A Note on an Equation Related to the Pell EquationAuthor(s): Morris NewmanSource: The American Mathematical Monthly, Vol. 84, No. 5 (May, 1977), pp. 365-366Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2319968 .

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1977] MATHEMATICAL NOTES 365

References

1. R. Baire, Sur les series a termes continus et tous de meme signe, Bull. Soc. Math. France, 32 (1904) 125-128. 2. , Leqons sur les fonctions discontinues, Gauthier-Villars, Paris, 1905. 3. L. E. Clarke, On marginal density functions of continuous densities, this MONTHLY, 82 (1975) 845-846. 4. E. W. Hobson, The Theory of Functions of a Real Variable; Vol. I, 3rd ed., Cambridge University Press,

1927, (Dover reprint, New York, 1957). 5. , The Theory of Functions of a Real Variable; Vol. II, 2nd ed.9 Cambridge University Press, 1926,

(Dover reprint, New York, 1957). 6. A. Verbeek, Lower semi-continuity of probability density functions, preprint, available from the author.

DEPARTMENT OF MATHEMATICS, P.M.B. 1154, UNIVERSITY OF BENIN, BENIN CITY, NIGERIA. SCHAEPMANLAAN 9, AMERSFOORT, NETHERLANDS.

A NOTE ON AN EQUATION RELATED, TO THE PELL EQUATION

MORRIS NEWMAN

Abstract. It is shown that the diophantine equation x2 - dy2 = - 1 has solutions, provided that d = PlP2 ' pr

where r is 2 or odd and PI, p2. p2 are distinct primes congruent to 1 modulo 4 such that (pipi) = - 1, i$ j.

Let d > 1, d square-free. The conditions under which the diophantine equation

(1) x2-dy2= -1

possesses solutions are still not fully known, although of course, it is known that the Pell equation

(2) x2- dy2,= 1

always has non-trivial solutions. If d = p, where p is a prime 1 modulo 4, then (1) has a solution; and if d = pq, where p, q are distinct primes 1 modulo 4 such that (p/q) = - 1, then (1) also has a solution, as was shown by Dirichlet in [1]. Dirichlet also treated the case when d is the product of three distinct primes. The theorem in this note is a direct generalization of Dirichlet's results and has apparently not been noticed previously.

We shall prove

THEOREM. Let r be 2 or odd. Let pi, P2, *. p, be distinct primes such that

(3) pi--lmod 4, 1 -c-i-r,

(4) (pi /pi) = - , C i, r, if7 j.

Put d = p1p2 ... Pr. Then the diophantine equation (1) has a solution.

Proof. Assume that (1) has no solutions. Let (f, 1) be the fundamental solution of (2). We have from (3) that {2 -2+ 1 mod 4, which implies that 5 is odd and ,1 even. Thus we have

. += n2 2d1 2d2 4

where did2 = d, and ((f - 1)/2d1, (, + 1)/2d2) = 1. It follows that

e L u2 = v22 d2V2- diU2= 1. 2d, 2d,

We note that uO, since (E 1. Suppose now that neither d1 nor d2 is 1. Since r is either 2 or odd, either d1 or d2 must be the

product of an odd number of primes. Suppose that d, is the product of an odd number of primes, and let p be any prime dividing d2. Because of (4) we have that (dilp) = - 1, which contradicts the fact

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366 PETER KLEINSCHMIDT [May

that di u2 - 1 mod p. Similarly, the case when d2 is the product of an odd number of primes may also be shown to be impossible. It follows that d1 = 1, d2 = d, or d1 = d, d2= 1. But di = 1, d2= d gives u2 - dv2 = - 1, which contradicts the assumption that (1) has no solutions. Thus di = d, d2 = 1, and so v2- du2= 1. We have

u_ 2 2 + 2U2 =_ U 2d' 2' _ 4d 4

Hence uv = 7/2, and so u < . But this contradicts the fact that (4 ?) is the fundamental solution of (2). Hence the assumption that (1) has no solutions is false, and the theorem is proved.

Reference

1. G.L. Dirichlet, Abh. Akad. Wiss. Berlin, (1834) 649-664, Werke 1, 219-236.

NATIONAL BUREAU OF STANDARDS, WASHINGTON, DC 20234.

RESEARCH PROBLEMS

EDITED BY RICHARD Guy

In this Department the Monthly presents easily stated research problems dealing with notions ordinarily encountered in undergraduate mathematics. Each problem should be accompanied by relevant references (if any are known to the author) and by a brief description of known partial results. Manuscripts should be sent to Richard Guy, Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada, T2N 1N4. (From July 1976 to June 1977: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England.)

WHEN IS THE GRAPH OF A TRIANGULATION UNIQUELY 4-COLORABLE?

PETER KLEINSCHMIDT

A graph is said to have an n-coloring if there is a partition of its vertices into n classes, called the color classes, in such a way that no adjacent vertices belong to the same color class. A graph with exactly one n-coloring is called uniquely n-colorable. Uniquely colorable graphs have been studied by Harary, Hedetniemi and Robinson in [31 and by Chartrand and Geller in [2]. It is well known that the 4-color-conjecture is equivalent to the conjecture that the graph of every triangulation of the plane possesses a 4-coloring.

One may ask which graphs are 'extreme' with respect to this conjecture, i.e., graphs of triangulations which are uniquely 4-colorable.

We consider a special class of such graphs, the edge-graphs of stack polytopes (see [1], [4] and [5]). A 3-dimensional polytope is called a stack polytope if its boundary complex is isomorphic to a complex obtained from the boundary complex of a tetrahedron by successively applying barycentric subdivi- sions to triangles. We can now formulate a conjecture for uniquely 4-colorable graphs.

CONJECTURE. The graph of a triangulation of the plane is uniquely 4-colorable if and only if it is combinatorially isomorphic to the graph of a stack polytope.

It is obvious that the graph of a stack polytope is uniquely 4-colorable. To show the more interesting part of the conjecture, one would have to prove that no 4-connected planar graph, which

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