Related papers: Triangular Numbers With a Single Repeated Digit
According to a popular belief, the decimal digits of mathematical constants such as {\pi} behave like statistically independent random variables, each taking the values 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 with equal probability of 1/10. If…
It is shown that there exist infinitely many triangular numbers (congruent to 3 mod 12) which cannot be the distance between two perfect numbers.
In the note, the authors give a unified proof of Identities~67, 84, and~85 in the monograph "M. Z. Spivey, The Art of Proving Binomial Identities, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2019; available online…
It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was extended by F. Irwin and others to deal…
A permutiple is a natural number that is a nontrivial multiple of a permutation of its digits in some base. Special cases of permutiples include cyclic numbers (multiples of cyclic permutations of their digits) and palintiple numbers…
A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in NT/9906016) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the…
It is known that, for any positive non-square integer multiplier $k$, there is an infinity of multiples of triangular numbers which are triangular numbers. We analyze the congruence properties of the indices $\xi$ of triangular numbers that…
A number $N$ is a triangular number if it can be written as $N = t(t + 1)/2$ for some nonnegative integer number $t$. A triangular number $N$ is called square if it is a perfect square, that is, $N = d^2$ for some integer number $d$. Square…
Padovan sequence is a ternary recurrent sequence defined by the recurrence relation $P_{n+3}=P_{n+1}+P_{n}$ with initial terms $P_{0}=P_{1}=P_{2}=1.$ In this study it is shown that $114,151,200,265,351,465,616,816,3329,4410,7739,922111$ are…
We present a, hopefully, elementary mathematical treatment of the computational aspects of congruent numbers, such that an amateur could understand the problem and perform their own calculations.
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with both…
For integers $n\ge 0$, an iterated triangulation $Tr(n)$ is defined recursively as follows: $Tr(0)$ is the plane triangulation on three vertices and, for $n\ge 1$, $Tr(n)$ is the plane triangulation obtained from the plane triangulation…
Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$…
The problem we are dealing with is the following: find two sequences $a_n$ and $b_n$ such that the average of the first $b_n$ triangular numbers (starting with the triangular number 1) is still a triangular number, precisely the $a_n$-th…
We have computed a table of the triangle sides of all congruent numbers less than 10,000, which improves and extends the existing public table. We give some background on properties of the triangle sides, and explain how we computed our…
In 1966, Alan Sutcliffe published an introductory paper on "numbers which are multiplied when their digits are reversed". Two years later T. J. Kaczynski extended Sutcliffe's work to show that there exists a 3 digit number in base n with…
In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an…
Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…
In 1973, Neil Sloane published a very short paper introducing an intriguing problem: Pick a decimal integer $n$ and multiply all its digits by each other. Repeat the process until a single digit $\Delta(n)$ is obtained. $\Delta(n)$ is…
This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent…