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Multiplying balancing numbers. (English) Zbl 1257.11004

A positive integer \(n\) is called a balancing number by Behera and Panda if \[ 1+\cdots+(n-1)=(n+1)+\cdots+(n+r) \] holds for some positive integer \(r\). Behera and Panda gave several interesting properties of balancing numbers. Later several authors investigated this topic and gave a few generalizations.
In this article the author studies a further generalization of balancing numbers. The idea is due to Behera and Panda.
A positive integer \(n\) is called a multiplying balancing number if \[ 1\cdot2\cdot\ldots\cdot(n-1)=(n+1)\cdot\ldots\cdot(n+r) \] for some positive integer \(r\). The number \(r\) is called the balancer corresponding to the multiplying balancing number \(n\). The cobalancing numbers have a similar definition.
A positive integer \(n\) is called a multiplying cobalancing number if \[ 1\cdot 2\cdot\ldots\cdot (n-1)\cdot n=(n+1)\cdot\ldots\cdot(n+r) \] for some positive integer \(r\). The number \(r\) is called the cobalancer corresponding to the multiplying cobalancing number \(n\). Using the concept of K. Liptai, F. Luca, Á. Pintér and L. Szalay [Indag. Math., New Ser. 20, No. 1, 87–100 (2009; Zbl 1239.11035)] the author gives a further generalization.
Let \(m,k,l\) be fixed positive integers with \(m\geq 4\). A positive integer \(n\) with \(n\leq m-2\) is called a \((k,l)\)-power multiplying balancing number for \(m\) if \[ 1^k\cdot\ldots\cdot(n-1)^k=(n+1)^l\cdot\ldots\cdot(m-1)^l. \]
In this paper the author gives a few results on the balancing numbers above. At first he proves that the only multiplying balancing number is 7 with balancer 3, moreover there is no multiplying cobalancing number. Later the author proves that there is only one \((k,l)\)-power multiplying number which is \(n=7\), \(m=11\) and \(k=l\).

MSC:

11A41 Primes
11A51 Factorization; primality

Citations:

Zbl 1239.11035
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