Abstract:
We survey classical results and recent developments, old and new problems, conjectures and ideas
selected from the endless theme of iterated application of the fundamental rule of addition. The multitude of forms created by this spring, like the commandment ''Let there be light''
from the first day of creation, is emphasized by the role played by the prime numbers.
The subject sounds harmoniously between poetry and astronomy or geometry, finding its origin in the
East at Pingala (4-2nd century BC) and in the West at Apollonius of Perga (3rd century BC).
Our work is divided in two parts: the present paper is mostly dedicated to playing with
numbers, while the second one, which will follow in a companion paper, is based on geometrical
motives.
Key Words: Pascal triangle, Sierpinski
gasket, fractals, Collatz conjecture, Thwaites conjectures,
Ducci game, absolute differences, Gilbreath conjecture, greatest prime factor,
continued fractions.
2000 Mathematics Subject Classification: Primary: 11-02,
Secondary: 11B65,
11L05,
11B50,
11B85.
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