~ _ml~'"~ ) dx=(ml, m2), ~,- I d d , m~+l , i=2,3 ..... k-l,

(1)

A R K I V F O R MATEMATIK Band 3 nr 2 4

Communicated 26 October 1955 by T. NAGELL

A theorem concerning the least quadratic residue and

n o n - r e s i d u e

By LARs

FJELLSTEDT

The purpose of this paper is to prove the following

T h e o r e m : Denote by v2* (p; 2) the least odd prime number which ks quadratic non-residue modulo the prime p. Then /or P > Po

~* (p ; 2) < 6- log p.

Denote by ~* (p; 2) the least odd prime number which ks quadratic residue modulo the prime p. Then /or p > Po

n* (p; 2) < 6. log p.

We shall require the following result which we do n o t prove:

L e m a ~ a . I[ the system

x ~ - b 1 (mod ml), x - ~ b 2 (mod ms) . . . x ~ b ~ (mod m~), b~>0, is solvable, its positive solutions are given by

where

- m 1 m 2 m 1 m 2 . . . m k _ l . . m 1 m 2 . . - mk

x = b x + m l t l + - - ~ l t 2 + ' " A d i d ___-/SS-~-~k-I-~ • "" ~ - z d x d~ ... dk-1

~ _[ml~'"~ )

d x = ( m l , m2), ~ , - I d d , m~+l , i = 2 , 3 . . . k - l ,

\ I " " i-i

t,

O < _ t l < m t + l

di and t > 0 an integer.

P r o o f o f t h e t h e o r e m . I f we assume y)*(p; 2 ) = p n , p~ denoting the ruth prime in the sequence 2, 3, 5, 7 . . . p satisfies

(2)

L. FJELLSTEI)T, The least quadratic residue and non-residue Thus

p ~ l , l l (mod 12), p - - l , 4 (mod 5), etc . . . . P u t t i n g N = 3 . 5 - 7 . . . p , , there exist v = c p ( N ) / 2 '~-1 integers at with

O < a l < 4 N , ( a ~ , 4 N ) = l , i = 1 , 2 . . . v

a n d with t h e p r o p e r t y t h a t e v e r y prime 79 satisfying (1) belongs to one of t h e arithmetical progressions

4 N t + a l , 42Vt + a ~ . . . . , 4 / ¥ t + a~.

I f we choose for each of t h e primes pt, i = 2 , 3 . . . n, one of t h e possible congruence conditions modulo pt or 4p~, we get e x a c t l y one residue class modulo 4 N which is therefore one of the n u m b e r s ak. L e t us assume t h a t we h a v e chosen xo, 0 < x 0 < 4 N , such t h a t

x o = b ~ (rood p*), xo~-b a (mod p~) . . . x o ~ b , (rood 79*), bt>O, w h e r e

p~ for p ~ = l (rood 4), P~=- 4pt for i0i~3 (mod 4).

W e m a y of course assume t h a t this system is solvable. P u t t i n g b = M i n (b~, b a . . . b,) a n d assuming t h a t b~,, b~ .. . . b~ k are all the integers b~ for which

a n d p u t t i n g also a n d

we h a v e

b~, = bt, = . . . . bt k = b, P = Pt," P~ "'" Pt~

P if P ~ m ~ l (rood4), r e = l , 2 . . . k, 4 P otherwise,

xo--b (mod P*).

I f we p u t P . Q = N when Q > 1, and define

Q* / Q if p j = l (rood4) when p J Q ,

= ( 4 Q otherwise, we also have, according to the ]emma,

Xo=--a (mod Q*),

where a is an integer such t h a t b < a < Q*. Using the l e m m a once more we get

Q*

x 0 = b + P * t o, 0 < t o < ( p , , Q , )

p * x o = a + Q* t 1, 0 g t 1 < ( p , , Q.).

(3)

ARKIV FSR MATEMATIK. B d 3 n r 2 4

I f t 1 > 0 it follows f r o m t h a t

P Q

**= 4 N . (P, Q)**

xo ~ dV~.

(2)

I f ~1 = 0 w e p r o c e e d in t h e following w a y . T h e n u m b e r /~ of different p r i m e f a c t o r s in P is e i t h e r ~_ n / 3 or it is < n/3. I f k ~_n/3, w e h a v e for 8 - - I n / 3 ] Xo > P* > Pl P ~ " " P~. (3) A s s u m i n g n e x t t h a t It<n~3 we define for all possible c o m b i n a t i o n s of r rlifferent p r i m e f a c t o r s p~v, q = l , 2 . . . r, of Q

a n d

Q Q (iv,, i t . . . iv,) =

P ~ " Ptv,.""" Pitt

Q* (i~ . . . ivr ) = {

Q(i v . . . ivr ) if t h i s integer h a s o n l y p r i m e divisors =~1 (meal 4),

4 Q (i v . . . i~r ) otherwise.

F o r t h e s e integers Q (~v .. . . ivr) we h a v e t h e congruences * "

Q ( , ... ivr)), 0 < c (i v . . . *vr( < Q (*v ... ~ )

x o=-c(i v ... ivr ) (mod * i , " * " i

a n d a s k f o r t h e least i n t e g e r r w i t h t h e p r o p e r t y t h a t for one c(iv, . . . ivr ) a t l e a s t

x 0 > c (i v .. . . ivr ). (4)

I t is e a s y t o see t h a t r_~ [ ( n - k ) / 2 ] . I n fact, s u p p o s e we h a v e t w o congruences

{

^{x ~ a} (rood A), 0 < a < A

x--b ( m o d B ) , ' 0 < b < B a * b (5)

w h e r e A a n d B a r e p r o d u c t s of different p r i m e s a n d (A, B) = 1, a n d suppose t h a t

x---c(modAB), m a x ( a , b ) < c < A B . (6)

I f t h e t o t a l n u m b e r of p r i m e f a c t o r s in A B is m, o n e of t h e integers A a n d B contains _~ [ m / 2 ] p r i m e factors. I f we cancel, in all possible w a y s , [ m / 2 ] p r i m e f a c t o r s of A B, t h u s obtaining n e w integers A*, B*, (A, B * ) = (A*, B ) = 1, t h e n for a t least one such p a i r we c a n n o t h a v e

x - - - - c ( m o d A * B * ) , O < c < A * B * ,

w i t h t h e s a m e integer c as in (6). Since w e m a y a s s u m e x o > Pn (otherwise we s h o u l d h a v e P > X o + 4 N ) , this a r g u m e n t obviously applies in o u r case.

(4)

L. FJELLSTEDT, The least quadratic residue and non-residue

* i " = Q * *

Thus it follows t h a t for a modulus Q ( g ... ~gr) with the property (4) we have

x 1 = c* + T . Q**, T > 0.

Since the number of different prime factors in Q** is at least

we have, for s----[3/3],

n - l c - r > ~ - i

Xo ~- Pl " P2 "" Ps (7)

I t results from (2), (3) and (7) t h a t in all cases

X O > R = T I " P 2 ' ' " P$"

I f we had Q = I , p would be > 4 N . F r o m

2 2 1 1

log R=v~(ps)>~p8 > -3s log s>-gn log n > ~Pn, we get

6. log p > 6. log x 0 > p , .

~ ~0,

Hence the first part of our theorem is proved.

Starting from

C):(;) ... (V)

^{- - + 1}

instead of starting from (1) the second part is obtained in exactly the same

w a y .

The best results previously obtained concerning this question are the following:

~v* (p; 2) < pa (log p)*, ~ = ¹ p ~ + 1 (rood 8) and P>Po.

2 V ; -

This was proved b y Vinogradov [1] in 1927. A. Brauer [2] and T. Skolem [6]

proved using elementary methods

y ~ * ( p ; 2 ) < C . p ~15, p~--±3, - 1 (rood8), C a constant.

I n 1954 Ankeny [3] proved

y , * ( p ; 2 ) < p ~, ~ > 0 , p ~ 3 (rood4) and P>Po.

Using the extended l~iemarm hypothesis several authors, Linnik, ErdSs, Ankeny etc., have obtained bounds for ~v* (p;2). The best one of these results is, as far as I know, the following (Ankeny [4]):

~* (p; 2) = 0 ((log p)2).

(5)

ARKIV FOR NIATEMATIK. B d 3 n r 24 On the other hand it has been proved b y Sali6 [5] and others t h a t

H*(P; 2 ) > c . l o g p

~* (p; 2) > c. log p

for infinitely m a n y primes p. Hence our result is in a sense the best possible.

Actually Sali6 proves only the first inequality. I t is however easy to see t h a t the second one can be proved b y the same method.

R E F E R E N C E S

1. Yr~OGRAI)OV, On t h e b o u n d of t h e least non-residue of n t h powers. Trans. Amer. M a t h . Soc. 29, 218-226 (1927).

2. BRAUER, A., ~ b e r d e n k l e i n s t e n q u a d r a t i s c h e n Niehrest. M a t h . Zeitsehrift 33, 161-176 (1931).

3. ANm~z~Y, N. C., Q u a d r a t i c residues. D u k e Math. J . 21, 107-112 (1954).

4. - T h e least q u a d r a t i c non-residue. Ann. of Math. (2) 55, 65-72 (1952).

5. S ~ r k , H., (~ber d e n kleinsten positiven q u a d r a t i s e h e n N i c h t r e s t n a c h einer P r i m z a h l . Math. Nachr. 3, 7-8 (1949).

6. SKOLEM, T., On t h e least odd positive q u a d r a t i c non-residue modulo p. D e t Kongel.

Norske ¥ i d . Selsk. F o r h . 27:20 (1954).

Tryck~ den 30 januari 1956

O'ppsala 1956. Almqvist & Wiksells Boktryckeri AB