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- Title
Computers As a Novel Mathematical Reality: III. Mersenne Numbers and Sums of Divisors.
- Authors
Vavilov, N. A.
- Abstract
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.
- Subjects
EUCLID; PYTHAGORAS, ca. 570 B.C.-495 B.C.; PRIME numbers; MATHEMATICAL functions; NUMBER theory; EULER theorem; COMPUTERS; CATALAN numbers; PYTHAGOREAN theorem
- Publication
Doklady Mathematics, 2023, Vol 107, Issue 3, p173
- ISSN
1064-5624
- Publication type
Article
- DOI
10.1134/S1064562423700783