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AN EXTENSION OF AN OLD PROBLEM OF
DIOPHANTUS AND EULER
And rej Dujella
Dept. of Math., University of Zagreb, Bijenicka cesta 30,10000 Zagreb, Croatia
(Submitted January 1998)
Diophantus studied the following problem: Find three (rational) numbers such that the
product of any two increased by the sum of those two gives a square. He obtained the solutions
{4,9,28} and {jl,,-y, ^ } (see [3]). Euler treated the same problem with four numbers (see [2]).
He found the solution {^, ^ , ^ , y}. Indeed, we have
65 9
224 224
65 .5
224 2
9
|
65
224
65 { 5 f l 5 V
224 2 1,8 J '
5,9
224 2
9 =fl3lY
_6f>_._9_+ 65 . 9
224 \22AJ ' 224 56 224 56
}
224
5
f13T
-(?J=
2
V 8;
?
(79
V.H2
_?_._?_ + . 9
224 56 224
9
_9_.A+A+.5
Z"7"12
56 2
56
(
56
51
U12
2 V4
In the present paper we will construct the set of five numbers with the above property.
Let {xh ..., xm) be the set of rational numbers such that xtXj +xi-hxJ is a perfect square for
all \<i <j<m. Since
XjXj + Xi + Xj = (Xf +l)(Xj
+1)-1,
if we put xtr + 1 = axf, / = 1,..., m, we obtain the set {a/r,..., am} with the property that the product of
its any two distinct elements diminished by 1 is a perfect square. Such a set is called a (rational)
Diophantine m-tuple with the property D(-l) (see [4], p. 75). If az's are positive integers, such a
set is also called a P_x-set of size m. The conjecture is that there does not exist a P^-set of size 4.
Let us mention that in [1], [6], and [7] it was proved that some particular P_ r sets of size 3 cannot
be extended to a P^-set of size 4. In [5], some consequences of the above conjecture were
considered.
We will derive a two-parametric formula for Diophantine quintuples and, as a consequence,
we will obtain a rational Diophantine quintuple with the property D(-l).
We will consider quintuples of the form {A, B, C, D, x2} with the property D(ax2), where A,
B, C, D, x, and a axe integers. Furthermore, we will use the following simple result known
already to Euler: If BC+n = k2, then the set {B, C, B + C ± 2k} has the property D(n).
Therefore, if we assume that
BC + ccx2 = k2,
A = B + C-2k,
D = B + C + 2k,
then the set {A, B, C, D, x2} has the property D(ax2) if and only if AD + ax2 is a perfect square.
Hence, we reduced the original (2) = 10 conditions to only two conditions:
312
(b2-a)(c2~a)
+ ax2 = k2,
(1)
(a2-a)(d2-a)
+ ax2=y2.
(2)
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AN EXTENSION OF AN OLD PROBLEM OF DIOPHANTUS AND EULER
Our assumptions
(b2 -a) + (c2 - a)-2k = a2 -a,
(b2 - a) + (c2 - a) + 2k = d2 - a
imply that 4k = (d + a)(d - a). Let d + a = 2p and d - a = 2r. This implies that k-pr
b2 +c2-a
and
= | ( a 2 +d2) = p2 +r2.
(3)
Let us rewrite condition (2) in the form (ad - a)2 - aid - a)2 -y2 - ax2. Thus, we may take
y-ad-a,
x-d-a-2r.
(4)
Substituting (3) and (4) into (1), we obtain
p2r2 - b2c2 - Aar2 - a(b2 + c2 - a) = a(3r2 -p2).
(5)
At this point we make the further assumption [motivated by (3) and (5)]:
h + c = p+r.
(6)
pr-hc = f,
(7)
pr + bc = 2(3r2-p2).
(8)
a = 4p2 + 4pr-l2r2.
(9)
Now (3) implies
and (5) implies
Adding (7) and (8) yields
From (6) and (7), we conclude that b and c are the solutions of the quadratic equation
z2-(p+r)z + (pr-^j
= 0.
The discriminant of this equation has to be a perfect square. Thus,
(p-r)2 + 2a = q2.
(10)
Substituting (9) into (10) we have, finally,
(3p+r)2-24r2
= q2.
(11)
Hence, we reduce our problem to the solving of (11). However, the general solution of
the equation u2 - 24v2 = w2 with (u, v,w) = l is given by
u = e2+6f2,
v = ef, w = \e2-6f2\
u = 2e2+3f2,
v = ef, w = \2e2-3f2\
or
(see [8], p. 225). Thus, we have proved
Theorem 1: If e = 0 (mod 3) or e = f (mod 3), then the set
{l(e 2 + 6 * / - 1 8 / 2 ) ^
f\e-2f)(5e
1999]
+
6f)^(e2+4ef-6f2)(6f2+4ef-e2X4e2/2}
313
AN EXTENSION OF AN OLD PROBLEM OF DIOPHANTUS AND EULER
has the property D(fe2f2(e2-ef-3f2)(e2+2ef-l2f2)),
andtheset
{±(9f2+6ef-2e2)(2e2+2ef-f2),±e2(5f-2e)(2e
2
f (e+f)(5e
has the property D(fe2f2(e2
2
2
2
+ 3f),
2
- 3/), 1 ( 3 / + 4ef - 2e )(2e +4ef - 3f ),
-ef-
4e2f}
3/ 2 )(4e 2 + 2ef - 3/ 2 )).
Substituting e = 5 and / = 2 in (12), we obtain the following two corollaries.
Corollary 1: The set {^,^-,j^,^,^}
£>(-l).
ls a
rational Diophantine quintuple with the property
Corollary 2: Thefivenumbers —jjj, ^ , ^J, 9, ^ have the property that the product of any two of
them increased by the sum of those two gives a perfect square.
REFERENCES
1. E. Brown. "Sets in Which xy + k Is Always a Square." Math. Comp. 45 (1985):613-20.
2. L. E. Dickson. History of the Theory of Numbers 2:518-19. New York: Chelsea, 1966.
3. Diophantus of Alexandria. Arithmetics and the Book of Polygonal Numbers, pp. 85-86, 21517. Ed. I. G. Bashmakova. Moscow: Nauka, 1974. (In Russian.)
4. A. Dujella. "On Diophantine Quintuples." ActaArith SI (1997):69-79.
5. A. Dujella. "On the Exceptional Set in the Problem of Diophantus and Davenport." In
Applications of Fibonacci Numbers 1. Dordrecht: Kluwer, 1998.
6. K. S. Kedlaya. "Solving Constrained Pell Equations." Math Comp 67 (1998):833-42.
7. S. P. Mohanty & A. M. S. Ramasamy. "The Simultaneous Diophantine Equations 5y2 - 20 =
x2 and 2y2 +1 = z2" J. Number Theory 18 (1984):356-59.
8. T. Nagell. Introduction to Number Theory. Stockholm: Almqvist; New York: Wiley, 1951.
AMS Classification Number: 11D09
314
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