Quadratic Reciprocity and Primes of the form x2+ny2
Fang X.
1
and Tan V.
2
Department of Mathematics
Faculty of Science, National University of Singapore
2 Science Drive 2, Singapore 117543
ABSTRACT
Nearly 400 years ago, Fermat studied when a prime p could be represented by the form p
= x2 + y2, p = x2 + 2y2 and p = x2 + 3y2. Years later, Euler became extremely interested in
the problems and he successfully gave the proofs. That’s not the end of the story, he
further conjectured more about when a prime p could be represented by the form x2 + ny2
where n > 3.
In this paper, we start with the quadratic residue and proof of quadratic reciprocity law。
Then we discuss primes of the form of x2 + ny2 with the modern version of Euler’s proof
of the case n = 1
p = x2 + y2, x, y  Z if and only p ≡ 1 mod 4.
We also discuss some of Euler’s theorems concerning p = x2 + ny2 for n > 3. The proofs
of these remarkable results involved Quadratic Reciprocity, Quadratic Forms, and some
elementary Genus Theory.
1. Quadratic Reciprocity.
The central theme of this section is to prove Quadratic Reciprocity. Let p and q be
1
2
fangxingyuan@nus.edu.sg, Year 2 math student
mattanv@nus.edu.sg, Associate Professor, Department of Mathematics, NUS
distinct odd primes, so that both of the Legendre symbols (p/q) and (q/p) are defined. It is
natural to inquire whether the value of (p/q) can be determined if that of (q/p) is known.
That is what Quadratic Reciprocity law tells us.
( p / q)(q / p)  (1)
(
p 1 q 1
)(
)
2
2
(1)
We prove the Quadratic Reciprocity law with elementary number theory mainly based on
Gauss’ Lemma, which is:
Let p be an odd prime and let gcd(a, p)= 1. If n denotes the number of integers in the set
with element a, 2a ,…, (p-1)a/2, whose remainders upon division by p exceed p/2, then
(a/p) = (-1)n.
2.
(2)
p = x2 + ny2 and the Quadratic Reciprocity
In this section, we give a modern version of Euler’s proof of Fermat’s theorem that for
primes p = x2 + y2, where x and y are integers if and only if p ≡ 1 mod 4.
The strategy is a two step proof with Descent step and Reciprocity step:
Descent Step: If p divides x2 + y2, gcd(x, y)=1, then p can be also written as the form
a 2  b 2 with gcd(a, b) = 1.
Reciprocity Step: If p ≡ 1 mod 4, then p divides x2 + y2, gcd(x, y) = 1.
Then, we restate Quadratic Reciprocity in order to relate it to our basic question of when
prime p could be represented as the form x2 + ny2.
The restatement is:
If p and q are distinct odd primes, then (q/p) = 1 if and only if p ≡  b2 mod 4q for some
odd integer b.
Next, we define a homomorphism  : ( Z / DZ )*  {1} such that  ([1]) = (D/p) for
odd primes p not dividing D.
Finally, we get the result that the three statements are equivalent: (i)p divides x2+ny2,
gcd(x, y)=1. (ii) (-n/p) = 1, and (iii) [ p]  ker(  )  ( Z / 4nZ )* ,
and we use the result to prove the Reciprocity step of the case when n = 2 and 3.
3. Quadratic Forms
In this section, we start with quadratic forms f(x, y) = ax2 + bxy + cy2. Then, we discuss
numbers that can be represented by a given form. We touch on some elementary Genus
Theory and prove the Descent Step we raised in section 2.
For the n’s, of which principal genus consists of only one class, we deduce from the
results of this section to find more conditions for p that can be represented as the form
x2 + ny2 where the 4n contains only one class in its principal genus as follows:
p  x 2  6 y 2 if and only if p  1,7 mod 24
(3)
p  x 2  10 y 2 if and only if p  1,9,11,19 mod 40
(4)
p  x 2  13 y 2 if and only if p  1,9,17, 25, 29, 49 mod 52
(5)
p  x 2  15 y 2 if and only if p  1,19,31, 49 mod 60
(6)
p  x 2  21y 2 if and only if p  1, 25,37 mod84
(7)
p  x 2  22 y 2 if and only if p  1,9,15, 23, 25,31, 47, 49, 71,81mod88
p  x 2  30 y 2 if and only if p  1,31, 49,79 mod120 .
(8)
(9)
Finally, we discuss Euler’s conjectures concerning n = 5 and 14:
p  x 2  5 y 2 if and only if p  1,9 mod 20
(10)
2 p  x 2  5 y 2 if and only if p  3,7 mod 20
(11)
 x 2  14 y 2 
p 2
if and only if p  1,9,15, 23, 25,39 mod 56
2
2 x  7 y 
3p  x 2  14 y 2 if and only if p  3,5,13,19, 27, 45mod 56 .
(12)
(13)
REFERENCES
[1] Burton, David. M. (1989), Elementary Number Theory, New York :
McGraw-Hill.
[2] Cox David. A. (1989), Primes of the Form x2 + ny2, New York : Wiley.
[3] Ireland. K and Rosen. M. (1986), A Classical Introduction to Modern Number
Theory, Springer-Verlag
[4] Rotman. J. J. (2006), A First Course in Abstract Algebra: With Applications,
Pearson Prentice Hall