Diophantine Problems in Many Variables: The Role of Additive Number Theory

Computers as Fresh Mathematical Reality. II. Waring Problem

Computer Tools in Education ◽

10.32603/2071-2340-2020-3-5-55 ◽

2020 ◽

pp. 5-55

Author(s):

Nikolai Vavilov ◽

Keyword(s):

Number Theory ◽

Classical Problem ◽

Additive Number Theory ◽

Natural Numbers ◽

Additive Number ◽

Waring Problem ◽

History Of ◽

Almost All ◽

Xix Century

In this part I discuss the role of computers in the current research on the additive number theory, in particular in the solution of the classical Waring problem. In its original XVIII century form this problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers n can be written as sums of s non-negative k-th powers, n=x_1^k+ldots+x_s^k. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that almost all n can be expressed in this form. In the XX century this problem was further specified, as for finding such G(k) and the precise list of exceptions. The XIX century problem is still unsolved even or cubes. However, even the solution of the original Waring problem was [almost] finalised only in 1984, with heavy use of computers. In the present paper we document the history of this classical problem itself and its solution, as also discuss possibilities of using this and surrounding material in education, and some further related aspects.

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Computers as novel mathematical reality. III. Mersenne numbers and divisor sums

Computer Tools in Education ◽

10.32603/2071-2340-2020-4-5-58 ◽

2020 ◽

pp. 5-58

Author(s):

Nikolai Vavilov ◽

Keyword(s):

Number Theory ◽

Additive Number Theory ◽

Additive Number ◽

Mersenne Numbers ◽

Perfect Numbers ◽

Divisor Sums ◽

Mersenne Primes ◽

Mathematical Reality

Nowhere in mathematics is the progress resulting from the advent of computers is 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 behaviour more that 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 inaccessible. A theorem due to Euclid and Euler reduces classification of even perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced manually, 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 what was constructed by humans over several millenia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson

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