AbstractLet ε be the fundamental unit of a real quadratic field of discriminant d and let integers Vn and Un be defined by (Vn = Und)⧸2=εn. It is well known that if p is any odd prime, then p|Um, where m = p − χ and χ is the value of the Legendre symbol (dp). In this paper several formulas are derived for the value of Um⧸p (modp) for the case in which p ∤ d

Williams, H.C.

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Some formulas concerning the fundamental unit of a real quadratic field 

Discrete Mathematics 92 (1991) 431-440 
North-Holland 
431 
H.C. Williams* 
Department of Computer Science, Uniuersity of Manitoba, Winnipeg, Manitoba, 
Canada R3T 2N2 
Received 19 September 1989 
Dedicated to Professor R.G. Stanton on the occash of his 68th biiday. 
Abstract 
Williams, H.C., Some formulas concerning the fundamental unit of a real quadratic field, 
Discrete Mathematics 92 (1991) 431-440. 
Let E be the fundamental unit of a real quadratic field of discriminant d and let integers V, and 
U” be defined by 
(V” + u&ii/2 = &“. 
It is well known that it p is any odd prime, then p 1 U,,,, where m =p - x and x is the value of 
the Legendre symbol (d/p). In this paper several formulas are derived for the value of 
CJJp (modp) for the case in which p i d. 
1. IntcotIuction 
Let E = (V + Ufi)/2 be the fundamental unit of the real quadratic field X with 
fundamental discriminant d and define integers V,, U, by 
V,Hlti 2” =(V+;~)“_ 
U-1) 
There is an extensive literature on numbers like V, and U,, (the Lucas functions 
with Q = *l). We refer the reader to Chapter XVII of Dickson [2j and to 
Lehmer [4-51 for some of this material. One particular result concerning U, is 
that for any odd prime p we have 
UP_, = 0 (modp), (1.2) 
where 2 is the value of the Legendre symbol (d/p). 
* Research supported by NSERC of Canada grant #A7649. 
0012-365X/91/$03.50 @I 1991- Elsevier Science Publishers B.V. All rights reserved 
432 H. C. Williams 
The result (1.2) is analogous to the classical result of Fermat that 
ap =u (modp) 
for any a E Z. If p t a, it is a problem going back to Abel as to what the Fermat 
quotient 
ap-l- 1 
P 
is modulo p. Eisenstein (cf. [2, p. 1051) showed that 
2P-’ - 1 
P 
_Pg (-l)‘+’ 
i=l 
---+modp)=~i=~~?+if(modp) (1.3) 
and Lerch (cf. [2, p. 1091) proved (We use [(u] to denote the integer part of (Y and 
we use l/i (modp) to denote an integer i’ such that ii’ = 1 (modp).) that 
Y-l-1 
P 
= -$t$lf(modp), 
i 1 
Y-‘-l lzp4 I 
P 
=-2lgi-2 zl i(modp) 
(l-4) 
(l-5) 
On examining Chapter IV of 123 we see that many other results concerning 
Fermat quotients have been developed, including the general result of Mirimanoff 
aP-l- 1 
E _‘$’ (ip’) (mod p) 
, 
P i=l 2 
(1.6) 
where by {x} we denote the least positive residue of x modulo a and by p’ we 
denote an integer such that p’p = 1 (mod d). However, few results seem to be 
known about what we can term the Pellian quotient CJpJp. 
Andrews [l] and Williams [S] have given some formulas concerning Up-,/p 
when d = 5 (the Fibonacci quotient). In fact in [8] the author stated that no other 
results concerning U,_,/p (even for d = 8) seemed to be known. In fact Kiselev 
and Slavutskii [3] had derived some formulas concerning Pellian quotients. One 
of these is 
2hV,‘U,,_ dp-l Id i 2 1 i 
P =- ;l (T)(i) &$modp) 
(1.7) 
which holds when p t d. Here h is the class number of X and m =p - x. The 
method used in [3] to derive (1.7) makes use of the properties of the Bernoulli 
polynomials and a congruence result of Voronoi. However, it is also possible to 
obtain results like (1.7) by using the general p-adic class number formula of 
Leopoldt [6] (see also Washington [7]) as it applies to real quadratic fields. 
The purpose of this paper is to derive some other formulas for c/,/p when 
p # d. We will use the same starting place as in [3], but we will employ different 
Concerning the jbdamental unit of a real quadratic field 433 
techniques to get a result which is analogous to (1.6). We will also present some 
special formulas for the cases of d = 8, 12, 5. These results are similar to (i.sj, 
(1.4) and (1.5). Our approach, throughout this discussion, will be elementary. 
2. Preliminary results 
By p we will denote any odd prime which does not divide d. On putting n =p 
in (1.1) we find that VP = V and c/, = XV (modp). By using these results and (1.1) 
it is easy to show that 
V, = 21V(x-‘)‘2 (modp), (2.1) 
where N = EE = N(E) = f 1. Also, by using (1.1) again with n = 2mk, we get 
U,,E4kv;1vm (modp)m 
P 
Putting k = h we see that 
2hv,‘v, v,, 
P 
=F (modp); 
thus, we now examine I]tnh/p (modp). 
A form of the Gauss-Dirichlet formula for h gives us 
p = IL (1 - 5;“) 
l-Ib Cl- Cb) *
(2.2) 
(2.3) 
Here 5 is any primitive dth root of unity and the products are taken over the 
ranges of a, b such that 1 s a, b s d - 1 and the Kronecker symbols (d/a) = 1, 
(d/b j = -1. Now it is well known (see, for example, [‘?, p. 121) that 
2 whend=8, 
M= n (1-<j)= d when d is a prime, 
lGj=Sd 
(j.d)=l 1 otherwise; 
hence, by (l.l), (2.3) and (2.4), we get 
fl Vzmh = ctih - E-~~ = M-m 
( 
n (1 - cb)2” - fl(1 
b 0 
We will use the notation 
cr=fl(modk) 
for ty, /3 E Z[c] and k E H+ to denote that 
cu=fJ+ky 
(24 
t”)“). (2.5) 
(2.6) 
(2.7) 
434 H. C. Williams 
for some y E E[c]. We remark that if (2.6) holds and a! - /3 E Z, then y in (2.7) 
must also be a rational integer. We will also specify 5 to be exp((2jci)ld). Under 
these conventions we can prove the following lemma. 
Lf3nma 2.1. Let 
f(x) ==kl (q)(-l)Y. 
i=r 1 
If 
Yl = c f (cm - CT, a Y2 = Tf K”)(l - P)-‘, 
then 
ti (y: - yl) = pS, 
where S E Z and 
S = $1 pan? (modp). . . (2.8) 
Proof. If we let G(ck) = EfZ: (d/i)g” and use the identity 
d-l 
d(1 - f”)-’ = 
then 
(2-9) 
4Y2 - YI) = ‘g p~lj(f)(-l)iG(*P~+l). 
j=l i=l 
Now it is well known that the Gauss sum G(Ck) = (d/k)G(1;) = (d/k)fi; hence, 
Since i(f) E&-l)‘+’ (modp2) for any i E Z such that 1 <i <p - 1, the lemma 
follows. cl 
We are now able to give the main result of this section 
Theorem 2.1. Zf S k defmed as in Lemma 2.1, we have 2dhLI, Vi’ = pS (mod p’). 
Proof. We note that (1 - 5”)” = 1 - Ckp + f (ck), where f (x) is defined in Lemma 
2.1 and f(x) =pg(x) with g(x) E Z[x]; hence, we have 
d n (1 - CY = n (1 - FP)(d + yi) (modp2), 
d 6 (1 - t;“)” = nf (1 - cbp)(d + y2) (modp2). 
b 
Concerning the findatnental unit of a real quadratic field 435 
Also, 
i 
n(l-9”) whenX=l, 
iiJ(l-C-Y= a 
v(l-cb) whenX=-1, 
when x = 1, 
J-J (1 - CbP) = 
1 
v (1 - Cb) 
b 
n(l-9”) whenX=-1; 
D 
thus, from (2.5) we get d2A4p-‘(~2”h - E-~) = d2(y2 - yi)(yr + y2 + 
2) (modp’). Now, by (2.9) 
y1 + y2 = ‘2’ ‘~‘j~)(-l)i ‘2:’ gWj+l) E z 
j=l i=l k=l 
(k.d)=l 
and P 1 y1 + ~2; thus, fi Mp-‘(+’ - E --2mh) s 2a (72 - j/i) (mod p2). The 
theorem now follows from this result, (24, Lemma 2.1 and (2.2). Cl 
3. The main resuIts 
In this section and the next we will provide various simplifications of formula 
(2.8) for S. We first require two simple lemmas. 
Lemma 3.1. CkdL2] (d/k) = 0. 
Proof. Follows easily from the facts (d/k) = (dl(d - k)) and cf=l (d/k) = 0. 0 
Lemma 3.2. CUE: k(dlk) = 0. 
Proof. We have 
by Lemma 3.1. Cl 
We will use the notation {x} to represent the least non-negative residue of z 
modulo d. Note that {-k} = d - {k} and {md+k}=(k). As j= 
0, 1,2, . . . ) d - 1, then, for any fixed i, pj + i runs through a complete set of 
436 H. C, Williams 
residues modulo d. Hence, we can write (2.8) as 
S = -‘g’ f lzl {p’(k - i)} (modp), 
i=l 
where pp’ = 1 (mod d). Now {p’(k - i)} =p'(k - i) - d[p'(k - i)/d]; thus, 
by Lemma 3.2. We can write 
where s = {p'i}. If j<s, then [(j-s)/d]=-1; if j>:s, then [(j-s)/d]=O; 
hence, 
+-l d p’(k-i) 
’ (k)[ d k=l ]=-(;)z(;)’ 
From these results we get 
(3. I) 
it follows from (3.1), (2.1) and Theorem (2.1) that we have the following 
theorem. 
Theorem 3.1. 
humlp G _xN(X-‘)f2 zl by (mod p), 
where 
N= N(E) and &9,(i) = q 
This result concerning the Pellian quotieiii is analogous to (1.6). 
We now turn our attention to pk(i), where we define &(i) by 
(kV d k&(i) = A4 5 0 (3.2) j=1 
with (k. d) = 1 and kk’ = 1 (mod d). From this definition we immediately get 
&k(i) = &(i) when k = n (mod d) and /3,(j) = &(i) when j = i (mod d). We also 
have the following lemma. 
Concerning the fundumenfai unit of a real quadratic field 
Lemma 3.3. &(i) = -/3k(k - i). 
437 
Proof. We have 1 - k’i = {k’(k - i)} (mod d); hence, if we put s = {k’i}, we get 
z cf, = P& - 9. 
Since Cf:; (d/j) = -zfSi (d/(d - t)), we get our result. 0 
We will also require the following. 
Lemma 3.4. &(i) = -&(d - i) - (d/i)(d/k). 
Proof. We note that (d - i)k’ = -ik’ (mod d); hence, {(d - i)k’} = d - {ik’} and 
where s = {ik’}. Since &(i) = C,S=,’ (d/j), (d/s) = (dlik’) = (dli)(dlk) and 
J$‘:t (d/j) = 0, the lemma follows. 0 
Corollary 3.1. &(d - kj) = -&Jk(j + 1)) = -E{=, (d/i). 
Proof. From the lemma we have 
We conclude this section with Theorem 3.2. 
Theorem 3.2. 
hU,,, fp = -2xNtx+= (P-L)Ry (modp)_ c 
i=l 
Proof. Follows by observing that 
p$ fip(i) = @s 
is1 1 i=l i 
+ (p$2 P,(p -j) ~ 2 @r P,(i) 
j=l p-i i=l 
7 (moW. 
by Lemma 3.3. Cl 
438 H.C. Williams 
4. Some special results 
In this section we will derive some remarkably simple formulas for UJp in the 
cases where d = 5, 8, 12. First, however, we observe that we can write the result 
of Theorem 3.1 as 
where p-i 
I-l d 
p + i (mod d), 
Ai = 
p-i 
i I 
- -1 p=i(modd). 
d 
We also have the following lemma. 
Lemma 4.1. I’ we define 
~ = (d - GP'))P + i 
d 
and 
Vi=jUi+Ai, 
then 
Proof. We have 
(modp). q 
The main result of this section is Theorem 4.1. 
Theorem 4.1. 1’ we define y(n) = CY=‘=, (d/i), then 
d-l 
2 ,(i)go&- _%,“I,,,, y([dj’pl)(modp). 
i=l k [p/d]+1 i 
Proof. Let 
(4. I) 
(4.2) 
(4.3) 
(4.4) 
Concerning the fundamental unit of a real quadratic field 439 
By Lemma 4.1 we have 
Asi=1,2,3,... , d - 1, then k = d - {ip’} also runs through this same range of 
values; hence, 
Here pk = pi and ok = vi for i = d - {pk}. Indeed from (4.2), (4.3) and (4.4) we 
get 
(4.5) 
Thus, by Corollary 3.1 and Lemma 3.1 we get 
Now by using (4.5) it is easy to verify that p&_l-_s =p - a,, o&_l-_s =p - ps. It 
fdiows that 
-dZ’ c ldF1 y(k) ,% f + d-‘z’m2 y(d - 1 - s) ‘g* 1 (mod p)_ 
k=l s=l j=p--a, / 
Since y(d - 1 - s) = -y(s) and y([d/2] - 1) = 0 when 2 1 d, we get 
-daY = “;T’ y(k) j$ ; + I@-$;’ Y(S) j; 
I 
; 
IV--JWl 
=2 kz, y(k)$(mod~)- 
For any given j (< p) there is a unique value of k (> d) such that 
In fact it is easy to deduce that this value of k must be [djlp]. Thus, we find that 
=’ E -2 k=[pld]+l 
2 IK+fWl y([dj/p]) 
j 
(modp). Cl 
Putting this result together with (4.1) we get the rather remarkable formuia 
hum/p s; @-I)” Il(d-yP’d y([dj/p]) (mod PI- 
k=ip/dj+l i 
(4.6) 
440 H. C. Williams 
In the special cases of d = 5, 8, 12, we know that h = 1. For d = 5, we have 
N = - 1, @-1)‘2 = 1 and 
(4.7) 
a formula much simpler than those given in [l] or [S]. When d = 8 we also have 
N= -1; hence 
Finally, when d = 12, we have N = 1 anti 
(4.8) 
References 
[l] G.H. Andrews, Some formulas for the Fibonacci sequence with generalizations, Fibonacci Quart. 
7 (1%9) 113-130. 
[2] L.E. Dickson, History of the Theory of Numbers, Vol. 1 (Chelsea, New York, 1952). 
[3] A.A. Kisekv and I.S. Slavutskii, On the number of classes of ideals and a quadratic field and its 
rings (in Russian), Dokl. Akad. Nauk SSSR 126 (1959) 1191-1194. 
[4] D.H. Lehmer, On the multiple solutions of the Pell equation, Ann. of Math. 30 (2) (1928) 66-72. 
[S] D.H. Lehmer, An extended theory of Lucas’ functions, Ann. of Math. 31 (2) (1930) 419-448. 
[6] H.W. Leopoldt, Eine p-adische Theorie der Zetawerte II. Die p-adische F-Transformation, J. 
Reine Angew. Math. 2741275 (1975) 224-239. 
[7] L.C. Washington, Introduction to Cyclotomic Fields (Springer, Berlin, 1982). 
[8] H.C. Williams, A note on the Fibonacci quotient F,_,/p, Canad. Math. Bull. 25 (3) (1982) 
366-370. 


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Some formulas concerning the fundamental unit of a real quadratic field

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