========================= MATHEMATICS ==============================

Consider the problem on the representation of natural numbers n ³ 1 by sums of the form

n_i ³ m, (1)

where m ³ 0, r ³ 2, s ³ 1 - are fixed integers.

In the early 1980s, using methods of Cognitive Comuter Graphics (CCG) methods [3 - 5], I saw (in the direct sense of the word) for the first time and formulated the generalization of the CWP to the case of any m ³ 0 and gave the complete solution to the Generalized Waring Problem (GWP) in the form of the theorem, cited in the right column of the Table 1 [3-7].

Here, for arbitrary values of the parameters m ³ 0, r ³ 2, and s ³ 1, we introduce, by definition, the following two families of sets of natural numbers:

,

,

G(m,r) = Arg { |N(m,r,s)| < ¥ }, g(m,r) = Arg { |N(m,r,s)| = |Z(m,r)| },

Table 1. The Logical Isomorphism of the Classical and Generalized Waring Problems.

G(0,r) º G(r), g(0,r) º g(r), and Z(0,r) = Æ for all r ³ 2.

Thus, for any m³ 0, r³ 2 and s³ g(m,r), the structure of the sets N(m,r,s) is completely determined by the structure of the corresponding sets, Z(m,r), - the so-called invariant sets of GWP.

The pythograms of the invariant sets displayed in the Fig. 1 are read (more precisely, are counted) from left - to right and from top to bottom, (in this case, starting from n=1). If a square is black then the corresponding natural number n Î Z(m,r), and if a square is white, then n Ï Z (m,r).

In [6], the general theorem on the finiteness of the invariant sets Z(m,r) was proved for any m³ 1, r³ 2. However, in practice, it is frequently more convenient to use the constructive finiteness criterion for invariant sets, Z(m,r) that is given by the next theorem [6,8,14].

THEOREM 1. For any m ³ 1 and r ³ 2,

IF $ n*Q(n*), where Q = (i) & (ii) & (iii). and

THEN " n> n*P(n), where P(n) = " n Ï Z(m,r) ",

PROOF. Assume that $ n*Q(n*), but " n > n* P(n) is incorrect. Then there exists a least positive integer, say t ³ 1, such that

1) Ø P(n*+k+t), i.e. n*+k+t Î Z(m,r), and

2) n* + j Ï Z(m,r) for each j = 1, 2, ..., k+t-1, - because t=min,

i.e. if n Î D = [n*+1, n*+k+t-1], then n Ï Z(m,r).

n₂= n₁-q=(n*+k+t)-q < n*+k+t-1 - because q > 2 for any m ³ 1, r ³ 2, and

n₂= n₁-q=n*+(k-q+t) ³ n*+1 - because k-q ³ 0, which follows from the condition (iii) and t ³ 1 according to our assumption. (In what follows, the symbol Þ is read as "follows", for short.)

Thus, n*+1 £ n₂ < n*+k+t-1 Þ n₂ Î D Þ n₂ Ï Z(m,r), - by virtue of (ii).

n₂ = n₁ - q = Þ n₁ = q+ Þ n₁ = Þ

However, the last statement contradicts to the consequence (i) of our assumption. The contradiction obtained proves the theorem. ÿ

$ n*Q(n*) ® " n>n*P(n). (2)

The extraordinariness of this implication is that its antecedent is a singular statement, but its consequent is a general statement. However, it is obvious, that now to prove the finiteness of invariant sets by means of Theorem 1, it is sufficient to find at the pythogram of the corresponding invariant set Z(1,r) such the element zÎ Z(1,r), - a black square, - which is followed by no less than q(1,r) =2^r -1 white squares (naturally, if such a black square exists, since Theorem 1 does not guarantee (!) the existence of such natural numbers). Taking into account, that by m=1

Thus, once having found the numbers n*(1,r) for r=2,3,4, we proved actually the truth of the singular antecedents of implication (2). From this, by modus ponens rule, we immediately infer that " n>n*P(n), - where, by virtue of Theorem 1, P(n) = " n Ï Z(m,r) ", and, hence, the corresponding invariant sets are finite, - What occurs for all n£ n*, can be directly seen in the corresponding pythograms. More strictly, we can see (and, if desired, can even write explicitly) all elements of the corresponding invariant sets Z(1,r), r = 2, 3, 4.

Fig.1. CCG-Images (the so-called pythograms) of the invariant sets of Generalized Waring's Problem, Z(1,r), r=2,3,4. These sets are the new number-theoretical objects that were discovered with the aid of the Cognitive Computer Graphics approach [6].

1. It is necessary to prove the general statement " n³ 1 P(n).

$ n*Q(n*) ® " n>n*P(n), (3)

is constructed (is devised!), where the number-theoretical predicates P and Q = f(P) are distinct, i.e. Q ¹ P.

4. The truth of the antecedent $ n*Q(n*) of implication (3) is proved; i.e., a natural number n* posessing the number-theoretical property Q is found ( by searching and testing on a computer or even by hand, if you please!).

5. If we succeed in finding such a unique number n*, then the truth of the antecedent $ n*Q(n*) and the truth of the implication $ n* Q(n*) ® " n>n* P(n), which have been proved , implay (by modus ponens) the authentic truth of the consequent, i.e. certain truth of the general statement " n> n* P(n).

6. The truth of the predicate P(n) is checked for all n £ n*:

(a) If " n£ n* P(n), then we have proved " n³ 1 P(n);

(b) Otherwise, we find (explicitely!) all elements of the finite exclusive set, N¨ = { 1£ n£ n* : Ø P (n)}, and thus,

" n³ 1 P(n), except for n Î N¨ , where N¨ = { 1£ n£ n* : Ø P(n)}.

In conclusion, we emphasize, that there are no analogues of the method of superinduction in modern mathematics. In particular, this method cannot be reduced to the method of complete mathematical induction, which, as is known, always deals with a unique predicate [13]. The method of superinduction differs essentially from methods based on a direct (analytical!) existence of a threshold element n₀ such that $ n_0" n>n₀ P(n) (see, e.g., the famous solution to CWP for biquadrates [11], which does not contain the implicative part of the form (3) at all), as well as from methods of classical deduction, which infer a less general statement from general one and, as a rule, do without computers. At the same time, construction (at present, invention!) and an analytical proof of implication (3) " from a singular statement to a general one, as well as a computer search for a unique number n*(m,r) followed by computetions (or an inspection of CCG-pythograms in the simplest cases) of the corresponding number-theoretical objects are inherent and unremovable stages of the method of superinduction. Combined together, these stages distinguish the method of superinduction from all known methods of proofs of general mathematical statements of the form " nP(n).

3. Zenkin A.A., Generalization of A.Wieferich's Theorem on the Case of Natural Summands. - Doklady AN USSR, 1982, ň.264, No.2, 282-285. {Translated in "Sov. Math. Dokl.", Vol. 25, 616-620 (1982).}

e-mail: alexzen@com2com.ru