shows

that there are just two nonnegative integral representations pq — Q, given

by the pairs

(^.(E + i,^ ) an d (l,,)_(»±i,«^).

Now we will move in the opposite direction and start with an integer

value for x, say x — n 2, and ask this question: Must there exist an

integer k satisfying 0 k n — 2 and primes p, q such that

n2

—

k2

— pq? If

the answer is "yes," then (if p q, say) we must have p = n — k and q = n+k

by the Fundamental Theorem of Arithmetic, and hence 2n — p + q. In other

words, an affirmative answer to this question on integral representations by

the binary quadratic form Q(x,y) —

x2

—

y2

would affirmatively settle the

celebrated Goldbach Conjecture: Every even integer 4 is the sum

of two primes.

Conversely, if the Goldbach Conjecture is true and for each integer n 2

there are primes p q such that 2n = p + g, then p — n — k and q = n + k

for some integer k satisfying 0 k n — 2 and hence

n2

—

k2

= pq. So

the Goldbach Conjecture and our quadratic forms conjecture are actually

equivalent. Note that a supplementary conjecture on Q(x,y) —

x2

—

y2

that there are infinitely many values of n for which k — 1 "works" (i.e.,

n2

— 1 = pq for some primes p and q) is equivalent to the Twin Prime

Conjecture: There are infinitely many primes p such that p + 2 is

also prime.