Computers as a novel mathematical reality. III: Mersenne numbers and sums of divisors. (English) Zbl 1532.11005
Summary: Nowhere in mathematics is the progress resulting from the advent of computers as apparent as in the additive number theory. In this part, we describe the role of computers in the investigation of the oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behavior more than 2500 years ago. A description of the trajectories of this function – perfect numbers, amicable numbers, sociable numbers, and the like – constitute the contents of several problems stated over 2500 years ago, which still seem completely impenetrable. The theorem of Euclid and Euler reduces classification of even perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced by hand, and since 1952 all of them have been discovered by computers. Using computers, now we construct hundreds or thousands times more new amicable pairs daily than were constructed by human beings over several millennia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson. 374 Refs.
For Parts II and IV, see [the author, ibid. 107, No. 3, 143–172 (2023; Zbl 1532.11004); ibid. 107, No. 3, 205–241 (2023; Zbl 07786636)].
For Parts II and IV, see [the author, ibid. 107, No. 3, 143–172 (2023; Zbl 1532.11004); ibid. 107, No. 3, 205–241 (2023; Zbl 07786636)].
MSC:
| 11-03 | History of number theory |
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 97F60 | Number theory (educational aspects) |
Keywords:
Mersenne primes; divisor sums; perfect numbers; amicable numbers; sociable numbers; aliquot sequences; Catalan-Mersenne conjecture; Catalan-Dickson conjecture; Guy-Selfridge conjectureReferences:
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