Adding or multiplying together the digits of an integer is an activity a curious child might feel like doing. But they probably have no idea that it is the source of problems still unsolved in 2025!
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Apply the same algorithm to the digits of a number, then repeat the process with the digits of the result, and you encounter the notion of persistence… and the questions it raises.
Unlike some composite numbers, it is hard to tell whether a number is prime just by looking at it. So what happens to a prime number if we make a change to its digits — for example, by permuting them or removing some of them? Can it stay prime? Or, on the contrary, does it stop being prime?
Conway loved playing with number sequences. His explorations led him to formulate conjectures that are elementary to state but formidable to tackle.
Algorithms for constructing integer sequences from a given number are diverse. Many were devised in the 20th century by well-known mathematicians (Ulam, Kaprekar, Sloane...) and sometimes produce surprising results, some of which remain conjectures.
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