Internat. J. Math. & Math. Sci.
Vol 9 o. 2 (1986) 261-266
261
SEQUENCES IN POWER RESIDUE CLASSES
DUNCAN A. BUELL
Department of Computer Science
Louisiana State University
Baton Rouge, LA
70803
and
RICHARD H. HUDSON
Department of Mathematics
University of South Carolina
Columbia, SC 29208
(Received June 13, 1985)
Using A. Well’s estimates the authors have given bounds for the largest prime
ABSTRACT.
such that all primes
P0
For
have sequences of quadratic residues of length
PO
the smallest prime having
8
m
P
m
m.
3(mod 4)
consecutive quadratic residues is
The reason for this phenomenon is investigated in this paper and
th
power
the theory developed is used to successfully uncover analogous phenomena for r
and
P0
(mod 4).
E
residues,
2, r
r
even.
KEY WORDS AND PHRASES. Sequences of consecutive r
zeros and ones, linear least squares fit.
power residues, random sequences
of
I.
INTRODUCTION.
In [I] the authors used Well’s estimates [2] to give explicit bounds for the largest
prime
P0
such that all primes
assigned length
m
primes having
is
m.
P0
P
have sequences of quadratic residues of pre-
Unexpectedly, the authors discovered that for all m < 8 the smallest
consecutive quadratic residues are congruent to
3(mod r) and that P 0
l(mod 4).
In this paper we develop a theory which suggests that primes
E
3(mod 4) can be
expected to have a longest sequence of quadratic residues (or non residues) which is, in
the mean, approximately one longer that the longest sequence for primes
conparable size.
Data in Table
l(mod 4) of
support this theory.
Using our theory we predicted that primes
5(mod 8) can be expected to have a
longest sequence of quadratic residues which is, in the mean, about I/2 longer than the
longest sequence for primes
in Table 2.
of comparable size.
This is supported by data
In Table 5 we observe that (as aconsequence) the smallest primes
having a sequence of 1,2,3
E
l(mod 8)
5(mod 8) rather than
E
l(mod 4)
I0 consecutive quadratic residues, respectively, are all
l(mod 8).
DUNCAN A. BUELL and RICHARD H. HUDSON
262
The data in Tables 3 and 4 support our theory regarding the approximate magnitude
th
2
power residue sequences, r
of the mean excesses in the lengths of the longest r
and even, for primes E r+l (mod 2r) versus the corresponding mean lengths for prime
E
6,8.
l(mod 2r); data is supplied for r
Additional supporting’material can be obtained from the first author in [3] where
th
r
powers (mod p) is
data for all cosets formed with respect to the subgroup of
presented and the mean value of the longest sequence in any coset is observed to be
closely approximated by logrp for all integers r _> 2, 70,000 < p < 300,000; see Table 6.
2.
NOTATION.
For each fixed
power residues modulo the prime
let m(g) be the smallest prime
th
We
which, as customary, we take to be E l(mod r).
p
and m(g) the largest prime such
such that n(p)
p
n(p)
that
3.
denote the length of the longest sequence of r
n(p)
let
r
RANDOM SEQUENCES OF ZEROS AND ONES.
Consider all n-character strings of
E
shall call a substring of
t
zeros and
n-t
For convenience we
ones.
or more consecutive zeros an E-substring.
If
event which is a function of a variable x and which happens with probability
X
shall say that
Let
happens almost surely if
R(n,,t)
P(X)
tends to
as
denote the number of n-character strings with
x
t
X
is an
P(X) we
tends to infinity.
zeros and n-t ones
which do not contain and -substring.
THEOREM 3.1
R(n+l,E,t)
(b)
R(n,E,t)
> O, we have
t >
For n
(a)
R(n,,t) + R(n,,t-l)
R(n-,,t-l),
i>O
The recursion for part (a) is easy.
PROOF.
If we append a "l" to and n-character
string with no -substring then we get an (n+l)-character string with no -substring,
accounting for the addend R(n,,t).
If we append a "0" to an n-character string with
no -substring then we get an (n+i)-character string which has no -substring unless the
original string ended with
-1
zeros, accounting for the second and thrird addends on
the right-hand-side of formula (a).
To prove (b) we use the two variable generating function
F(x,y)
R(n,,t)
(x n-t)
(yt).
Using the recursion in (a) and taking into account the "initial conditions" we get
F
xF + yF
xyF
+
y
Thus
F(x,y)
where
(l-yE)/(1-y).
G(y)
(;
l-y
l-x-y+xy E
G
l-xG
The remainder of the proof is easily supplied by the
reader using simple algebra to extract coefficients.
Let
s(n,E,t)
0
other digits,
denote the number of n-character strings with
through
r-l,
with no E-substring.
following theorem as its proof is entirely similar to the above.
THEOREM 3.2
(a)
For
n
s (n+l
r
t
E > 0 we have
r
r
t
zeros and
n-t
We may omit the proof of the
SEQUENCES IN POWER RESIDUE CLASSES
Sr(n,,t)
(b)
v(n)
Let
THEOREM 3.3
w(n)
n().
(b)
If
rt then
(c)
Let
n
v(n)
logr(n)+w(n)
zeros.
t
)(
)(I
Then
+ n(t-)
v(n)
rt then
n
Then
almost surely does not happen.
be the event that there exists no -substrlng.
2
P(E 2)
If
w(n) denote any
tends to infinity.
n
be the event that there exists at least one -string.
P(X I) <
X
tends to infinity as
Consider an arbitrary n-character string having
X
Let
(d)
(-l)i(r-l) n-t+l-i’n-t+l(
i )(.n-in_t)
denote the length of the longest -substring and
function such that
(a)
i=O
263
logrn-W(n)
almost surely happens.
PROOF.
(a)
The probability that at least one -substrings.
n(t-)
P(X I)
(n-+l)-
Thus
(n-+l)(a),
n
t
where
t (t-l)... (t-+l)
n(n-l)... (n-+l)
a
Henceforth, we assume the easy inequality
t-I
t
for
n
n-i
t > i
n
a
from which it follows that
(b)
Let
n
If
log
n()
P(X I)
r
-w(n)
(c)
n(r-
+ w(n),
r(n)
X
-
P(X I) < n(t/n)
so that
(a).
Then
).
r
then
tends to zero.
Let
(t/n)
<
and apply part
rt
P(X I) <
0,
r
-w(n)
/n.
As
be the number of -substrings.
n
tends to infinity,
Using an appropriate version of
Chebyshev’s inequality we have
P(X2)
As before,
E(X2)
E2(X)
<
E(X)
E(X
I.
(n-+l)(a).
2)()
2(
n-2+2
2
-I
+ 2
Then
(n-2
i=l
2(
It is not difficult to see that
n-2
(t-2)
n-2+i,
+l+i)(t_2+i)
n-2+2
2
+
n-
(n-+l)(t_).
-I
)a2
+
(n-2+l+i)a2_i + (n-+l)(a)
2
i=l
P(X2)
-I.
(n-)a2
DUNCAN A. BUELL and RICHARD H. HUDSON
264
Using the aforementioned inequality we
have
(n-2+2) (n-2+I) (a2)
2
<
I,
(n-+l)a
a2_i
a
a2
<
n-
i
a2
so that
2-< (_-)
Cosequently,
n-2+2
2
2
"
)a2+2 i=]
P(X 2) <
(n-2+I+i) a2-i
-I
(n-+l) 2 a 2
so that
2
i=l
P(X2
(n-2+I+i) a2_i
(n-+l)
2
a
[ a2-i
2
..,2
(.-i
<
2
i=l
a
(i=l
(n-)
or
<
P(X2)
n2
(-+I)
n-
i=l (t--)
2
i
n-
(n-+l)(t-)
t- i
Summing the geometric series completes the proof of (c).
(d)
Let
n
P(X2)
Let
=log
r
For large
n
<
2r
(r-I
n-r
(r-l)
(r) (I +
(n(r-i)
n-w(n).
(I +
so that by part (c)
rt
n-r
Then there is a constant
)
<
exp(
2 (r_
m
such that
< m log n
so that
2
< exp .m log2n
[n-2m
n-r
log n
this is easily bounded by 2 so that
P(X 2)
4r
(n(r_l)) (r)
4r
-w(n)
(n(r_l)) riO grn)r
4r
(_l)r
-w(n)
0
as
n
completing the proof of Theorem 3.3.
4. CONSEQUENCES OF 3.
We can visualize coset membership formed with respect to the subgroup of r th powers
(mod p) as a string of digits, each digit from the range 0 to r-l, with 0 correth
sponding to the r
powers. These strings are clearly not randomly determined. Nonetheless the theory in 3 suggests that to the extent that n(p) can be thought of as a
random variable we should expect its mean value to be on the order of
logrp. Specifically,
in Table 6 we give coefficients a and b produced by a linear least squares fit
of
n(p)
to
n(p)
Tables
2
m In p+b.
to 6 illustrate the central feature of this note.
then we have
(p-l)/2r
r
th
If
powers (and similarly for the other
r
is even and
r-1
cosets formed
with respect to the subgroup of r th powers (mod p)) in the interval
to (p-l)/2 if
p
l(mod 2r) and in the interval
to p-1 if p
r+l(mod 2r). Data in the
following tables suggest that mean lengths of n(p) for primes
r+1(mod 2r)
exceed those for primes E 1(mod 2r) by in 2/in r for r
even and
2. The mean difference of I, e.g., for r=2 translates into an expected value
for n(p) for
p
3(mod 4) equal to the expected value for N(q) for a prime
l(mod 4) approximately twice the size of p.
265
SEQUENCES IN POWER RESIDUE CLASSES
n(p)
The theory which allows us to conjecture approximate mean differences in
between the residue classes produces estimates which are startlingly close to actual
data given the obvious differences between r th power residue distributions and random
However, our theory represents only a partial first hypothesis, and
character strings.
a superior explanation for the phenomenon described here may well be developed. None-
theless statistical tests (see [4] for the standard Z-test) provides better than 99%
confidence that
l(mod 2r)
primes
r+l(mod 2r), r=2,4,6,8, constitute
and primes
two entirely different sample populations.
Of course this note raises a number of questions, many very difficult.
2
r
m()
%
it is not at all obvious for large
3(mod 4)
and
m(E)
E
1(mod 4)
%
Even for
whether the probability that
increases or decreases or even has a limit as
goes to infinity.
5. COMPUTATION.
Originally computations were done in VS FORTRAN (IBM’s FORTRAN 77) on
the IBM 3033N of the System Network Computer Center at Louisianna State University.
The computation was redone, also in FORTRAN, on the VAX 11/780 running VMS of the
Department of Computer Science at L.S.U.
Most of the statistical results were obtained
using SAS on the IBM 3033.
Table i
Primes
l(mod 4)
r
2
Primes E 3(mod 4)
Range
# p
Mean
Dev.
#p
Mean
Dev.
<70000
140000
210000
280000
350000
420000
3449
12.738
14.938
15.731
16.234
16.566
16.886
2.4q6
3029
2914
2790
2762
2709
3485
3046
2883
2835
2783
2704
13.809
16.007
16.742
17.245
17.572
17.897
2.445
1.905
1.907
1.923
1.946
1.941
1.995
1.969
1.992
2.046
1.982
Table 2
Primes
=- l(mod 8)
r
5(mod 8)
Rang.
# p
Mean
Dev.
#p
Mean
Dev.
<70000
140000
210000
280000
1820
1606
1549
1482
6.359
7.545
7.954
8.172
1.438
1.173
1.170
1.083
1860
1620
1535
1508
6.985
8.078
8.506
8.672
1.302
0.959
Primes
l(mod 12)
r
n
r/En 2
i
1.071
1.069
1.011
1.011
1.006
1.011
4
Primes
Table 3
Difference
in Mean
1.024
0.967
Difference
in Mean
r/En 2
0.5
.626
.533
.552
.500
6
Primes
7(mod 12)
Range
#p
Mean
Dev.
#p
Mean
Dev.
<70000
140000
210000
280000
1830
1601
1537
1499
4.892
5.737
6.064
6.253
1.073
0.857
0.864
0.822
1851
1624
1548
1480
5.307
6.206
6.438
6.680
1.019
0.794
0.763
0.766
Difference
in Mean
En r/En 2
.415
.469
.374
.427
386..
DUNCAN A. BUELL and RICHARD H. HUDSON
266
Table 4:
Primes
-= l(mod
16)
Range
#p
Mean
Dev.
70000
140000
210000
280000
909
803
776
739
4.183
4.923
5.173
5.361
1.011
0.857
0.776
0.872
i
2
3
4
5
6
7
8
9
m()
5
37
29
41
1153
2689
32233
103529
i01
269
389
661
1301
4621
4261
i0
#p
Mean
Dev.
911
803
773
743
4.535
5.296
5.542
5.681
0.821
0.693
0.683
0.730
r
%n r/En2--.333..
.352
.373
.369
320
4
m()(mod 8)
m()(mod 8)
1
1
1
1
1
5
5
5
5
5
5
5
5
5
5
m
Regression n(p),
70000 < p < 300000
Table 6:
Difference
in Mean
9(mod 16).
Primes
Table 5:
m()
8
r
r
m
b
2
3
4
5
1.436
0.910
0.726
0.612
-1.088
-0.970
-0.608
-0.622
np
+__b
i/En r
1.443
0.910
0.721
0.621
We wish to express sincere thanks to Andrew Thomason, who has declined to accept co-authorship of this paper, for insight, help and particulary for
ACKNOWLEDGMENTS.
suggestions regarding the theorems in 3.
REFERENCES
i.
BUELL, D. A., and HUDSON, R. H., "On Runs of Consecutive Quadratic Residues and
and Quadratic Nonresidues", BIT 24 (1984), 243-247.
2.
WELL, A.,
3.
BUELL, D. A., and HUDSON, R. H., "Sequences in Power Residue Classes", L.S.U. Computer Science Technical Report, Baton Rouge, LA.
4.
WALPOLE, R. E., Introduction to Statistics, MacMillian, New York, 1968.
Algbriques
et les Varietes qui s’en Deduisent",
"Su les Courbes
Actualites Math. Sci. No. 1041 (Paris, 1945), deuxime partie, Section IV.