INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A19

TRIANGULAR NUMBERS IN GEOMETRIC PROGRESSION

Yong-Gao Chen

Department of Mathematics, Nanjing Normal University, Nanjing 210097, P. R. China

                           ygchen @ njnu.edu.cn

Jin-Hui Fang^{1}

Department of Mathematics, Nanjing Normal University, Nanjing 210097, P. R. China

Received: 1/11/07, Accepted: 4/9/07, Published: 4/12/07

Abstract

In [R. K. Guy, Unsolved Problems in Number Theory, 3rd ed. Springer Verlag, New York, 2004, D23], it is stated that Sierpinski asked the question of whether or not there exist four (distinct) triangular numbers in geometric progression. Szymiczek conjectured that the answer is negative. Recently M. A. Bennett [Integers: Electronic Journal of Combinatorial Number Theory 5(1) (2005)] proved that there do not exist four distinct triangular numbers in geometric progression with the common ratio being a positive integer. In this paper we prove that there do not exist four distinct triangular numbers in geometric progression. Thus Sierpinski’s question is answered and Szymiczek’s conjecture is confirmed.

In [4, D23], it is stated that Sierpinski asked the question of whether or not there exist four (distinct) triangular numbers in geometric progression. Szymiczek conjectured that the answer is negative. Recall that a triangular number is one of the form T_{n} = \frac{n(n+1)}{2} for n ∈ ℕ. The problem of finding three such triangular numbers is readily reduced to finding solutions to a Pell equation(whereby, an old result of Gerardin[3] (see also[2], [5]) implies that there are infinitely many such triples, the smallest of which is (T_{1}, T_{3}, T_{8})). Recently M. A. Bennett[1] proved that there do not exist four distinct triangular numbers in geometric progression with the ratio being positive integer. In this paper, we extend Bennett’s result to the rational common ratio and prove that there do not exist four distinct triangular numbers in geometric progression. Thus Sierpinski’s question is answered and Szymiczek’s conjecture is confirmed.

Theorem There do not exist four distinct triangular numbers in geometric progression.

Proof. Suppose that there exist four distinct triangular numbers T_{n_{1}}, T_{n_{2}}, T_{n_{3}}, T_{n_{4}} in geometric progression. Let q be the common ratio. It is obvious that q > 0 and q ≠ 1. Without loss of generality, we may assume that 0 < q < 1. Let a = 8T_{n_{1}}. Then

1Supported by the National Natural Science Foundation of China, Grant No.10471064.

8T_{n_{2}} = aq, 8T_{n_{3}} = aq^{2}, 8T_{n_{4}} = aq^{3}.

Let m_{i} = 2n_{i} + 1 (i = 1, 2, 3, 4). Then

a + 1 = m_{1}^{2}, aq + 1 = m_{2}^{2}, aq^{2} + 1 = m_{3}^{2}, aq^{3} + 1 = m_{4}^{2}.  (1)

Let

q = \frac{b_{1}}{a_{1}} , a_{1}, b_{1} ∈ ℤ, (a_{1}, b_{1}) = 1, a_{1} ≥ 1.

Because aq^{3} is positive integer, we have a_{1}^{3} | ab_{1}^{3}. Noting that (a_{1}, b_{1}) = 1, we have a_{1}^{3} | a. Let a = a_{1}^{3}a_{0}, a_{0} ∈ ℕ. By (1) we have

m_{1}^{2} − a_{1}^{3}a_{0} = 1, m_{3}^{2} − b_{1}^{2}a_{1}a_{0} = 1.  (2)

Because a = m_{1}^{2} − 1 and a = a_{1}^{3}a_{0} ∈ ℕ, we have a_{1}a_{0} is not a perfect square.

Let x_{0} + y_{0}\sqrt{a_{0}a_{1}} be the basic solution of Pell equation x^{2} − a_{0}a_{1}y^{2} = 1. Then by (2) and the theory of Pell equations, we have

m_{1} + a_{1}\sqrt{a_{0}a_{1}} = (x_{0} + y_{0}\sqrt{a_{0}a_{1}})^{k},
m_{3} + b_{1}\sqrt{a_{0}a_{1}} = (x_{0} + y_{0}\sqrt{a_{0}a_{1}})^{l}.

where k, l are all positive integers. By 0 < q < 1 and (1) we have m_{1} > m_{3} and a_{1} > b_{1}. So k > l ≥ 1.

If k = 2, then m_{1} + a_{1}\sqrt{a_{0}a_{1}} = (x_{0} + y_{0}\sqrt{a_{0}a_{1}})^{2}. Thus we have a_{1} = 2x_{0}y_{0}. So x_{0} | a_{1}. Since x_{0}^{2} − a_{0}a_{1}y_{0}^{2} = 1, we have x_{0} = 1, a contradiction with x_{0} + y_{0}\sqrt{a_{0}a_{1}} being the basic solution of Pell equation x^{2} − a_{0}a_{1}y^{2} = 1. If k ≥ 3, then m_{1} + a_{1}\sqrt{a_{0}a_{1}} = (x_{0} + y_{0}\sqrt{a_{0}a_{1}})^{3}. Thus a_{1} > \binom{k}{3}x_{0}^{k−3}a_{1}a_{0}y_{0}^{3}, which is obviously impossible. □

References

[1] M. A. Bennett, A Question of Sierpinski on Triangular Numbers, Integers: Electronic Journal of Combinatorial Number Theory 5(1) (2005).

[2] L. E. Dickson, History of the Theory of Numbers, Vol.II, p. 36, Carnegie Inst., Washington, D. C. 1920.

[3] A. Gerardin, Sphinx-Oedipe 9 (1914),75,145-146.

[4] R. K. Guy, Unsolved Problems in Number Theory, 3rd ed. Springer Verlag, New York, 2004.

[5] K. Szymiczek, L’equation uv = w^{2} en Nombres Triangulaires (French), Publ. Inst. Math. (Beograd) (N.S.) 3(17) (1963), 139-141.

[6] K. Szymiczek, The Equation (x^{2} − 1)(y^{2} − 1) = (z^{2} − 1)^{2}, Eureka 35 (1972), 21-25.