On equal values of pyramidal and polygonal numbers

June 15,1998

Indag. Mathem., N.S., 9 (2), 183-185

On equal values of pyramidal

and polygonal

numbers

by B. Brindza, A. Pint& and S. Turj6nyi Mathematical Inslitute of Kossuth Lajos University, P. 0. Box 12, Debrecen, H-4010, Hungary

Communicated

by Prof. R. Tijdeman at the meeting of January 27,1997

I. INTRODUCTION

There are several classical results related to the equal values of these combinatorial numbers mentioned in the title. It is known that the Yth pyramidal number of order n and the xth polygonal number of order m can be calculated as Y+l Pyr’ n = ,-((a

-

2)Y2 + (5 - n)v)

and Pal,

m-2 =2x

2

--x

m-4

2

(see Dickson [D, p. 41). The equation (1)

Pyr{

= Poli

in positive integersx,Y

was resolved by Erdiis [El, Avanesov [Al] and Avanesov [A21 in the special cases (m, n) = (4,3), (3,3), (3,4), respectively. The purpose of this paper is to show that (1) has only finitely many solutions apart from some effectively determinable exceptional cases. Moreover, as an example, the equation (2)

Pyr{

= Polj(,

that is Y3 + y2 = x2 +x

will be resolved. For further references related to these kind of Diophantine problems we refer to [Ml, [C] and [PI.

183

Theorem. Let m and n be rational integers greater than 2. Apart from some effectively determinable exceptional pairs (m, n), all the solutions of (1) satisfy

max(x,y) < C, where C is an eflectively computable constant depending only on m and n. Remarks. We note that for (m, n) given by (x,y) = (t(2t2 - l), 2t2 is no other exceptional pair (m, n) The case n = 2 leads to a family of is a solution to

= (4,5) there are infinitely many l), (t E N) and our conjecture is for which (1) has infinitely many Pell-type equations. The pair x =

solutions that there solutions. 1, y = -2

y(y + 1) = (m - 2)x2 - (m - 4)x, therefore, it has infinitely many solutions. 2. PROOF

OF THE

THEOREM

By using the transformations z = 2(n - 2)(m - 2)y, t = (n - 2)(m - 2)[2(m - 2)x - (m - 4)]

the equation (1) yields (3)

j&(z)=~~+6(m-2)z~-4(n-5)(n-2)(rn-2)~z +6(n-2)2(m-2)2(m-4)2

=6t2,

and it has finitely many solutions in positive integers z, t if and only if the discriminant off,,n(z) (denoted by discCf&)) is nonzero. A straightforward calculation gives disc&+)

= -4(m - 2)4(n - 2)*[243m4(n2 - 4n + 4)

- 648m3(5n2 - 17n + 12) - 16m2(4n4 - 68n3 - 624n2 + 1807n + 253) + 64m(4n4 - 68n3 - 219n2 + 430n + 1225) - 64(4n4 - 68n3 - 219n2 + 430n + 1225)]. Moreover, the polynomial in the square brackets is irreducible computer). Applying Runge’s theorem (cf. [HS] and [WI) for disc&+)

(verified by

= 0

we obtain the Theorem. If disc(J&) # 0, then (3) is an elliptic equation and a known result of Baker [B] completes the proof. 3. AN EXAMPLE

Resolving the equation (2) in positive integers we have three solutions, namely, 184

(x,~) = (l,l), (3,2) and (98,21); h ence 1,6 and 4581 are the only triangular numbers which are penta-pyramidal. Indeed, after a linear transformation we get the elliptic equation (4)

v3+2v2+2=2u2inintegersu,vwithu>3andv>2,

and if the real zero of the polynomialf(x) = x3 + 2x2 + 2 is denoted by 8 then the ring of integers in the field K = Q(d) is a U.F.D. Furthermore (1,13,0*} forms an integral power basis for ad and Q = -8* - 28 + 1 is a fundamental unit. Using standard algebraic number-theoretical arguments (4) can be reduced to the Thue equation m4 - 4nm3 - 2n2m2 + n4 = l(m > 0, n > 0), which can be resolved by the program package KANT (see [K]). REFERENCES

WI

Avanesov,

E.T. - The Diophantine

equation

3y(y + 1) = x(x + 1)(2x + 1). Volz. Mat. Sb.

Vyp. 8 (1971) 3-6. (Russian)

WI

Avanesov,

E.T. - Solution

(1966/67).

PI

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Arith.

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Diophantine

equation

y2 = ax3 + bx* + cx + d. J. London

Math. Sot. 43,

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[Cl

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J.H.E.

- The Diophantine

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185