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.
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