.ND ""
.EQ
delim $$
define sstar % size 10 * %
define yy % bold y %
define zz % bold z %
define pprime % size 10 prime %
define brsl % "\o'|\(sl'" %
.EN
.nr Au 0
.nr Pt 1
.SA 1
.TL
On the Density of Sequences of Integers the Sum
of No Two of Which is a Square\0\0\0II.
General Sequences
.AU "J. C. Lagarias" JCL MH 11216 4531 2C-358
.AU "A. M. Odlyzko" AMO MH 11216 7286 2C-370
.AU "J. B. Shearer*" JBS MH "" "" ""
.TM ""
.AS 1
.P
This paper studies the maximal density
attainable by a sequence $S$ of positive integers
having the property that the sum of any two distinct elements
of $S$ is never a square.
It shows there is a constant
$N sub 0$ such that for all
$N~>=~N sub 0$ any set
$S~\(ib~ [1,N]$ having
this property must have
$| S |~<~.475N$.
The proof uses the Hardy-Littlewood
circle method.
.rs
.sp 2.5i
_____________
.VL 2
.LI "\u*\d"
Current address:
University of California, Berkeley, CA 94720.
.LE
.AE
.OK ""
.MT 4
.H 1 "Introduction"
.ls 2
.P
P. Erdo\*:s and D. Silverman (see [4]) posed
the problem of determining the maximal density
attainable by a set
$S~=~ "{" s sub i "}"$ of positive integers
having the following property.
.FS *
Current address:
University of California, Berkeley, CA 94720.
.FE
.sp
.I
PROPERTY NS.
$s sub i~+~s sub j$ is not a perfect square
for all $i != j$.
.R
.P
J. P. Massias [8] observed that the set
$S sub 1$ of integers consisting of all
$x~==~1$ (mod 4)
together with all $x~==~14,~26,~30$ (mod 32)
has property $NS$ and density
$11 over 32$.
In a previous paper [6] we proved the following
result.
Let $d(S)$ denote the natural density of
a sequence $S$.
.sp
.I
Theorem A.
Let $S$ be a union of arithmetic progressions
(mod $N$) having property $NS$.
Then the density $d(S)~<=~11 over 32$ with
equality possible only if $32|N$.
For all other $N$,
$d(S)~<=~1 over 3$.
.R
.P
In this paper we bound the maximal
upper asymptotic density
.DS 3
.EQ
d bar (S) ~=~{lim~ roman "sup"} from {N ~->~inf} ~1 over N |S ~\(ca~
[1,N] |
.EN
.DE
attainable for an arbitrary
sequence $S$ having property $NS$.
It is easy to see that this density cannot
exceed $1/2$, for any square $n sup 2$ excludes
$half$ the positive integers smaller than $n sup 2$,
since at most one element of each pair
$(k,~n sup 2~-~k )$
can be in a set $S$ having property $NS$,
while for each $epsilon~>~0$ there is a
square between $x$ and $(1 + epsilon ) x$
for all sufficiently large $x$.
.P
Our main result applies to finite sets.
Let $S$ denote a finite set with all elements
$<=~N$ which has property
$NS$, and let
.DS 3
.EQ (1.1)
d(N)~=~max from S~
|S| over N
.EN
.DE
denote the maximum density of such a
set in $[1,N]$.
Our main result is the following.
.sp
.I
Theorem B.
There exists an absolute constant $N sub 0$ such that
for all $N~>=~N sub 0$,
.R
.DS 3
.EQ (1.2)
d(N)~<=~.475.
.EN
.DE
.P
Theorem B is proved using the Hardy-Littlewood
circle method, based on an idea
used in [6].
It immediately implies that the upper
asymptotic density $d bar (S)$ of an
infinite sequence having property $NS$ must
satisfy
.DS 3
.EQ (1.3)
d bar (S)~<=~.475.
.EN
.DE
The bound (1.2) can be improved
by extending the methods used in this paper,
but we see no hope of attaining an
upper bound near to $11 over 32$ without
some new ideas.
(In fact, it may well be that sequences
with $d bar (S)~>~11 over 32$ exist.)
.P
The methods of this paper also apply to the analogous problem of bounding
the maximal density attainable by
a sequence $S~=~"{" s sub i "}"$ of positive
integers having the following property.
.sp
.I
Property NP(k).
$s sub i~+~s sub j$ is not a perfect $k sup th$
power for all $i != j$.
.R
.P
For $k~=~p-1$ where $p$ is an odd prime the
set
$S sub p~=~"{" x~:~x~==~i$ (mod p),
$1 ~<=~ i ~<=~ p-1 over 2 "}"$
has property $NP(k)$, since
$x sup k~==~0$ or 1 (mod p)
for all $x$.
$S sub p$ has density $1 over 2~-~1 over 2p$.
By an adaptation of the method of this
paper it can be shown that for any sequence $S$
with Property $NP(k)$,
.DS 3
.EQ
d bar (S)~<=~
1 over 2~-~c sub 0 (k) ,
.EN
.DE
where $c sub 0 (k)$ is a positive absolute
constant depending on $k$.
.P
It is interesting to note that the density behavior
of sets having property $NS$ differs
completely from that of sets $S$ having the
following property
.sp
.I
Property DS.
$s sub i~-~s sub j$ is not a perfect
square whenever $i~!=~j$.
.R
.P
Sa\*'rko\*:zy [9] has shown that any sequence
having property $DS$ must have density zero;
he shows the number of elements
$<=~x$ in such a set is
$O left ( { x~( log log~x ) sup 2/3} over {( log~x ) sup 1/3} right )$.
.P
In another related direction,
Erdo\*:s [3] has proposed the following
problem:
Given a sequence $n sub 1 ~<~n sub 2 ~< "..."$ of positive integers
with $n sub i+1 / n sub i ~->~1$ as $i ~->~inf$, and such that
the $n sub i$ are uniformly distributed modulo $d$ for
every $d$, does it follow that if
$S~=~"{" s sub i "}"$ is an infinite sequence
of positive integers for which
$s sub j~+~s sub k~!=~n sub i$ for all
$i,j,k$, then
$d bar (S)~<~1 over 2$?
.H 1 "Proof of the Main Theorem"
In this section we prove Theorem B assuming certain
results proved by the circle method in later
sections.
.sp
\f2Proof of Theorem B\f1.
Let $S$ be a subset of $[1, N]$
having Property $NS$.
We wish to bound $| S |$ from above.
Let $G * (N)$ denote the
graph having $N$ vertices labelled 1 through $N$,
and in which $ "{" i, j "}"$ is an edge 
if and only if
.DS 3
.EQ (2.1)
i~+~j~=~k sup 2
.EN
.DE
for some integer $k$.
Note that $G * (N)$ contains loops
at those vertices $i$ with $i~=~1 over 2~k sup 2$.
If $alpha * (N)$ denotes the independence
number of the graph $G * (N)$ then
.DS 3
.EQ (2.2)
d(N)~<=~{alpha * (N)} over N ~+~N sup -1/2 ~,
.EN
.DE
since if $S~\(ib~[1,N]$ has Property NS, it consists of an
independent set in $G*(N)$ plus some integers $j$ such
that $2j ~=~k sup 2$, and there can only be $<=~N sup 1/2$ of those.
Thus the problem is that of bounding the
independence number of $G * (N)$.
.P
We may compare $G * (N)$ with the
graph $G(N)$ of [6], in which
$"{" i, j "}"$ was an edge if and only if
.DS 3
.EQ
i~+~j~==~k sup 2~~~~~~~roman {(mod~N)} .
.EN
.DE
All edges of $G * (N)$ are edges of $G(N)$,
but $G * (N)$ has $O ( N sup 3/2 )$ edges, while
for any $epsilon~>~0$ 
and all $N~>~N sub 0 ( epsilon )$,
$G(N)$ has at least
$N sup {2 - epsilon}$ edges. 
In [6] we showed the independence number
$alpha (N)$ of $G(N)$ satisfies
$alpha (N)~<=~11 over 32 N$.
We may expect a weaker bound here since
$G * (N)$ has many fewer edges than $G(N)$ and
hence, presumably, a larger independence
number.
.P
As in [6] we view the problem of
calculating the independence number
$alpha (G)$ of a graph $G$ as that of
solving the following $0-1$ integer
programming problem:
Maximize
.DS 3
.EQ (2.3)
Z~=~sum from i=1 to n~x sub i
.EN
.DE
subject to
.DS 3
.EQ (2.4)
x sub i~+~x sub j~<=~1 ,
.EN
.DE
if $"{" i,j "}"$ is an edge of $G$.
We will obtain an upper bound for (2.3)
by first replacing the constraints (2.4)
with a set of weaker constraints implied by
(2.4), and second by treating the
resulting program as a linear program
with the added constraints
.DS 3
.EQ (2.5)
0~<=~x sub i~<=~1~,
.EN
.DE
for all $i$.
.P
The weaker constraints we consider are obtained
as follows.
If $H$ is a subgraph of $G$ the constraints (2.4)
for $G$ imply
.DS 3
.EQ (2.6)
sum from {i member V (H)}~x sub i~<=~
alpha (H) ~,
.EN
.DE
where $V(H)$ is the set of vertices
of $H$.
For example, if $H$ is a
$(2s+1)$-cycle, then
$alpha (H)~=~s$.
We will use a subset of the constraints
.DS 3
.EQ (2.7)
sum from {i member V ( C sub 2s+1 )}~
x sub i~<=~s ~,
.EN
.DE
where $C sub 2s+1$ is any
$(2s+1)$-cycle in $G * (N)$.
.P
We can produce a large number
of explicit
$(2s+1)$-cycles in $G* (N)$
using simple identities involving sums of
squares.
Let $y sub 0 , . . . , y sub 2s$ be
$2s+1$ integers such that
.DS 3
.EQ (2.8)
sum from i=0 to 2s~y sub i~==~0~~~~~~~~roman {(mod~2)}~.
.EN
.DE
For $0~<=~k~<=~2s$ set
.DS 3
.EQ (2.9)
2n sub k~=~sum from i=0 to 2s~(-1) sup i~
y sub k+i+1 sup 2
.EN
.DE
where the subscripts on the $y sub k+i+1$ are
interpreted (mod $2s+1$).
The congruence (2.8) is a necessary
and sufficient condition that all the
$n sub k$ be integers.
A calculation shows that
.DS 3
.EQ (2.10)
n sub k~+~n sub k+1~=~y sub k+1 sup 2
.EN
.DE
for $0~<=~k~<=~2s-1$, and that
.DS 3
.EQ (2.11)
n sub 0~+~n sub 2s~=~
y sub 0 sup 2~.
.EN
.DE
Consequently if all the $n sub k$'s
are positive then
$(n sub 0 , n sub 1 , . . . , n sub 2s )$
is a $(2s+1)$-cycle in
$G * (N)$.
We label this cycle $C ( y sub 0 , y sub 1 , . . . , y sub 2s )$.
A sufficient condition to guarantee that all of
$(n sub 0 , n sub 1 , . . . , n sub 2s )$ be
positive is that all the $y sub i$ be
nearly equal in size.
Such a condition is given by
.DS 3
.EQ (2.12)
(1- epsilon )~M~<=~y sub i~<=~M ,~~~~
0~<=~i~<=~2s~,
.EN
.DE
for any $M~>~0$ and any $epsilon$ with
.DS 3
.EQ (2.13)
0~<~epsilon~<~1 over s+1~.
.EN
.DE
To see this, we note that (2.9), (2.12), and (2.13)
imply
.DS 3
.EQ
2n sub k~>~
(s+1)~(1- epsilon )~M~-~sM~>=~0~.
.EN
.DE
Next note that the constraint
(2.7) corresponding to
$C( y sub 0 , y sub 1 , . . . , y sub 2s )$ is
.DS 3
.EQ (2.14)
sum from k=0 to 2s~x sub n sub k~<=~s~.
.EN
.DE
.P
We now consider the linear program $L sub s$
having the objective function (2.3) and the
constraints (2.5) and (2.14) for all
$C( y sub 0 , . . . , y sub 2s )$ satisfying (2.12),
for a \f2fixed\f1 value of $s$.
Let $yy~=~(y sub 0 , . . . , y sub 2s )$.
If we add up the constraints
$C( yy )$ in (2.14) weighted with nonnegative
weights $w( yy )$ we obtain
.DS 3
.EQ
sum from n=1 to N~
r(n)~x sub n~mark =~sum from yy~
w( yy )~
left ( sum from k=0 to 2s~x sub n sub k right )
.EN
.sp
.EQ (2.15)
lineup <=~
s left ( sum from yy~w ( yy ) right )~,
.EN
.DE
where
.DS 3
.EQ (2.16)
r(n)~lineup =~sum from yy~
m sub n ( yy )~w ( yy )~,
.EN
.DE
and $m sub n ( yy )$ is the number of times
$n$ occurs as a component in the vector
$(n sub 0 , . . . , n sub 2s )$ corresponding
to $yy$ via (2.9).
Note that we have the identity
.DS 3
.EQ (2.17)
sum from n=1 to N~
r(n)~=~(2s+1)~sum from yy~w( yy )~,
.EN
.DE
since each $yy$ produces a vector of
$2s+1$ $n sub j$'s.
If we can find nonnegative weights 
$w ( yy )$ such that (2.16) gives
.DS 3
.EQ (2.18)
r(n)~>=~1,~~~~~1~<=~n~<=~N ,
.EN
.DE
then
.DS 3
.EQ
Z~mark <=~sum from n=1 to N~
r(n)~x sub j
.EN
.sp
.EQ
lineup <=~
s( sum from y~w ( yy ) )
.EN
.sp
.EQ (2.19)
lineup <=~
s over 2s+1~
left ( sum from n=1 to N~
r(n) right )
.EN
.DE
using (2.15) and (2.17).
If furthermore
.DS 3
.EQ (2.20)
r(j)~<=~mu ,~~~~~1~<=~j~<=~N ,
.EN
.DE
we obtain the upper bound
.DS 3
.EQ (2.21)
Z~<=~s over 2s+1~ mu ~N ~,
.EN
.DE
i.e., $d(N)~<=~s over 2s+1~mu$.
In linear programming terms
the $w ( yy )$ are dual variables and
(2.18) are the conditions that
the $w( yy )$'s be a dual feasible solution.
Dual feasible solutions always provide an
upper bound on the primal problem's
objective function, which in this case is
(2.19).
.P
We have now transformed the problem to that
of finding a "good" choice of the
nonnegative weights $w ( yy )$ so as to
make all the $r(j)$ nearly equal as given
by (2.18) and (2.20).
Now the formula (2.16) for $r(j)$ shows
that it is a weighted sum over those
$yy$ such that
.DS 3
.EQ (2.22)
2n~=~
sum from i=0 to 2s~
(-1) sup i~y sub i sup 2~,
.EN
.DE
for some $k$, i.e., over representations of
$2n$ by the indefinite diagonal quadratic form
.DS 3
.EQ (2.23)
Q( zz )~=~sum from i=0 to 2s~
(-1) sup i~z sub i sup 2~.
.EN
.DE
We can count the number of weighted integral
representations of such a form satisfying
certain side conditions using the
Hardy-Littlewood circle method.
Let $r sub {M, epsilon} sup sstar (2n)$
denote the number of ordered $(2s+1)$-triples
of integers $zz ~=~(z sub 0 , z sub 1 , . . . , z sub 2s )$
such that $2n~=~Q( zz )$ and
.DS 3
.EQ (2.24)
(1 - epsilon ) M~<=~z sub i~<=~M
.EN
.DE
for $0~<=~i~<=~2s$, and such that all the
$z sub i$ are distinct.
Using the circle method, in Section 3 we will prove the
following result.
.sp
.I
Theorem C.
Let $s >= 2$, and suppose
$0~<~epsilon~<~1 over 4(s+1)$.
There is a constant $delta pprime~>~0$ such that
.DS 3
.EQ (2.25)
r sub {M , epsilon} sup sstar (2n)~=~
M sup 2s-1~G sub s (2n)~
f( 2n over {M sup 2} )~+~
O ( M sup {2s-1- delta pprime} ) ~,
.EN
.DE
where the $O$-constant depends on $s$ and
$epsilon$, but not on $n$ and $M$.
Here:
.R
.VL 10 5
.LI (i)
.I
For all positive integers $n$, $G sub s (2n)$
satisfies
.DS 3
.EQ (2.26)
c sub 1 (s)~>=~G sub s (2n)~>=~c sub 0 (s)
.EN
.DE
for certain constants $c sub 1 (s)$ and $c sub 0 (s)$.
.R
.LI (ii)
.I
$f(t)$ is a continuously differentiable function
which is nonnegative and not identically zero.
It vanishes outside
the interval $I sub {epsilon , s}$ given by
.DS 3
.EQ (2.27)
1~-~2(s+1) epsilon~+~
(s+1) epsilon sup 2~<=~t~<=~
1~+~2s epsilon~-~s epsilon sup 2~.
.EN
.DE
.LE
.R
.P
In this theorem $G sub s (2n)$ is the \f2singular series\f1;
it is defined by (3.18) below.
The proof of Theorem C shows that we can take
$delta pprime ~=~1 over 12$,
a constant independent of $M,~s$, and $epsilon$.
In addition we note that the conditions on
$epsilon$ and $s$ insure that
.DS 3
.EQ
I sub {epsilon , s}~\(ib~
[ 1 over 2 , 2 ]~.
.EN
.DE
The constant $c sub 0 (s)$ is strictly positive for
$s~>=~2$, as will be seen in Section 4.
.P
We can use Theorem C to obtain weights
$w( yy )$ so as to obtain
$r (n)~approx~r sub {M, epsilon} (2n)$.
However $r sub {M, epsilon} (n)$ fluctuates
greatly in size in the interval
$0~<=~n~<=~N$ due to the term
$f ( n over {M sup 2} )$.
We damp out these fluctuations
by choosing weights that involve a further
averaging over the parameter
$M$.
We first set
.DS 3
.EQ (2.28)
w ( yy , M)~=~M sup -2s
.EN
.DE
provided that
.DS 3
.EQ (2.29)
M( 1 - epsilon )~<=~ y sub i ~<=~M,~~~0 <= i <= 2s ,
.EN
.DE
and that all the $y sub i$ are distinct.
Otherwise we set $w( yy , M)~=~0$.
Our choice of weights is
.DS 3
.EQ (2.29)
w ( yy )~=~{ sum pprime} from M~
w ( yy , M )~,
.EN
.DE
where the prime in this and later summations indicates
it is over all integers $M$ in the
range
.DS 3
.EQ (2.30)
epsilon~ sqrt N~<~
M~<~
(3 - epsilon )~sqrt N .
.EN
.DE
.sp
.I
Lemma 2.1.
For $s~>=~2$ and the choice of weights
$w ( yy )$ given by
(2.29), (2,30), there are positive constants
$c sub 2$ and $delta pprime pprime$ such that
.DS 3
.EQ (2.31)
r(n)~=~c sub 2~
G sub s (2n)~+~O(N sup {- delta  pprime pprime} )
.EN
.DE
for $epsilon~N~<~n~<~(1- epsilon )~N$. 
In any
case
.DS 3
.EQ (2.32)
r(n)~<=~c sub 2 G sub s (2n)~+~O(N sup {- delta  pprime pprime} )
.EN
.DE
for $1~<=~n~<=~N$.
The $O$-symbol constants depend on $s$ and $epsilon$,
but not $N$.
.sp
Proof.
.R
Recall $Q( yy )~=~SIGMA from i=0 to s (-1) sup i y sub i sup 2$
and let
.DS 3
.EQ
w * (2n)~mark =~
{sum pprime} from { pile { yy above Q( yy ) = 2n}}~~w( yy )
.EN
.sp
.EQ
lineup =~
{sum pprime} from M~
sum from { pile { yy above Q ( yy ) = 2n}}~~
w(y,~M)
.EN
.sp
.EQ
lineup =~ {sum pprime} from M~M sup -2s~
r sub {M, epsilon} sup sstar~(2n)
.EN
.sp
.EQ
lineup =~ {sum pprime} from M
"{" M sup -1 ~G sub s (2n)~f ( 2n over {M sup 2})~+~
O ( M sup {-1- delta  pprime} ) "}"
.EN
.sp
.EQ (2.33)
lineup =~{sum pprime} from M
~M sup -1~G sub s (2n)~f ( 2n over {M sup 2} )~+~
O(( epsilon sqrt N ) sup {- delta  pprime} )~,
.EN
.DE
using Theorem C.
Since $epsilon > 0$ is fixed, we can
replace the error term in
(2.33) by $O( M sup {-~ 1 over 2 delta  pprime} )$.
Now
.DS 3
.EQ (2.34)
{sum pprime} from M~
M sup -1~f ( 2n over {M sup 2} )~=~
int from {epsilon sqrt N} to
{(3- epsilon ) sqrt N}~
t sup -1~f( 2n over {t sup 2 }) dt~+~
O( N sup {-~ 1 over 2} ) ,
.EN
.DE
using
.DS 3
.EQ
| f pprime (t) |~<=~
K sub 0 ( epsilon , s )~,~~~~
t~epsilon~(- inf , inf )~.
.EN
.DE
The change of variables $u~=~ t over {sqrt 2n}$ yields
.DS 3
.EQ (2.35)
{sum pprime} from M~
M sup -1~f ( 2n over {M sup 2} )~=~
int from { epsilon sqrt {N over 2n}} to {(3- epsilon ) sqrt {N over 2n}}~
u sup -1~f( u sup -2 ) du~+~
O ( N sup {-~ 1 over 2} ) .
.EN
.DE
For $epsilon ~N~<=~n~<=~(1- epsilon )~N$,
the range of integration in (2.35)
includes $[ 1 over 2 ,~2 ]$, hence
using Theorem C (ii) we have
.DS 3
.EQ (2.36)
{sum pprime} from M~M sup -1 ~
f( 2n over {M sup 2} )~=~
c sub 3~+~O ( N sup {-~ 1 over 2} ) ,
.EN
.DE
where
.DS 3
.EQ
c sub 3~=~int from {1/2} to 2~
u sup -1~f(u sup -2 )~du .
.EN
.DE
Then (2.33) and (2.36) yield
.DS 3
.EQ (2.37)
w * (2n)~=~c sub 3~G sub s (2n)~+~
O ( M sup {- delta  pprime pprime} ) ,
.EN
.DE
with $delta pprime pprime~=~min ( 1 over 2 delta pprime ,~1 over 2 )$,
for $n$ in the range $epsilon~N~<=~n~<=~(1- epsilon )~N$, and
.DS 3
.EQ (2.38)
w sup sstar (2n)~<=~
c sub 3~G sub s (2n)~+~
O(N sup {- delta  pprime pprime} )
.EN
.DE
for $1~<=~n~<=~N$.
.P
A counting argument now shows that
.DS 3
.EQ (2.39)
r(n)~=~(2s+1)~w sup star (2n)~.
.EN
.DE
To prove (2.39), let
$sigma sub i$ be the cyclic permutation acting
on $y$ by
.DS 3
.EQ
sigma sub i (y sub j )~=~
y sub {i+j}~,
.EN
.DE
where subscripts are interpreted (mod $2s+1$).
Note that the definitions (2.28)-(2.30) guarantee
that
.DS 3
.EQ (2.40)
w ( sigma sub i ( yy ) )~=~
w( yy )
.EN
.DE
for all $sigma sub i$,
$0~<=~i~<=~2s$.
The condition that $yy$ has distinct
coordinates implies that
.DS 3
.EQ (2.41)
sigma sub i ( yy )~!=~yy
.EN
.DE
for $0~<=~i~<=~2s$.
Now each $w( yy )$ weights $2s+1~n sub i$'s,
(see (2.9)) but only the weights corresponding
to $n sub 0$'s are counted in
$w sup sstar (n)$.
The permutation $sigma sub i$ sends $yy$ to
$sigma sub i ( yy )$, and $n sub 0 ( sigma sub i ( yy ))~=~n sub i ( yy )$.
This gives a one-to-$(2s+1)$ weight-preserving map
(by (2.40)) from the set
$"{" ( yy ,~n sub 0 ( yy ))~|~roman all~ yy "}"$
\f2onto\f1
$"{" ( yy ,~n sub i ( yy ))~|~roman all~yy$,
$0~<=~i~<=~2s "}"$,
which proves (2.39).
.P
The lemma follows from (2.37)-(2.39),
with $c sub 2~=~c sub 3 (2s+1)$. \(sq
.P
Lemma 2.1 shows that the choice of weights
(2.29) has eliminated almost all fluctuations in the
resulting $r(n)$, except those due to the
singular series.
By increasing $s$ we can reduce the fluctuations
in the singular series given by (2.26).
In Lemma 4.3 we prove that for
$s~=~7$ and all $n$,
.DS 3
.EQ (2.42)
1.0085~>=~G sub 7 (n)~>=~0.9915 .
.EN
.DE
Now choosing $s~=~7$ and applying (2.42) with
Lemma 2.1 and (2.19), we obtain
.DS 3
.EQ
0.9915~c sub 2 Z~mark <=~
sum from n=1 to N~c sub 2~
G sub 7 (2n)~x sub n
.EN
.sp
.EQ
lineup <=~sum from n=1 to N~
r(n) x sub n~+~
1.0085~c sub 2 (2 epsilon N )~+~
O(N sup {1- delta pprime })
.EN
.sp
.EQ
lineup <=~7 over 15~
left ( sum from n=1 to N~r(n) right )~+~
1.0085 c sub 2 (2 epsilon N )~+~
O(N sup {1- delta pprime} )
.EN
.sp
.EQ
lineup <=~7 over 15 ~c sub 2
left ( sum from n=1 to N~G sub 7 (2n) right )~+~
2.017~c sub 2 epsilon N~+~
O(N sup {1- delta pprime} )
.EN
.sp
.EQ (2.43)
lineup <=~
c sub 2 (1.0085 )
( 7 over 15~+~2 epsilon ) N~+~
O( N sup {1- delta pprime} ) ~.
.EN
.DE
The term
$2.017~ c sub 2 epsilon N$ arises from
$n$ in the intervals
$[0,~epsilon N]$ and
$[(1- epsilon ) N,~N]$ where (2.32)
was used.
Choosing $epsilon~=~.0001~<~1 over 4(s+1)$
and dividing by $c sub 2$,
we obtain
.DS 3
.EQ
Z~<=~.4747~N~+~
O(N sup {1- delta pprime} ) .
.EN
.DE
For sufficiently large
$N~>~N sub 0$ it follows that
.DS 3
.EQ (2.44)
Z~<=~.475 ~N ,
.EN
.DE
the desired result.
$square$
.sp
\f2Remark\f1.
There are a number of ways to improve the above bound.
For example, it is easy to see that
.DS 3
.EQ
sum from n=1 to N~G sub s (2n)~=~
N~+~O(N) .
.EN
.DE
If we used this in the inequalities leading
to (2.43) instead of (2.42) we would obtain
.DS 3
.EQ
Z~<~.4707~N .
.EN
.DE
Further improvements are possible because $G sub s (2n)$
depends largely on the residue classes of $2n$ modulo
small prime powers, and so cannot be close to its
lower bound too often.
To get substantial improvements, however, we would
need to take $s$ much smaller than 7.
(The circle method as we use it here works only
for $s~>=~2$, but since we only need results
that hold for most values of $n$, rather than all $n$,
we could modify the method to work for $s=1$ as well.)
For small $s$, however, the $G sub s (2n)$ factors
oscillate wildly, and to smooth out the oscillations
we would need to consider weights $w( yy , M)$ that
depend on the congruence properties of
$y sub 0 , "...", y sub 2s$.
This can be carried out (cf. [7]), but the proofs
are quite cumbersome, and since it seems that they
would not yield a bound close to $11 over 32 ~N$, we
have not pursued this subject further.
.H 1 "Application of the circle method"
In this section we prove Theorem C
following a version of the
circle method 
incorporating improvements of I. M. Vinogradov, which is
described in Davenport [2, pp. 9-48].
Since this proof is a relatively routine
variant of the circle method, we shall
only sketch the details.
.sp
\f2Proof of Theorem C\f1.
We shall first estimate the number
of representations $r sub {M, epsilon} (n)$
of $(y sub 0 , . . . , y sub 2s )$ of 
.DS 3
.EQ (3.1)
n~=~sum from i=0 to 2s~(-1) sup i y sub i sup 2
.EN
.DE
for which
.DS 3
.EQ (3.2)
(1- epsilon ) M~<=~
y sub i ~<=~M ,~~~0 <= i <= 2s ,
.EN
.DE
where the $y sub i$ need not be distinct.
.P
We suppose $s~>=~2$ and $epsilon$ are fixed,
$0~<~epsilon~<~1 over {4(s+1)}$.
$M$ will be a large variable
integer, and we set
$M sub 1~=~ [ M(1- epsilon ) ]$.
.P
The circle method involves study of
the trigonometric sum
.DS 3
.EQ (3.3)
T( alpha )~=~
sum from {x= M sub 1} to M~
e( alpha x sup 2 ) ,
.EN
.DE
where
.DS 3
.EQ
e( alpha )~=~exp (2 pi alpha )~.
.EN
.DE
Clearly we have
.DS 3
.EQ (3.4)
r sub {M, epsilon} (n)~=~
int from 0 to 1~T( alpha ) sup s+1~
T( - alpha ) sup s~
e (-n alpha ) d alpha~.
.EN
.DE
We estimate this integral by dividing the
interval [0,1] into major and minor
arcs.
We take a parameter $delta$, to be chosen
later, which satisfies $0~<~delta~<~1 over 10$,
and define the major arcs
$m sub a,q$ 
to be the sets $m sub a,q$ with $1 <= q <= M sup delta$
and $(a,q)~=~1,~1 <= a <= q$, where
.DS 3
.EQ (3.5)
m sub {a,q}~=~
"{" alpha :~0 <= alpha < 1,~|~
alpha~-~a over q ~|~<~
M sup {-2 + delta} "}"~.
.EN
.DE
(We consider $alpha$ modulo 1 here.)
Let $U$ denote the union of the major arcs and let
$V$ be its complement, the minor arcs.
.P
We obtain the minor arcs estimate
as in Davenport [2, p. 20].
(Note Davenport's $s$ is our $2s+1$.)
.sp
.I
Lemma 3.1.
(Minor Arcs Estimate)
We have
.DS 3
.EQ (3.6)
int from V~|~
T( alpha )~| sup s+1~|~
T( - alpha ) | sup s~d alpha ~=~
O(M sup {2s-1- delta sub 1} ) ,
.EN
.DE
for a fixed $delta sub 1~>~0$
depending on $delta$.
.R
.P
An examination of Davenport's proof shows we
may take $delta sub 1~=~delta /2~+~eta$
for any fixed $eta~>~0$. 
.P
We next treat the major arcs.
Analogously to [2, Lemma 4, p. 22] we obtain:
.sp
.I
Lemma 3.2.
Let $alpha~member~m sub {a,q}$, and set
$beta~=~a/q~-~alpha$.
We have
.DS 3
.EQ
T( alpha )~=~
q sup -1~S sub {a,q}~
I( beta )~+~O (M sup {2 delta} )~,
.EN
.DE
where
.DS 3
.EQ (3.7)
S sub a,q~=~sum from k=1 to q~
e ( a over q k sup 2 )
.EN
.DE
is a Gaussian sum, and where
.DS 3
.EQ 
I( beta )~=~int from {M sub 1} to M~
e( beta t sup 2 )~dt .
.EN
.DE
.R
.P
Summing up over all the major arcs and
making a change of variables
leads to the following.
(See [2, Lemma 5, p. 23])
.sp
.I
Lemma 3.3.
(Major Arcs Estimate).
We have
.DS 3
.EQ
int from U~T( alpha ) sup s+1~
T( - alpha ) sup s~
e(-n alpha ) d alpha~mark =~
M sup 2s-1~
G(n, ~M sup delta )~J( n, M sup delta )
.EN
.sp
.EQ (3.8)
lineup +~O ( M sup {2s-1- delta sub 2} )~,
.EN
.DE
where $delta sub 2~=~1-5 delta~>~0$, and where
.DS 3
.EQ (3.9)
G(n,~M sup delta )~mark =~
sum from {q <= M sup delta }~
sum from { pile { a=1 above (a,q) =1}} to q~
q sup -2s-1~|
S sub a,q | sup 2s~
S sub a,q~e ( - na over q )
.EN
.DE
and
.DS 3
.EQ (3.10)
J(n, M sup delta )~=~
int from {| gamma | < M sup delta}~
H( gamma ) sup s+1~
H( - gamma ) sup s~
e ( - {n gamma} over {M sup 2} )~d gamma ~,
.EN
.DE
where
.DS 3
.EQ (3.11)
H( gamma )~=~int from {1- epsilon} to 1~
e ( gamma t sup 2 ) dt~.
.EN
.DE
.R
.P
We next approximate
$J(n, M sup delta )$.
We define
.DS 3
.EQ (3.12)
f(u)~=~
int from {- inf} to inf~
H( gamma ) sup s+1~
H( - gamma ) sup s~
e( - gamma u ) ~d gamma ~.
.EN
.DE
This integral converges, since
we have
.DS 3
.EQ (3.13)
H( gamma )~=~O ( | gamma | sup -1 )
.EN
.DE
as $gamma~->~inf$, a fact checked by letting
$x~=~t sup 2$ in (3.11) and integrating
by parts.
Comparing (3.12) and (3.11) using the
bound (3.13), absolute value estimates yield
.DS 3
.EQ (3.14)
J(n,~M sup delta )~=~
f( n over {M sup 2} )~+~
O( M sup {-2 delta s} ) ~.
.EN
.DE
.P
We claim $f(u)$ is a real-valued nonnegative
function which vanishes outside the interval
$I~=~I sub {s, epsilon}$ defined by
.DS 3
.EQ (3.15)
1-2 (s+1) epsilon~+~
(s+1) epsilon sup 2~<=~
u~<=~1~+~2 s epsilon~-~s epsilon sup 2~.
.EN
.DE
To see this, we note using (3.11) that
$H( gamma )$ is the Fourier integral transform
.DS 3
.EQ
H( gamma )~=~int from {- inf} to inf~
h( u)~e ( gamma u )~du
.EN
.DE
of
.DS 3
.EQ (3.16)
h(u)~=~left "{"
matrix {
ccol {1 over 2 u sup {- 1 over 2 ~~} above 0~~}
lcol { (1- epsilon ) sup 2~<=~u~<=~1 , above roman elsewhere,}
}
.EN
.DE
and $H( - gamma )$ the Fourier integral transform of
$h sup sstar (u)~=~h(-u)$.
But (3.12) says $f(u)$ is the inverse Fourier
integral transform of
$H( gamma ) sup s+1 H( - gamma ) sup s$,
which implies that $f(u)$ is given
by the repeated convolution
.DS 3
.EQ (3.17)
f(u)~=~
[ h sub 1 (u) star . . . star h sub s+1 (u) ]~star~
[h sub s+2 ( u ) star . . . star h sub 2s+1 (u) ] ~,
.EN
.DE
where
.DS 3
.EQ
h sub i (u)~=~ left "{"
matrix {
ccol { h(u) above h sup sstar (u) }
lcol { 1~<=~i~<=~s+1, above s+2~<=~i~<=~2s+1.}
}
.EN
.DE
Using the definitions of $h(u)$ and
$h sup sstar (u)$, the expression (3.17)
shows $f(u)$ is real and nonnegative,
and that it vanishes outside the
interval (3.15).
The fact that $s~>=~2$ shows that
$f(u)$ in (3.17) is continuously differentiable,
since $h(u)$ and $h sup sstar (u)$ are
piecewise continuous.
.P
We next approximate
$G(n,~M sup delta )$ by the singular
series $G sub s (n)$ defined by
.DS 3
.EQ (3.18)
G sub s (n)~=~
sum from q=1 to inf~
q sup -2s-1
sum from { pile { a=1 above (a,q) =1}} to q ~|
S sub a,q | sup 2s~S sub a,q~e ( -~ na over q ) ~,
.EN
.DE
provided this sum converges absolutely.
Since $S sub a,q$ is a Gaussian sum,
we have
.DS 3
.EQ (3.19)
S sub a,q~=~( a over q ) S sub 1,q ,
.EN
.DE
where $( a over q )$ is the Jacobi symbol, and we can
rewrite $G sub s (n)$ as
.DS 3
.EQ (3.20)
G sub s (n)~=~sum from {q=1} to inf~
q sup -2s~| S sub 1,q | sup 2s~A sub q (n) ,
.EN
.DE
where
.DS 3
.EQ (3.21)
A sub q (n)~=~q sup -1
sum from { pile { a=1 above (a,q)=1}} to q~
S sub a,q~e ( -~ na over q )~.
.EN
.DE
The quadratic Gaussian sum is explicitly evaluated
to be (e.g., [1, Theorem 4.15]):
.DS 3
.EQ (3.22)
S sub 1,q~=~
left "{"
matrix { 
ccol{ (1+i) sqrt q above sqrt q above 0 above i sqrt q}
lcol { q == 0~(mod ~4), above q == 1~(mod~4), above q == 2~(mod~4), above q == 3~(mod~4).}
}
.EN
.DE
Hence
.DS 3
.EQ 
| S sub a,q |~<=~sqrt 2q .
.EN
.DE
Absolute value estimates show that
.DS 3
.EQ
| A sub q (n) |~<~sqrt 2q .
.EN
.DE
and hence that (3.18) converges for $s >= 2$, and
that for all $n$
.DS 3
.EQ (3.23)
| G sub s (n) |~<~
c sub 1 (s)~=~2 sup s+1~
zeta ( s-~ 1 over 2 )~.
.EN
.DE
Similar absolute value estimates give
.DS 3
.EQ (3.24)
G sub s (n,~M sup delta )~=~
G sub s (n)~+~O( 2 sup s+1
M sup {- delta ( s- 3 over 2 )} )~.
.EN
.DE
.P
Combining lemmas 3.2 and 3.3 with (3.14),
(3.23) and (3.24) we obtain
for $s~>=~2$ that
.DS 3
.EQ (3.25)
r sub {M, epsilon} (n)~=~
M sup 2s-1~G sub s (n)~
f( n over {M sup 2} )~+~
O ( M sup {2s-1- delta pprime} ) ,
.EN
.DE
where $delta pprime~=~min [ 2 delta s ,~
delta (s-~ 3 over 2 )~+~eta ,~
1-5 delta , ~delta over 2 + eta ]$ for
any $eta~>~0$.
We can choose $delta pprime~=~1 over 12$ by
taking $delta$ slightly exceeding $1 over 6$.
Here $G sub s (n)$ and $f(x)$ satisfy
(i), (ii) of Theorem C,
by (3.15) and an earlier remark.
.P
Theorem C will be proved if we show that
for $s~>=~2$,
.DS 3
.EQ (3.26)
r sub {M, epsilon} sup sstar (n)~=~
r sub {M, epsilon} (n)~+~
O( M sup 2s-2+1/10 ) .
.EN
.DE
Let $r sub ij (n)$ denote the number of solutions to
(3.1), (3.2) with
$y sub i~=~y sub j$, so that
.DS 3
.EQ (3.27)
r sub {M, epsilon} (n)~-~
r sub {M, epsilon} sup sstar (n)~<=~
sum from i=0 to 2s ~~
sum from {j~!=~i}~
r sub ij (n) .
.EN
.DE
But $r sub ij (n)$ is exactly the number of solutions
to 
.DS 3
.EQ
Q sub ij ( yy )~=~n ,
.EN
.DE
where $Q sub ij$ is a diagonal quadratic form
in either $2s$ or $2s-1$ variables
and all the variables satisfy
.DS 3
.EQ
M(1- epsilon )~<=~
y sub i~<=~M.
.EN
.DE
If we now fix all but 2 of the variables, we will
have $O(M sup eta )$ solutions for each $eta >0$, which
yields the desired result.
\(sq
.H 1 "The Singular Series"
In this section we evaluate the singular series
$G sub s (n)$ in Theorem C,
in order to obtain the bound (2.44).
Again the method is standard, as in [2].
.P
We first examine the expressions $A sub q (n)$
given in (3.21).
.sp
.I
Lemma 4.1.
We have
.DS 3
.EQ (4.1)
A sub q (n)~=~
{phi (q)} over q~
sum from x=1 to q~
{ mu ( {q} over {(q, x sup 2 -n )} ) } over
{phi ( {q} over {(q, x sup 2 - n)} )}~.
.EN
.DE
Moreover, $A sub {q sub 1 q sub 2} (n) ~=~A sub q sub 1 (n) A sub q sub 2 (n)$
if $(q sub 1 , q sub 2 ) ~=~1$.
.R
.sp
\f2Proof\f1.
From (3.21) we obtain
.DS 3
.EQ
A sub q (n)~mark =~
q sup -1 sum from k=1 to q~
sum from { pile { a=1 above (a,q) = 1}} to q~
e ( a over q ( k sup 2 - n ))
.EN
.sp
.EQ (4.2)
lineup =~
q sup -1~sum from k=1 to q~
c sub q ( k sup 2 - n ) ,
.EN
.DE
where $c sub q (m)$ is Ramanujan's sum.
This sum can be evaluated explicitly
[5, Theorem 272] as
.DS 3
.EQ (4.3)
c sub q (m)~=~mu ( q over d )
{phi (q)} over {phi ( q over d )} ,
.EN
.DE
where $d~=~(q,m)$.
This proves (4.1).
The multiplicativity of $A sub q (n)$ follows from the
multiplicativity of $mu (k)$ and $phi (k)$.
\(sq
.P
We can now represent
$G sub s (n)$ as an Euler product,
using (3.20) and noting that
$q sup -2s~|~S sub 1,q~| sup 2s$ is multiplicative.
We obtain
.DS 3
.EQ (4.4)
G sub s (n)~=~prod from p~
(1~+~xi sub p (n) )~,
.EN
.DE
where the product is over all primes $p$, 
and where by
(3.22) we find for odd $p$ that
.DS 3
.EQ (4.5)
xi sub p (n)~=~
sum from k=1 to inf~
p sup -ks~
A sub {p sup k} (n) ,
.EN
.DE
and that
.DS 3
.EQ
xi sub 2 (n)~=~
sum from k=1 to inf~
2 sup {-(k-1)s} A sub {2 sup k} (n)~.
.EN
.DE
.P
The next step is to evaluate explicitly
all the $A sub {p sup k} (n)$.
.sp
.I
Lemma 4.2.
If $p$ is an odd prime then
$A sub p (n)~=~( n over p )$ and for
$k~>=~1$,
.DS 3
.EQ
A sub {p sup 2k+1}~(n)~mark =~
left {
matrix {
ccol { left ( {n/p sup 2k} over {p} right ) p sup k above nothing above 0}
lcol { \f2if\f1 ~~~~p sup 2k | n , above nothing above otherwise,}
}
.EN
.sp
.EQ
A sub {p sup 2k} (n)~lineup =~
left "{"
matrix {
ccol {p sup k-1 (p-1) above nothing above -p sup k-1 above nothing above 0}
lcol { \f2if\f1~~~p sup 2k | n, above nothing above \f2if\f1~~~(p sup 2k -1) || n, above nothing above otherwise.}
}
.EN
.DE
If $p~=~2$ then
$A sub 2 (n)~=~0$ and for
$k~>=~1$
.DS 3
.EQ
A sub {2 sup 2k+1} (n)~mark =~
left "{"
matrix {
ccol { 2 sup k above nothing above -2 sup k above nothing above 0}
lcol { \f2if\f1~~n~==~2 sup 2k-2 ~~(mod~2 sup 2k+1 ), above nothing above \f2if\f1~~n~==~2 sup 2k~+~2 sup 2k-2~~(mod~2 sup 2k+1 ), above nothing above otherwise,}
}
.EN
.sp
.EQ
A sub {2 sup 2k} (n)~lineup =~
left "{"
matrix {
ccol { 2 sup k-1 above nothing above -2 sup k-1 above nothing above 0}
lcol { \f2if\f1~~n~==~0,~2 sup 2k-2~~(mod~2 sup 2k ), above nothing above \f2if\f1~~ n~==~2 sup 2k-1 ,~3.2 sup 2k-2~~(mod~2 sup 2k ) , above nothing above otherwise.}
}
.EN
.DE
.sp
Proof.
.R
These are derived from (4.1).
The only nonzero contributions to
$A sub q (n)$ for
$q = p sup k$ come from those $x$ with
$(q, x sup 2 -q )~=~q$ and
$(q,~x sup 2 -q )~=~p sup k-1$,
and so
.DS 3
.EQ (4.6)
A sub {p sup k} (n)~=~
p sup -1 "{"
p~A sub {p sup k} sup pprime (n)~-~
A sub {p sup k} sup {pprime pprime} (n) "}" ,
.EN
.DE
where
.DS 3
.EQ
A sub {p sup k} sup pprime (n)~mark =~
| "{"
x~:~ 1 ~<=~x~<=~p sup k~,~
x sup 2~==~n~(mod~p sup k ) "}" |.
.EN
.sp
.EQ
A sub {p sup k} sup {pprime pprime} (n)~lineup =~
| "{" x~:~
1~<=~x~<=~p sup k ~,~x sup 2~==~n~
(mod~p sup k-1 ) "}" |.
.EN
.DE
.P
Suppose $p$ is odd.
Then $A sub p sup pprime (n)~=~1,~2$ or 0
according as $p | n$, $n$
is a quadratic residue (mod p ),
or $n$ is a quadratic nonresidue (mod p),
respectively.
On the other hand
$A sub p sup {pprime pprime} (n)~=~p$ in all cases.
Hence $A sub p (n)~=~
( n over p )$ using (4.6).
We next treat $A sub {p sup 2} (n)$.
Here $A sub {p sup 2} sup pprime (n)~=~p , 0, 2, 0$
in the four cases $p sup 2 | n$, $p || n$,
$p brsl n$ and
$( n over p )~=~1$,
$p brsl n$ and $( n over p )~=~-1$, respectively.
Similarly $A sub {p sup 3} sup {pprime pprime} (n)~=~p, p, 2p , 0$.
If $p sup 2 brsl n$ then
$A sub {p sup k} sup pprime (n)~=~A sub {p sup k} sup {pprime pprime} (n)~=~0$.
If $p sup 2 | n$, however, then
.DS 3
.EQ
A sub {p sup k} sup pprime (n)~mark =~
p~A sub {p sup k-1} sup pprime ( n over {p sup 2} ) ,
.EN
.sp
.EQ
A sub {p sup k} sup {pprime pprime} (n)~lineup =~
p~A sub {p sup k-2} sup {pprime pprime}
( n over {p sup 2} ) .
.EN
.DE
The lemma follows for odd $p$.
The analysis for $q = 2 sup k$ is
similar and is omitted. \(sq
.P
We now estimate $G sub 7 (n)$.
(Note that somewhat better estimates can be obtained
by bounding $xi sub p (n)$ from above and below,
instead of estimating $| xi sub p (n) |$.)
.sp
.I
Lemma 4.3.
For all $n$,
.DS 3
.EQ (4.7)
1.0085~>=~G sub 7 (n)~>=~ 0.9915.
.EN
.DE
Proof.
.R
Using Lemma 4.2, we have
for odd $p$ that
.DS 3
.EQ
| A sub {p sup k} (n) |~<=~
left "{" ~
matrix {
lcol { p sup {k over 2 ~-~ 1 over 2} above nothing above p sup {k over 2}}
lcol {if~~2 brsl k, above nothing above if~~2|k.}
}
.EN
.DE
Applying this when $s~=~7$ in (4.5) we obtain
.DS 3
.EQ
| xi sub p (n) |~mark <=~
sum from k=1 to inf~
p sup {-7(2k-1) + (k-1)}~+~
sum from k=1 to inf~
p sup -7(2k)+k
.EN
.sp
.EQ (4.8)
lineup =~
{p sup -7} over {1-p sup -13}~+~
{p sup -13} over {1-p sup -13}~=~
{p sup 6~+~1} over {p sup 13~-~1} .
.EN
.DE
Also from Lemma 4.2,
.DS 3
.EQ
A sub {2 sup k} (n)~<=~
left "{" ~
matrix {
lcol { 2 sup {k over 2 ~-~ 1 over 2} above nothing above 2 sup {k over 2 ~-1}}
lcol { if~~2 brsl k, above nothing above if~~2|k.}
}
.EN
.DE
Hence using $A sub 2 (n)~=~0$ we obtain
.DS 3
.EQ
| xi sub 2 (n) |~mark <=~
sum from k=1 to inf~
2 sup {-7(2k) + k}~+~
sum from k=1 to inf~
p sup -7(2k-1)+(k-1)
.EN
.sp
.EQ
lineup <=~{2 sup -13} over {1-2 sup -13}~+~
{2 sup -7} over {1-2 sup -13}
.EN
.sp
.EQ (4.9)
lineup <=~
{2 sup 6 +1} over {2 sup 13 -1} ~.
.EN
.DE
Hence
.DS 3
.EQ
G(n) ~mark <=~ 8260 over 8195 ~prod from pile {p above p >= 3} ~
left ( 1+~ {p sup 6 +1} over {p sup 13 -1} right ) ~<=~
1.0085~,
.EN
.sp
.EQ
G(n) ~lineup >= ~8130 over 8195 ~prod from pile {p above p >= 3} ~
left ( 1-~
{p sup 6 +1} over {p sup 13 -1} right ) ~>=~0.9915 ~. ~~~~~~(sq
.EN
.DE
.SG gm
.bp
.ce
References
.sp
.RL
.LI
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.LI
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.LI
P. Erdo\*:s, Problem 268, Problems from
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.LI
P. Erdo\*:s and R. L. Graham,
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.LI
G. H. Hardy and E. M. Wright,
\f2An Introduction to the Theory of Numbers\f1
(4th Edition),
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.LI
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I.
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.R
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.LI
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.LI
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.LI
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.LE
.ls 1
.CS
.ls 1