At the very beginning of the 80s, I formulated and proved a so-called
Generalized Waring Problem (GWP) that is an extension of D.Hilbert's
solution to the Classical Waring's Problem (CWP) from the classical case
m=0 to any m=1,2,3,... (see [1] Zenkin A.A., Generalization of Wieferich's
Theorem to the case of Positive integral summands. - Soviet Math. Dokl.,
Vol. 25 (1982), No. 3, 616-620; and [2] Zenkin A.A., Some Extensions of
Pall's Theorem. - Computers&Math. with Applications, Vol. 9 (1983), No.4,
609-615).
In the framework of GWP, the so-called invariant sets, Z(m,r), play an
important role, where Z(m,r) is a set of natural numbers n>=1 that are not
representable as sums of s summands, ni^r-m^r, by any s>=1, ni>=m
{here and further: a^b is "a to the power b "}.
The following quite unusual theorem gives an effective and constructive
criterion for the finitude of Z(m,r).
THEOREM 1 (1979). For any fixed m>=1, r>=2,
IF there exists a single (!) natural number n* which is an element of
the set Z(m,r) and which is followed by k, k>= (m+1)^r-m^r, natural numbers
n*+1, n*+2, ..., n*+k, that are not elements of the set Z(m,r), THEN any
natural number n>n* is not an element of the set Z(m,r).
Or, shortly,
_En*Q(n*) ==> _An>n*P(n), (1)
{here and further: _E is the abbreviation for "Existence", _A is the
abbreviation for "for any"}
The mathematical proof (by the Reductio ad Absurdum method) of
Theorem 1 is a deductive proof of the authentic inductive inference
"from a single, _En*Q(n*), to a common, _An>n*P(n)".
Thus, in Mathematics, there is a reliable inference of a common
assertion from a single (!) assertion in contrast with the well-known
J.S.Mill's and all Natural Science's Paradigm: a common assertion may be
inferred from a partial assertion, but only as a plausible hypothesis.
At the same time, I discovered a quite different class of number-
theoretical problems that were solved by means of the same (1)-type
Theorem (see [1], and Theorem 4 in: [3] Zenkin A.A. Waring's problem:
g(1,4) = 21 for fourth powers of positive integers.- Computers and
Mathematics with Applications, Vol.17, No. 11, pp. 1503 - 1506, 1989).
All this allowed me to formulate the following new method for proving
common mathematical statements - the so-called Super-Induction (SI) method.
SUPER-INDUCTION METHOD.
1. It requires to prove a common mathematical statement: _A n>=1 P(n).
2. For the given mathematical predicate P(n) we construct (for the time
being - invent!) a new mathematical predicate Q = f(P) and formulate a
conditional mathematical statement (theorem) of the following special form:
"IF there exists a natural number n* such that the predicate Q(n*) holds
true, THEN for any natural number n>n* the predicate P(n) is valid", that
can be written shortly so:
_En*Q(n*) ==> _An>n*P(n), (1)
by the way, the mathematical predicates Q and P are different, i.e. Q =/= P.
3. Now, we must prove (if possible!) the mathematical theorem (1).
4. Then we must prove the truth of the single mathematical statement
_En*Q(n*), i.e. we simply search and check, by a computer, even one (!)
natural number n* such that Q(n*) is valid.
5. IF (!) we have found such unique number n*, and, consequently,
we have proved the single mathematical statement _En*Q(n*), and IF we have
proved the conditional mathematical theorem _En*Q(n*) ==> _An>n*P(n),
THEN we infer, by the classical modus ponens rule, the common mathematical
statement _An>n*P(n).
6. For all n < = n* we check the truth values of the predicate P(n):
A) if _An < = n* P(n), then we have proved the common mathematical
statement in its traditional form: _An >= 1 P(n);
B) otherwise, we find explicitly all the elements of the finite exclusive
set, Ne = { 1 < = n < = n* : not-P(n) } and therefore
7. We prove the common mathematical statement in its not quite usual form:
_An >= 1 P(n), except for n (- Ne, where Ne = { 1 < = n < = n* : not-P(n)}.
Note that in [3] I generalized, by means of SI-method, the famous
classical (m=0) Balasubramainan's et al. result g(0,4)=19 to the
non-classical case m=1: g(1,4)=21.
In the very end of 1997, Prof. A.Shinzel paid my attention to the
following Lemma of the German mathematician, H.E.Richert [see W.Sierpinski,
Elementary Theory of Numbers. - Warszaw, 1964, Chapter III. Prime numbers,
pp. 143-144.].
LEMMA 2 (H.E.Richert, 1949). If m1, m2, ... is an infinite sequence
of natural numbers such that for a certain natural number k the inequality
(18) m_(i+1) < = 2 m_i for i > k
holds and IF there exist an integer n* > = 0 and a natural number
s0 >= m_(k+1) such that each of the numbers
(21) n*+1, n*+2, ..., n*+s0
is the sum of different terms of the sequence m1, m2, ..., mk, THEN
every natural number > n* is the sum of different terms of the sequence
m1, m2, ...
Define the following two number-theoretical predicates:
P(n) = "n is the sum of different terms of the sequence m1, m2, ...",
and
Q(n) = Q1(n)&Q2, where Q1(n) = "n+j is the sum of different
terms of the sequence m1, m2, ..., mk for every j=1, 2, ..., s0"
and Q2 = "s0 > m_(k+1)",
Then we can rewrite H.E.Richert's Lemma 2 in the brief form:
LEMMA 2A. Let m1, m2, ... be an infinite sequence of natural numbers
such that for a certain natural number k the inequality
(18) m_(i+1) < = 2 m_i for i > k
holds and let s0 be a natural number such that s0 >= m_(k+1).
In that case,
IF there exists a number n* >= 0 such that Q(n*) holds true,
THEN for any n > n* P(n) is valid, or, in short symbolic notation,
_En*Q(n*) ==> _An>n*P(n), (1)
Using Lemma 2, Richert proved many interesting theorems on the
representability of natural numbers by sums of different prime numbers.
As is easy to see, Richert uses the Super-Induction method, but as an
empirical special mathematical method for a special mathematical problem.
So, today, German mathematician H.E.Richert is, of course, the author
of the discovery that in mathematics there exist mathematical statements of
the (1)-form.
But, the Math History question arises: may be, other examples of such
theorems are known in Mathematics before 1949?
At last, my next discovery consists of the following "impossible" thing.
If, for a given P(n), we assume that
Q(n*) = f(P) = [P(n*)& [ _An>n*[P(n) ==> P(n+1)]] and substitute such Q(n*)
in (1), then we obtain
PASCAL's THEOREM (XVII C.). For any mathematical predicate P(n)
_En*[P(n*)&[ _An>=n* [P(n) ==> P(n+1)]]] ==> _An>n* P(n). (1)
Thus, the classical complete mathematical induction B. Pascal's method
is a special case of the Super-Induction Method.
Taking into account the important role which Pascal Induction plays in
F.O.M., I think it will be interesting to investigate the Super-Induction
method from the same point of view.
There is a once more example of Super-Induction application for extremal
combinatorial problems on graphs in [Baranov V.I., Stechkin B.S. Extremal
Combinatorial Problems and Their Applications, page 106. - Translation from
the Russian Edition: Moscow, "Nauka", 1989].
Repeat once more, maybe, somebody knows other theorems of the form
_En*Q(n*) ==> _An>*P(n) in other area of Math.?
It would be very important in order to understand the inner nature of
the connection Q=f(P) in common case and of an applicability of the
Super-Induction method in other fields of Mathematics.
Now, am I right when I state:
1) German mathematician H.E.Richert is the author of the discovery of the
cardinally new type of mathematical theorems of the form:
_En*Q(n*) ==> _An>*P(n) ?
2) the mathematical theorems of the form, _En*Q(n*) ==> _An>*P(n),
is the discovery in the Inductive Logic: the deductive inference of the
authentic induction "from a single to a common"?
3) the SI-method is a cardinally new method for proving common mathematical
statements ?
4) the famous classical complete mathematical induction B.Pascal's method
is a special case of SI-method?
The latest publication concerning said above are:
[4] Zenkin A.A., Superinduction: A New Method For Proving General
Mathematical Statements With A Computer. - Doklady Mathematics, Vol.55,
No.3, pp. 410-413 (1997). Translated from Doklady Alademii Nauk, Vol. 354,
No. 5, 1997, pp. 587 - 589.
[5]. Zenkin A.A., Generalized Waring's Problem: G(m,r) < = G(0,r)+m+1
for any m >= 1, r >= 3. - Doklady Mathematics, vol 56, No. 1, pp. 499-501
(1997). Translated from Doklady Alademii Nauk, Vol 355, No. 2, 1997,
pp. 151-153.
[6]. Zenkin A.A., Generalized Waring's Problem: An Estimation of
G(m,r)-Function Via g(m-1,r)-Function By Means Of The Superinduction
Method. - Doklady Mathematics, vol 56, No. 1, pp. 597-600 (1997).
Translated from Doklady Alademii Nauk, Vol 355, No. 6, pp. 727-730.(1997).
Sincerely,
Alexander Zenkin
==========
Name: Alexander A. Zenkin
Position: Leading Scientific Co-Worker (Employee), Professor of
Computer Science.
Institution: Computer Center of the Russian Academy of Sciences,
Moscow, Russia.
Research interest: Number Theory, Logic, Foundations of Mathematics,
Cognitive Computer Visualisation of Mathematical Abstractions as a
technique for a new knowledge generating in Number Theory, Logic and
Set Theory.
More information: http://www.com2com.ru/alexzen