FOM: On one generalization of L.Shnirelman's Theory

Alexander Zenkin alexzen at com2com.ru
Wed Nov 10 12:44:10 EST 1999

```Dear [HM]- and [FOM]-list members,

Whether anybody knows a WEB-version of Hardy and Wright's "Introduction
to the Theory of Numbers" or, more precisely, of that paragraph of the
book where Hardy and Wright offer to consider the Classical Waring's
Problem, in general case, as a problem on the summing up of the  series

0^r, 1^r, 2^r, ┘ , n^r, ┘ ,  (1)

for a fixed r>=2 and a fixed number of summands s >=1. Unfortunately, I
have not that book, and have not today a possibily to use a library.
I need in that Hardy and Wright's consideration in connection with
the following [HM]- and [FOM]-problem.
In one of my previous messages to [HM] and [FOM] lists (Subject:
Reverse Mathematics and GAPs in THE CONTINUUM. Date: Thu, 12 Aug 1999
12:32:21) I described a so-called ADDITIVE Eratosthenes' sieve as the
following constructive algorithm.
Let

m1 , m2 ,  m3 , . . . , m_i , . . . , (2)

be any infinite sequence of strictly increasing integers such that m1 >=
0, for any i : m_(i+1) > m_i , and, possibly, with some other
restrictions to the elements and their increment velocity in (2).
Then we fixed a number s >=1 and strike out all possible sums of s
elements of (2) from the series of natural numbers,

1, 2, 3, ┘ , n, ┘ .    (3)

Denote N*(s) a set of all those natural numbers which remain
non-striked out in (3) after that operation.
It is obvious, that by s=1, the ADDITIVE Eratosthenes' sieve
"produces" a set N*(1) that is simply a complementary set to (2) in (3).
For example, if (2) is the set of all primes, then the ADDITIVE
Eratosthenes' sieve works as an "inverse" common multiplicative
Eratosthenes' sieve and "produces" a set N*(1) of all composite numbers.
It is trivially. But if (2) is (1) and r=2, then for s=2 the set N*(2)
is a set of all natural numbers that are not sums of two squares which
is described by Euler's Theorem (1749); for s=3 the set N*(3) is a set
of all natural numbers that are not sums of three squares which is
described by Gauss'es Theorem (1801). And so on. So, any particular case
of Classical Waring's Problem can be considered as an application of the
ADDITIVE Eratosthenes' sieve to (1) with a corresponding r and s.
Probably, one of the most unusual class of ADDITIVE Eratosthenes'
sieve applications arises when, for a given (2), s accepts any values:
s=1,2,3, ┘ . As I told earlier {see the same message}, if elements in
(2) are defined by the expression

(n_i)^r - m^r, all n_i >m, and m>0, r>1 are fixed integers,

then in such the case the ADDITIVE Eratosthenes' sieve algorithm allows
to construct any invariant set, Z(m,r), of the so-called Generalized
Waring's Problem, i.e, of the Problem on the summing up of the series

m^r, (m+1)^r, (m+2)^r, . . . (4)

for any fixed m >= 1, r >= 2. The finity of Z(m,r) sets is
constructively proved by means of the super-induction method with
computer graphics images (pythograms) of these sets { Subject: [HM]
Intuition, Logic, and Induction. Date: Sat, 06 Mar 1999 14:54:47}.
It is a good place to remark here, that the complete solution of the
Generalized Waring's problem, which proves that the arithmetic function,
g(m,r), - which is a natural analog of the famous Hilbert's function
g(r) = g(0,r) in the classical case m=0, - is finite for all m >=1,
r>=2, allows to state that any infinite sequence (4) is a set of
positive density with the finite degree g(m,r)  (in L.Shnirelman's
sense) for the natural numbers series (3) except for a finite set all
elements of which are defined explicitely with the help of the main
parameters, g(m,r) and Z(m,r), of the Generalized Waring's Probelm. As
is known, the Classical L.Shnirelman's Theory of sets densities up to
now was applicable to sequences (2) WITH ZERO only. So, it can be stated
now that that L.Shnirelman's Theory is applicable and could be
generalized in a complete volume to the case of the summing up of the
natural numbers series (1) WITHOUT ITS ANY INITIAL SEGMENT.
So, I believe that the ADDITIVE Eratosthenes' sieve is a quite
general effective algorithm for solution of a wide classes of Number
Theoretical problems, and, simultaneously, a quite common point of view
to the Additive Number Theory as a whole.

Once more ask you to inform me about that Hardy and Wright's citation
if it possible.

Best reagards,

AZ

= = = = = = == = = = = = == = = = = = == = = = = = =
Prof. Alexander A. Zenkin,
Doctor of Physical and Mathematical Sciences,
Leading Research Scientist of the Computing Center
of the Russian Academy of Sciences,
e-mail: alexzen at com2com.ru
= = = = = = == = = = = = == = = = = = == = = = = = =

```