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1980]
ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS
these cannot vanish.
165
Therefore9
So 5 CQ = 0 implies
Hence, by Favardfs Theorem, these Rn must be an orthogonal set with respect
to some weighting function w1(x) and some range [o9 d] if the integral be
considered a Stieltjes integrals
If C0 i1 0 S the Rn do not satisfy a three-term recursion formula (unless
n = 2) and by applying the contrapositive of the converse, we see that the Rn
cannot be an orthogonal set with respect to any weighting function and range.
REFERENCES
1.
2.
J. Favard. "Sur les polynomes de Tchebicheff." Comptes Rendus Acad. Sei.
Paris 200 (1935):2052-2053.
G. Szego. Orthogonal
Polynomials.
Colloquim Publication Volume XXIII.
New York: The American Mathematical Society9 1939.
ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS INCLUDING THE
ALGEBRAIC SUM OF TRIANGULAR NUMBERS
Research
ANDRZEJ WlgCKOWSKI
Fellow of the Alexander von Homboldt
at the University
of Saarbrucken
Foundation
The natural number of the form
tn
=
In + 1\
1/ , 1N
V 2 )= I n ( n ) s
where n is a natural number, is referred to as the nth triangular number.
The aim of this work is to give solutions of some equations and systems of
equations in triangular numbers.
1.
THE EQUATION ttx + tty = tta
It is well known that the equation
(1)
tx + ty = tz
has infinitely many solutions In triangular numbers txs ty9 and t z .
amples it follows immediately from the formula:
(2)
*(2n + l)fc+
t
^tnk + n=
For ex-
Wl)Hn*
We can ask whether there exists a solution of the equation:
The answer to this question is positive9 because there exist two solutions:
tt„+
tt??=
tts3
and
**,„, + t t l l , = t t 2 i e .
ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS
166
[April
The problem of finding all the solutions to equation (1) above will be
solved in a subsequent paper.
2.
TRIPLES OF TRIANGULAR NUMBERS, THE SUMS OR DIFFERENCES
OF ANY TWO OF WHICH ARE ALSO TRIANGULAR NUMBERS
The system of three equations:
(4)
t x
^~ ^ y
— U
tx
+ tz
=
ty + 2tz
U9
tV9
— tq ,
has infinitely many solutions in triangular numbers tx> ty9 tZ9 tU9 tv , and
tq.
This theorem can be proved by insertion of the following formulas into
equations (4):
x = n,
(5)
y = j(tn
- 3),
z = tn
l
- 1,
3
u = -^(tn
+ 1),
v = tn,
<7 = -2"(tn - 1 ) ,
where n is a natural number of the form 4/c + 1 or 4fc + 2 for natural k.
In particular, putting n = 14, we have:
# = 14, z/ = 51, z = 104,
w = 53, y = 105, <? = 156.
Since tq - tz
equations:
= t15S
tx
(D)
= t l l s = tW9
- t10k
iri t n e
+ ty ~ tU9
^x
+ ^3
ty
+ tz
=
we obtain a solution of the system of
numbers:
tlh
+ t51 = t53,
^14~^^'104=:^;105S
^y>
= £w,
t 5 1 + £10if
=
^116*
We see that there exists a triple of triangular numbers whose sums in pairs
are also triangular numbers. The problem of whether there exist three different triangular numbers, the sum of any two of which is a triangular number
was formulated by W. Sierpinski [1].
ThdOtiQJMi Suppose that x > y > z;
(7.1)
(7.2)
tx
*~ ty
=
tU9
tx
+ tM -
tv,
ty
~> t z
— tw I
tx "•" ty = tu ,
tx + tz - tv9
ty
(7.3)
then each of the systems of equations:
- tz
= tW9
tx "F ty
== tu ,
tx
- t,
= tv,
ty
+ tz
=-- tw,
where x + w9 y + V;
where x £ w, y f
v;
1980]
ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS
(7.4)
- *v =
~£x + U =
ty + tz =
£x
- ty =
*x - tz =
h + tz =
£ x
(7.5)
~tx
(7.6)
- ty =
^Us
s
VV
where x ± w9 z ± u;
t^W9
^U
5
Uv
s
~^W
5
where
x ^ w
9
y ^ v
9
z ^ u ;
til 9
tx + tz
-
ty
= t y 9 where x + W 9 y + V9
- tz
167
tv9
+ *V = t-U 3
£x ~ tz =
^ - tz = ^ W )
£x
(7.7)
^
5
- ty = ^U S
^x - tz = ty s
= ty)}
t w , where x ^ w
** - t* =
^x
(7.8)
9
y ^ v
9
z ^ u ;
has infinitely many solutions in triangular numbers tX9ty9
tZ9tU9
tV9andtw.
Vfuoofc We prove even more. Each of the following systems of equations has
infinitely many solutions in natural numbers x and y.
(8.1)
"^ 16x +2
+
^16x4-2
+
(8.2)
(8.3)
^15x + 3 "
(8.4)
^12x + 2
^12x+2
=
^\9x + 2
=
^12x+2
"*" ^ 9cc + 2
=
^16x+2
+
^13x+2
=
^16x+2
+
^12x42
=
^13x42
~
^12x+2
=
^16x42
+
^12x42
=
^9x42
=
^9x42
=
^16x42
~
^12x42
+
=
^ 9 x + 2»
^15x43
+
t5x
= ty9
^12x42
+
t5x
-
" ^ i 3 a . + 25
^20x + 3 '
ty,
^15x+3'
ty9
^20x+35
t
b x
l
^20x + 35
ty,
"^15x + 3 '
^ 1 3x + 2 ~
Or
^12x42
=
^5x'
^y,
+2
+
^9x + 2
=
^12x + 2
+
^9x + 2
=
t13x
^15x43'
168
[April
ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS
^12a: + 2
(8.5)
^ 9a; + 2
^15rc+3
(8.6)
^12x+2
+
^16tf + 2
~ "^13ar + 2
"^16ic+2
+
13a; + 2 »
^12a: + 2
^13£C + 2
~ ^12ar + 2
^52#+2
+
(8.7)
=
^# »
~
^20aM
=
^5a:»
^ 3 9a; + 2
^ 6 5a:+3 »
'36tf + 2
ty ,
^15:r+ 3
(8.8)
'9X + 2 '
h5a; + 3
tc
The systems of equations (8.1)-(8.8) are, respectively, equivalent to
the following equations, for which there exist initial solutions given below:
(9.1)
(9.2)
(9.3)
337a;z + 125a; - yl
425a;
2
175a;
+ 145a; - y
2
2
+ 35a; - y
= 3
o,
Uo = 0
Z/o
= 3
o,
2/o
= 3
y = 0,
o> yQ
2
+ 15a; - y
1408a; + 80a; - y2
2
56x2
y0
0,
200a;2 + 100a?'- y2
(9.8)
0,
y = -12,
y = -12,
(9.5)
(9.7)
= 3
y =
+ 110a; - y
Six
Uo
2
250a:
2
0,
2
(9.4)
(9.6)
y - -12,
y = -12,
2
+ 40a; - y2
o,
y =
o,
y = -6,
o
0,
z/0 = 0
0,
y0 = 2
From the theory Of Pell's equation (also referred to as Fermat's equation), it follows that if, simultaneously, k and m are natural numbers, 1, n9
and q are integers, then the product k • m is not a square, and if there exists an initial solution of the equation,
(10)
kx2 + lx - my2 - ny = q9
in integers x0 and y0, where lxQ + T T ]
+ lyQ + -y-J
f 0, then equation (10)
has infinitely many solutions in natural numbers x and y .• Applying this to
equations (9.1)-(9.8) we prove that all the systems of equations (8.1)-(8.8)
have infinitely many solutions in natural numbers x and y. This theorem is
thus proved.
1980]
ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS
169
Some years ago, A. Schinzel found the following proof for the statement
that there exist infinitely many triples of different triangular numbers for
which the sum of any two is a triangular number [private communication from
A. Schinzel].
SckLnzeZ1A
:
?K.00{ [unpubtl^k^d)
x
It is well known that the equation
2
- 424z/2 = 1
has infinitely many solutions, where x E 1 (mod 106) [in every solution, we
have ±x = 1 (mod 106)]. Putting
25
k = 5y -
YQ^(^
- 1) - 1,
1 = |(x - 1) - 50y + 2,
we find
3.
^5fc + 4
+
^9fc+6
+
^12fc+9
=
h3fe+10'
^12k+9
=
^15k+ll*
SYSTEMS OF EQUATIONS INCLUDING THE ALGEBRAIC SUM
AND THE PRODUCT OF TRIANGULAR NUMBERS
W. Sierpinski [1] has asked whether there exists a pair of triangular
numbers such that the sum and the product of these numbers are triangular
numbers. We have found some such systems of equations for which there exist
one or two solutions in triangular numbers, e.g.:
1.
V x "• ty
~tx ~~ ty — t u s
ty j
— tw ,
txty
(ID
(This solution was found by K. Szymiczek [2].)
2.
(12)
tx "F ty
^9
3.
+
^13
=
=
£# "" ^y
tU9
^16'
=
= tv s
txty
=
^9^13
tu,
\tx ' *-)~by — t UJ
^90'
^ 9
= ty,
txty
+
^ ^ 1 3
=
^91'
tx/ty
— 1 = tWJ
^21'^6
~ 1 = t^ .
(13)
^21 ~ ^6 = ^ 2 0 s
4.
(14)
^21^6
=
tx ~ ty = t^,
txty
=
5.
t# + t 3 = t u ,
tyj
~ ty5
txtz
^21 ~ ^6 = h 0 '
and
^98'
^21^6
=
^98
^63 ~ ^38
=
h o '
^63^38
=
^17285
tx + tz ~ tq,
+
^21
s
^35
^21^35
^63
+
^219
^63^219
ty - tz
=
S i '
=
^539s
=
^228'
=
^9855°
= tu,
(continued)
170
ON EULER'S SOLUTION TO A PROBLEM OF DIOPHANTUS—II
5.
continued
^29^69
6.
[April
^14i+9»
^168^69
^8 280 *
For the system of equations,
(16)
tx
+ ty
— ~tus
txty
~ tv
,
there exists also the solution:
^505
+
^531
=
^73 3 >
=
^50 5^531
^189980'
The author wishes to thank Professor Dr. Andrzej Schinzel for his valuable hints and remarks.
REFERENCES
1.
2.
W. Sierpinski. Liczby trojkatne
(Triangular
Numbers).
. PZWS Warszawa:
Biblioteczka Matematyczna, 1962. (In Polish.)
K. Szymiczek. "On Some Diophantine Equations Connected with Triangular
Numbers." Sekoja Matematyki
(Poland) 4 (1964):17-22. (Zeszyty Naukowe
WSP w Katowicach.)
ON EULER'S SOLUTION TO A PROBLEM OF DIOPHANTUS—11
JOSEPH ARKIN
Turnpike,
Spring Valley,
V. E. H0GGATT, JR.
State University,
San Jose,
and
E. G. STRAUS*
of California,
Los Angeles,
197 Old Nyack
San Jose
University
NY 10977
CA 95192
CA 90024
1. INTRODUCTION
In an earlier paper [1] we considered solutions to a system of equations:
x^xj
1 <_ i < j £ n.
+ 1 = yld ;
In this note we look at the generalized problems:
(1.1)
x^Xj
+ a = y\. , a + 0.
In Section 2 we apply the results of [1] to the solutions of (1.1).
Section 3 we consider the following problem: Find n x 2 matrices
I ax
\&1
a2
^2
...
• ••
In
an\
h
n
/
so that a^b- ± ajb-c = ±1 for all 1 £ i < j £ n.
In Section 4 we apply the
results of Section 3 to get two-parameter families of solutions of (1.1),
linear in a, for n = 4.
dation
*This author's
research
was supported
Grant No. MCS 77-01780.
in part
by National
Science
Foun-