Rev. Acad. Canar. Cienc., XIX (Núms. 1-2), 121-125 (2007) (publicado en septiembre de 2008)
A NEW PROOF OF THE UNIQUENESS OF THE SOLUTION
OF A DIOPHANTINE EQUATION ASSOCIATED WITH
THE NATURAL NUMBER 16
Abstract.
Isabel Fernández & José M. Pacheco*
Departamento de Matemáticas
Universidad de Las Palmas de Gran Canaria
This paper offers a new proof of the fact that a certain diophantine equation associated to the natural
number 16 has a unique solution in the domain of natural numbers. The proof derives in a straightforward
manner from the division ofthe naturals into odd and even numbers.
AMS 2000 MSC: 11099
Key words: "Book prooF', Diophantine equation, Natural numbers, Uniqueness.
l. lntroduction.
The fourth power of 2 is 24 = 16 , and in tum, 16 = 4 2 is the square -or second power- of
4. Therefore there exist at least two natural numbers x = 2,y = 4 satisfying the
diophantine equation xY = y x . A rather natural problem is to decide whether the pair
(2,4), found by simple inspection, is the only solution in natural numbers to the
equation. Indeed we are aware that ( 4,2) is a solution as well, but both can be identified
via the symmetry of the equation. Moreover, if we allow x =y, there exist infinitely
many trivial solutions, so from now on x-:;:. y will be supposed.
The study of the equation xY = y x and its solutions has been addressed rather often and
by many authors to a considerable degree ofgenerality, including solutions in Z / p and
in the complex domain. As an example, simple inspection determines as well the real
number 4.81047 ... in the form ¡-; = (-iY. See Euler 1748, Dickson 1966, Hausner
1961, Hurwitz 1967, Sved 1990 and references therein.
Our equation features uniqueness of solution only when considered over the natural
numbers. Even the simplest extension ofthe domain, viz. to the whole numbers domain,
has no unique solution, for (- 2,- 4) is a solution as well. Here the common value is also
related with 16: xY = yx = 16-1 •
The next extension, to the rational domain, is best studied by considering first the larger
real fiel d. In this case, under the hypothesis x < y, let us write y = mx, where m > l is
an otherwise arbitrary real number. Both members in the equation become:
x "'x = x x(rn- 1+ 1) =XX x x(rn- 1)
* Corresponding author. Address: Campus de Tafira Baja, 35017 LAS PALMAS, Spain.
pacheco@dma. ulpgc.es
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and:
From here it is straightforward to find that (observe: m = 2 yields the natural case!):
1
X= mm- 1'
y= mx= m"'-1
Elementary Calculus shows that for m ~ oo we ha ve x ~ 1 and y ~ oo. Therefore
there exist an infinite number of solutions, and ali of them belong to the part of the
curve -whose parametric equations are the above expressions- Iying in the plane
strip{(x,y): 1 < x <e}.
12
10
In this curve are
6
4
4 10
Fig. 1. Where real solutions live
In order to look for rational solutions we impose m = p , a fraction in lowest terms such
q
that p > q, and conditions are found for the solutions to be rational. A known result of
Euler (Euler 1748) -which, in a certain sense, is also an uniqueness result because ali
rational solutions are provided by these fonnulae (a proof is offered in an appendix)- is
obtained when p = n + 1 and q = n :
(n+l)n ( l)n (n+l)n+I ( l)n+I x= -- = l+ - ,y= -- = I+ -
n n n n
In this paper a new and elementary proof of the uniqueness of the solution over the
natural numbers is presented. To establish it, only a remark on parity is needed, so quite
possibly proofpresented could be the "Book proof' in the sense ofErdos (see e.g. Babai
and Spencer 1998, p. 65).
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2. The proof.
The main observation is a very simple one: If there is a solution(x,y)of xY = yx over
the natural numbers, then both x and y must be of the same parity. Should this not occur,
the left and right members would be of different parities and could not have the same
value. Let us write x for the smaller of the two numbers, so for sorne k E N, we obtain
y = x + 2k . By plugging this expression into the equation we obtain:
[ Yx =l]?=}[(x+2kY =l] l' x+2k '
X' X
or
( 2k )x
2k 1 +--;- =X
Under the radical symbol we recognise the familiar expression leading to the
exponential e2k in the limit when x ~ oo. This expression is monotonically increasing
and satisfies the estímate
( 1+ -2-;k-)x < e ik .
By remarking that the function "to obtain the 2k -th root" is a monotonic one, the
following estímate holds:
X= 2k ( 1+ -2-;k-)x <e= 2.718 ... ,
so the only natural candidates for x are l and 2. First, let us consider x = 1. For any k,
both members ofthe equation become:
X y =Jl+2k =J
and
Yx = (J + 2k )1 = J + 2k > J ,
Therefore, x = 1 does not yield a solution ofthe equation. Now we tum our attention
to x = 2 . Both members become:
X y = 2 2+2k = 2 2(l+k) = 4 1+k
/ = [2(1+k)f=4(1 +k)2
and their are equal when k = l, 16 being their common value. Thus we find again what
we found by simple inspection in the Introduction. No more natural pairs (x,y) can be
found, for the set of possible candidates for x is already exhausted. Nevertheless, the
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following argument will reinforce our conviction: For any natural k 2'. 2, the inequality
41+* > 4(1 + k )2 always holds -beca use the exponential grows fas ter than the second
degree polynomial- as the reader can easily check by induction on k. Therefore, we have
obtained the following theorem:
THEOREM: "16 is the only natural number that can be written in two different ways
x>' = yx, with x ::/:-y natural numbers. The solution to the diophantine equation is
provided by x = 2, y= 4 ( or, symmetrically, 4 and 2)"
References.
Babai, L., Spencer, J. (1998) Paul Erdos 1913-1996. Notices Am. Math. Soc. 45 (1) pp.
64-73.
Dickson, L. ( 1966) History of the Theory of Numbers, vol. 2. Chelsea, New York.
Euler, L. (1748) lntroductio in Analysin Jnfinitorum. T. Il, Chap. 21, Sect. 519.
(Springer Edition 1990).
Hausner, A. (1961) Algebraic number fields and the Diophantine Equation xY = y x.
Amer. Math. Monthly 68, pp. 856-861.
Hurwitz, S. (1967) On the rational solutions to m" = n'" with m ::/:- n. Amer. Math.
Monthly 74, pp. 298-300.
Sved, M. (1990) On the rational solutions of x>' = yx. Math. Mag. 63, pp. 30-33.
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Appendix: The Euler formulae provide ali positive rational solutions.
Let m = p in lowest terms such that p > q. We obtain that
q
~ p-q ~ p- q
1 ( ) q 111 ( )_!!_
X = m m- 1 = : , y = m m- 1 = :
and for these numbers to be rational ones , p and q must be (p - q) -powers of sorne
other natural numbers, say P and Q. This condition detennines the rationality of
solutions. For instance, for p = n + 1 and q = n the Euler formulae are obtained:
(n+ ¡)n ( ¡ )n (n+l)n+I ( l)n+I x= -- = !+- ,y= -- = 1+ -
n n n n
Therefore, for p - q = 1 there exist rational solutions. Do more rational solutions occur
for sorne pair p and q such that p - q > 1? The answer is in the negative: If we let
p - q > 1, then the following contradiction appears:
p-q=P"-q -Q"-q ~(Q+l)"-q -Q"-q ~l+(p-q)Q~l+(p-q)>p-q
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