LES TRANSFORMATIONS AGREABLES ET UNE NOUVELLE CLASSE DE NOMBRES NARCISSIQUES PARFAITS

Rene-Louis Clerc (1)

1 UT3 - Université Toulouse III - Paul Sabatier

Number theory (discrete iteration) PLEASANT TRANSFORMATIONS AND A NEW CLASS OF NARCISSISTIC PERFECT NUMBERS. This paper is situated between recreational mathematics on integers and iteration in N of discrete dynamical systems. We resume and specify here the notion of pleasant transformation that we had initially defined in ([9], [11]); the simplest of these applications are easily attached to the classic happy ([6]), unhappy and narcissistic perfect numbers or PPDI ([3]). We will first give some theoretical and practical properties on the pleasant transformations as well as a calculation technique to improve downwards the calculation times of the fixed points and other attractors of the associated iterations. We will define by composition with a power function a class of pleasant transformations of which a large number will have only one fixed point which will ensure a unique and remarkable property to the integer which represents it; we will show that there are a finite number of such remarkable integers. Various properties will express some of these representatives and their specific peculiarity; such as 754, the only number strictly greater than 1, which is equal to the sum of the squares of the digits of its power of 8, or the prime number 3877, the only integer strictly greater than 1 which is equal to the sum of the powers 38 of the digits of its square...but there is no integer greater than 1 that is equal to the sum of the digits of its power of 105. By a mixed method, theoretical and numerical, we will establish that there is no perfect narcissistic number of order > 56, which improves the classical bound which is 60. We will then define a new class of perfect narcissistic numbers, named r-PPDI or perfect rnarcissistic numbers and we will show that they are finite in number; we will exhibit the 10 smallest perfect r-narcissistic numbers. After some remarks in connection with prime numbers and constellations of prime numbers ([10], [12], [13]), we will give some examples of other pleasant transformations. Some transformations used here can be simulated online.

Domaines: Mathématiques [math]

Rene-Louis Clerc. LES TRANSFORMATIONS AGREABLES ET UNE NOUVELLE CLASSE DE NOMBRES NARCISSIQUES PARFAITS. 2023. ⟨hal-03619147v4⟩