INTEGER NUMBER SOLUTIONS OF LINEAR SYSTEMS Florentin Smarandache, Ph D Associate Professor Chair of Department of Math & Sciences University of New Mexico 200 College Road Gallup, NM 87301, USA E-mail:
[email protected]
Definitions and Properties of the Integer Solution of a Linear System Let’s consider n
aij x j
(1)
bi , i 1, m
j 1
a linear system with all coefficients being integer numbers (the case with rational coefficients is reduced to the same).
x 0j , j 1,n is a particular integer solution of (1) if
Definition 1. x j
x 0j
n
, j 1,n and
aij x 0j
bi , i 1, m .
j 1
Let’s consider the functions f j : Definition 2. x j
h
, j 1,n , where h
*
.
f j (k1,..., kh ), j 1,n is the general integer solution for (1) if:
n
aij f j (k1 ,..., kh ) bi , i 1, m , irrespective of k1 ,..., kh
(a)
Z;
j 1
(b)
Irrespective of x j
x 0j , j
1,n a particular integer solution of (1) there is
(k10 ,..., kh0 ) Z such that f j (k10 ,..., kh0 ) x j , j 1,n . (In other words the general solution that comprises all the other solutions.) Property 1. A general solution of a linear system of m equations with n unknowns, r(A) m n , is undetermined n m -times. Proof: We assume by reduction ad absurdum that it is of order r , 1 r n m (the case r 0 , i.e., when the solution is particular, is trivial). It follows that the general solution is of the form:
1
(S1)
x1
u11 p1 ... u1r pr
v1
: xn
un1 p1 ... unr pr
vn , uih , i Z
ph
parameters Z
We prove that the solution is undetermined n m -times. The homogeneous linear system (1), resolved in r has the solution: Dm1 1 Dn1 x1 xm 1 ... xn D D :
Dmm 1 Dnm xm 1 ... xn D D Let xi xi0 , i 1,n , be a particular solution of the linear system (1). Considering xm 1 D km 1 xm
: xn
D kn
we obtain the solution Dm1
x1
1
km
1
... Dn1 kn
x10
: Dmm 1 km
xm xm
D km
1
... Dnm kn
1
1
xm0
xm0
1
: xn
D kn
xn0 ,
kj
parameters Z
which depends on the n m independent parameters, for the system (1). Let the solution be undetermined n m -times: x1
c11k1 ... c1n m kn
m
d1
:
(S2)
xn
cn1k1 ... unn m kn
cij , di
Z, k j
m
dn
parameters Z
(There are such solutions, we have proved it before.) Let the system be: a11 x1 ... a1n xn b1
: am1 x1 ... amn xn
xi = unknowns I.
, aij , bi
The case bi
bm
Z.
0, i 1, m results in a homogenous linear system: 2
ai1xi ... ain xn 0; i 1, m . ai1 ci1k1 ... c1n m kn m d1
(S2)
0
ai1c11 ... ain cn1 k1 ...
... ain cn1k1 ... cnn m kn ai1c1n
m
... ain cnn
m
kn
m
dn
0
ai1d1 ... ain dn
m
kj
For k1 ... kn m For k1 ... kh 1
0
kh
ai1cih ... ain cnh ai1cih The vectors
... ain cnh
0,
ai1d1 ... ain d n 1
... kn
m
0 and kh 1
ai1d1 ... ain dd(n) i 1, m,
0.
0
h 1,n m .
c1h : Vh , h 1, n m : cnh are the particular solutions of the system. Vh , h 1,n m also linearly independent because the solution is undetermined n m -times V1 ,...,Vn m d is a linear variety that includes the solutions of the system obtained from (S2). Similarly for (S1) we deduce that U1s : , s 1, r Us : U ns are particular solutions of the given system and are linearly independent, because (S1) is V1 : undetermined n m -times, and V is a solution of the given system. : Vn Case (a) U1,...,Ur , V = linearly dependent, it follows that U1 ,..., U r is a free sub-module of order r n - m of solutions of the given system, then, it follows that there are solutions that belong to V1 ,...,Vn m d and which do not belong to U1 ,..., U r , a fact which contradicts the assumption that (S1) is the general solution. Case (b) U1,...,Ur , V = linearly independent. U1 ,...,U r +V is a linear variety that comprises the solutions of the given system, which were obtained from (S1). It follows that the solution belongs to V1 ,...,Vn
3
m
d and does
not belong to U1 ,...,U r +V , a fact which is a contradiction to the assumption that (S 1) is the general solution. II. When there is an i 1, m with bi 0 then non-homogeneous linear system
ai1xi ... ain xn b1, i 1, m ai1 c11k1 ... c1n m kn m d1 ... ain cn1k1 ... cnn m kn
(S2) it follows that ai1c11 ... ain cn1 k1 ... ai1c1n m ... ain cnn ai1d1 ... ain d n b1 ; For k1 ... kn m 0 For k1 ... k j 1 k j 1 ... kn m 0 and k j 1 ai1c1 j ... ain cnj
ain d1 ... ain d n
ai1c1 j ... ain cnj
0
ai1d1 ... ain d n
bi
;
m
kn
m
dn
bi
ai1d1 ... ain dn
m
bi
bi it follows that i 1, m,
j 1, n m .
c1 j Vj
: ,j cnj
1,n m , are linearly independent because the solution (S2) is
undetermined n m -times.
d1 (?!)
V j , j 1, n m , and d
: dn
are linearly independent. We assume that they are not linearly independent. It follows that s1c11 ... sn m c1n m . d s1V1 ... sn mVn m :
s1cn1 ... sn m cnn Irrespective of i 1, m : b1 ai1d1 ... ain dn
ai1 s1c11 ... sn m c1n
ai1c11 ... ain cn1 s1 ....
ai1c1n
m
m
... ain cnn
m
... ain s1cn1 ... sn m cnn m
sn
m
m
0.
Then, bi 0 , irrespective of i 1, m , contradicts the hypothesis (that there is an i 1, m , bi 0 ). It follows that V1,...,Vn m , d are linearly independent. V1 ,...,Vn m d is a linear variety that contains the solutions of the nonhomogeneous system, solutions obtained from (S2). Similarly it follows that G1 ,..., Gr +V is a linear variety containing the solutions of the non-homogeneous system, obtained from (S1). n - m r it follows that there are solutions of the system that belong to __________________________ “?!” means “to prove that” 4
V1 ,...,Vn m d and which do not belong to G1 ,..., Gr +V , this contradicts the fact that (S1) is the general solution. Then, it shows that the general solution depends on the n m independent parameters.
Theorem 1. The general solution of a non-homogeneous linear system is equal to the general solution of an associated linear system plus a particular solution of the nonhomogeneous system. Proof: Let’s consider the homogeneous linear solution: a11 x1 ... a1n xn 0 :
, ( AX
am1 x1 ... amn xn 0 with the general solution: x1 c11k1 ... c1n m kn
0)
d1
m
: xn
cn1k1 ... cnn m kn
x1
x10
dn
m
and
: xn xn0 with the general solution a particular solution of the non-homogeneous linear system AX b ; x1 (?!)
c11k1 ... c1n m kn
m
x10
d
: xn0
xn cn1k1 ... cnn m kn m d n is a solution of the non-homogeneous linear system. We note: a11 ... a1n x1 b1
A
:
am1 ... amn (vector of dimension m ), k1 K : ,C kn m
, X
:
, b
xn c11 ... c1n : cn1 ... cnn
AX A Ck d x 0 A Ck d We will prove that irrespective of
:
0 , 0
bm m
,d m
AX 0
5
: 0
d1
x10
: , x0 dn
:
b 0 b
xn0
;
y10
x1 :
xn yn0 there is a particular solution of the non-homogeneous system
k10
k1
Z ,
: kn
kn0
m
m
Z
with the property:
c11k10 ... c1n kn0
x1
m
d1 x10
y10
: cn1k10 ... cnn m kn0
xn
m
d1 xn0
yn0
y10 We note Y 0
:
.
yn0 We’ll prove that those k 0j (there are such X 0j
, j 1,n m are those for which A CX 0
d
0
because
x1
0
: xn 0 is a particular solution of the homogeneous linear system and X CK d is a general solution of the non-homogeneous linear system) A CK 0 d X 0 Y 0 A CK 0 d AX 0 AY 0 0 b b 0 . Property 2 The general solution of the homogeneous linear system can be written under the form: (SG) x1 c11k1 ... c1n m kn m (2)
:
xn cn1k1 ... cnn m kn m k j is a parameter that belongs to (with d1 d2 ... dn 0 ). Poof: (SG) = general solution. It results that (SG) is undetermined (n m) -times. Let’s consider that (SG) is of the form
6
x1
c11 p1 ... c1n
m
pn
d1
m
:
(3)
xn cn1 p1 ... cnn m pn m dn with not all di 0 ; we’ll prove that it can be written under the form (2); the system has the trivial solution x1 0 ; :
xn it results that there are p j
, j 1,n m ,
x1 (4)
0 c11 p10 ... c1n m pn0
m
d1
0
: xn
Substituting p j
kj kj p 0j pj p
0 j
cn1 p10 ... cnn m pn0
dn
m
0
p0j , j 1,n m in (3) pj kj
,
pj
p 0j
which means that that they do not make any restrictions. It results that x1 c11k1 ... c1n m kn m c11 p10
... c1n
m
pn0
m
d1
: xn
cn1k1 ... cnn m kn
cn1 p10
m
... cnn
m
pn0
m
But
ch1 p10 ... chn m pn0 m dh Then the general solution is of the form: x1 c11k1 ... c1n m kn m : xn k j = parameters
cn1k1 ... cnn m kn
0, h 1,n (from (4)).
m
, j 1,n m ; it results that d1
d2
... dn
Theorem 2. Let’s consider the homogeneous linear system: a11 x1 ... a1n xn 0
: am1 x1 ... amn xn r(A)
m , ah1,...,ahn
,
0
1 , h 1, m and the general solution 7
0.
dn
x1
c11k1 ... c1n m kn
m
: xn
cn1k1 ... cnn m kn
m
then ah1 ,..., ahi 1 , ahi 1 ,..., ahn
ci1 ,..., cin
m
irrespective of h 1, m and i 1, n . Proof: Let’s consider some arbitrary h 1, m and some arbitrary i 1, n ; ah1x1 ... ahi 1xi 1 ahi 1xi 1 ... ahn xn ahi xi . Because ah1 ,..., ahi 1 , ahi 1 ,..., ahn ahi it results that d
ah1 ,..., ahi 1 , ahi 1 ,..., ahn xi
irrespective of the value of xi in the vector of particular solutions. For k2 k3 ... kn m 0 and k1 1 we obtain the particular solution:
For k1 solution:
k2
x1 :
c11
xi :
ci1
xn
cn1
... kn
d | ci1
0 and kn
m 1
x1 :
c1n
m
xi :
cin
m
xn
cnn
d | cin
m
m
1 it results the following particular
;
m
hence
d | cij , j 1,n m
d ci1,...,cin
Theorem 3. If
x1
c11k1 ... c1n m kn
m
: xn
cn1k1 ... cnn m kn
8
m
m
.
, cij
k j = parameters
being given, is the general solution of the homogeneous
linear system
a11 x1 ... a1n xn
0 , r(A)
: am1 x1 ... amn xn then c1 j ,..., cnj
1,
m
n
0
j 1,n m .
Proof: We assume, by reduction ad absurdum, that there is j0 1, n m : c1 j0 ,..., cnj0
d
we consider the maximal co-divisor 0 ; we reduce to the case when the maximal codivisor is d to the case when it is equal to d (non restrictive hypothesis); then the general solution can be written under the form: x1 c11k1 ... c1' j0 dk j0 ... c1n m kn m :
(5)
xn
where d
cij0 ,.., cnj0 , cij0
cn1k1 ... cnj' 0 dk j0 ' ij0
' ij0
... cnn m kn ' nj0
d c and c ,..., c
m
1.
We prove that x1
c1' j0
: xn
cnj' 0
is a particular solution of the homogeneous linear system. We’ll note: k1 ' c11 ... cij0 d ... c1n m : C
:
: ' nj0
cn1 ... c
: d ... cnn
,k
k j0 :
m
kn
m
x C k the general solution. a11 ... a1n We know that AX
0
A(CK )
0, A
:
.
an1 ... amn We assume that the principal variables are x1,..., xm (if not, we have to renumber). It follows that xm 1,..., xn are the secondary variables. For k1 ... k j0 1 k j0 1 ... kn m 0 and k j0 1 we obtain a particular solution of the system
9
x1
c1' j0 d
: xn
c1' j0 d 0
c1' j0
A :
cnj' 0 d
c1' j0
d A :
cnj' 0 d
x1
A :
cnj' 0
0
c1' j0
:
cnj' 0
xn
cnj' 0
is the particular solution of the system. We’ll prove that this particular solution cannot be obtained by x1 c11k1 ... c1' j0 dk j0 ... c1n m kn m c1' j0 (6)
: cn1k1 ... cnj' 0 dk j0
xn xm
(7)
... cnn m kn
cm 1k1 ... cm' 1dk j0
1
... cm
cnj' 0
m
k
1, n m n m
cm'
1 j0
: xn
cn1k1 ... cnj' 0 dk j0
cm
1,1
... cm
: k j0
ch ,1 cm
1,1
... c
: ...
... cm 0.
:
d ... cm 0.
' nj
cnj' 0
m
1, n m
cnj ... cn ,n ' m 1 j0
: ch ,1
1j
: ...
... cnn m kn
m
1, n m
1 d
Z
:
c d ... cn ,n
m
(because d 1 ). It is important to point out the fact that those k j k 0j , j 1,n m , that satisfy the system (7) also satisfy the system (6), because, otherwise (6) would not satisfy the definition of the solution of a linear system of equations (i.e., considering the system (7) the hypothesis was not restrictive). From X j0 follows that (6) is not the general solution of the homogeneous linear system contrary to the hypothesis); then c1 j ,..., cnj 1, irrespective of j 1,n m . Property 3. Let’s consider the linear system a11 x1 ... a1n xn b1
: aij ,bi
am1 x1 ... amn xn bm , r(A) m n , x j = unknowns Resolved in , we obtain
10
x1
f1 xm 1 ,..., xn
:
, x1 ,..., xm are the main variables,
xm
fm xm 1 ,..., xn
where fi are linear functions of the form: cmi 1 xm 1 ... cni xn fi di where cmi j , di , ei
ei
,
; i 1, m, j 1,n m .
ei irrespective of i 1, m then the linear system has integer solution. di Proof: For 1 i m, xi , then f j . Let’s consider If
xm
1
u m 1km
: xn :
u n kn
x1
v1m 1km
1
1
... v1n kn
e1 d1
: em dm is the maximal co-divisor of the denominators of the fractions xm
vmm 1km
1
... vnm kn
a solution, where um 1 cmi j , i 1, m, j 1, n m calculated after their complete simplification. di vmi n
cmi j um j
di
j
is a solution undetermined (n m) -times which depends on
m independent parameters km 1,..., kn but is not a general solution.
Property 4. Under the conditions of property 3, if there is an ei0 ei i0 1, m : fi0 umi0 1 xm 1 ... uni0 xn with umi0 j , j 1, n m , and 0 di0 di0 system does not have integer solution. Proof: . xm 1,..., xn in , it results that xi0 Theorem 4. Let’s consider the linear system 11
then the
a11 x1 ... a1n xn
b1
: aij ,bi
ih
am1 x1 ... amn xn bm , x j = unknowns , r(A) m n . If there are indices 1 i1
... im
n,
1, 2,.., n , h 1, m , with the property:
a1i1 ... a1im :
0 and
:
ami1 ... amim b1 a1i2 ... a1im xi1
:
:
:
is divided by
bm ami2 ... amim . . .
a1i1 ... a1im 1 b1 xim
: : : is divided by ami1 ... amim 1 bm
then the system has integer number solutions. Proof: We use property 3 di , i 1, m; eih xi , h 1, m h
Note 1. It is not true in the reverse case. Consequence 1. Any homogeneous linear system has integer number solutions (besides the trivial one); r(A) m n . Proof: 0 : , irrespective of h 1, m . xi h
1 , it follows that the linear system has integer number Consequence 2. If solutions. Proof: xi : ( 1) , irrespective of h 1, m ; h
xih
.
12
