Z-property of natural numbers

Let's consider the representation of natural number N in Bergman's number system:

(1)

The representation of number N in form (1) is called the t-code of the natural number N.

Note that the discrete variables i in (1) take their values from the set {-¥, ..., -3, -2, -1, 0, 1, 2, 3, ..., +¥}.

Let's remember the wonderful formula deduced by the French 19th century mathematician Binet. This one connects Fibonacci and Lucas numbers to the golden ratio and has the following form:

(2)

where the discrete variables i take their values from the set {-¥, ..., -3, -2, -1, 0, 1, 2, 3, ..., +¥}.

Let's remember that Fibonacci and Lucas numbers are the infinite number series determined for positive and negative values of the indices of n and having the following values presented in Table 1.

Table 1.

n012345678910
Fn011235813213455
F-n01-12-35-813-2134-55
Ln213471118294776123
L-n2-13-47-1118-2947-76123

By investigating Table 1 we can find the following mathematical properties for Fibonacci and Lucas numbers:

2Fi+1 = Li + Fi

or

(3)

and also

Li+1 = Fi+1 + 2Fi.(4)

Then if we substitute the expression (2) instead ti in (1) we can represent (1) as the following:

(5)

where

(6)
(7)

Note that the binary numerals in the expressions (6), (7) coincide with the corresponding binary numerals in the expression (1) for the t-code of the natural number N.

Let's consider now the expression (5). This expression is highly extraordinarily. In fact, it follows from Table 1 that the sum B of Fibonacci numbers with the binary coefficients given by (7) and the sum A of Lucas numbers with the binary coefficients given by (6) are integer numbers always. But according to (5) the natural number N is equal to the half-sum of the integer number of A with the product of the integer number of B multiplied by the irrational number . And this is valid for arbitrary natural number N! There is a question: for what condition is it possible? The answer is very simple: this is possible only for the case if the term B in the expression (5) equals 0 identically and the term A is the even number, that is:

(8)

Taking into consideration the identity (8) the expression (5) can be represented also in the following form:

(9)

where A is defined by the expression (6) and B by the expression (8).

Taking into consideration the expressions (6) and (7) we can represent the expression (9) as the following:

(10)

But now we can use the expression (3) and then the expression (10) can be represented in the form

(11)

The expression (11) is called the F-code of N.

As the binary numerals of the expressions (1) and (11) coincide, it follows from here that the F-code of N is got from the t-code of the same natural number N by means of replacing the golden ratio power ti in the formula (1) by Fibonacci number Fi+1, where i = 0, ±1, ±2, ±3, ... .

Let'us represent now the F-code of N in the following form:

(12)

where the term B is defined by the expression (8). Then the expression (12) can be represented in the following form:

(13)

Taking into consideration the identity (4) the expression (13) can be represented in the following form:

(14)

The expression (9) is called the L-code of N.

As the binary numerals of the expressions (1), (14) coincide it follows from here that the L-code of N is got from the t-code of the same natural number N by means of replacing the golden ratio power ti in the formula (1) by the Lucas number Li+1, where i = 0, ±1, ±2, ±3, ... . It is clear that the L-code of N also can be got from the F-code of the same number N by means of replacing Fibonacci number Fi+1 in the formula (11) by Lucas number Li+1.

The abridged notation of the sums (1), (11), and (14) has the following form:

N = am am-1 ... a1 a2 a0 a-1 a-2 ... a-(m-1) a-m.(15)

The expressions (1), (11), (14) give three different methods of the binary representation of the same natural number N. The t-code (1) is the representation of the number N as the sum of the golden ratio powers, the F-code (11) is the representation of the same number N as the sum of the Fibonacci numbers and the L-code (14) is the representation of the same number N as the sum of the Lucas numbers. As this takes place all the above-considered methods of the number N representation have equal abridged notation (15).

Let's consider the abridged notation (15). We can see that the latter is separated by the comma into two parts namely the left part consisting of the digits with the nonnegative indices and the right part consisting of the digits with the negative indices. For example let's consider the code representation of the number 10 (the basis of decimal number system):

10 = 1 0 1 0 0, 0 1 0 1.(16)

For the t-code (1) code combination (16) has the following algebraic interpretation:

10 = t4 + t2 + t -2 + t -4.(17)

By using Binet's formula (2) we can represent the sum (17) as the following:

(18)

If we take into consideration the following correlations connecting the Fibonacci and Lucas numbers (see Table 1)

L-2 = L2; L-4 = L4; F-2 = -F2; F-4 = -F4.

we can get from the expression (18) the following result:

Let's consider now the interpretation of the code combination (16) as the F- and L-codes:

10 = F5 + F3 + F-1 + F-3 = 5 + 2 + 1 + 2;

10 = L5 + L3 + L-1 + L-3 = 11 + 4 - 1 -4.

However, let's return to the expression (8). Comparing the expression (8) with the expression (1) we can conclude that the expression (8) can be got from the expression (1) by replacing ti by Fi. But according to (8) such replacing brings automatically to 0!

This unusual natural number property arising at its representation in the t-code (1) is called the Z-property of natural number.

Let's demonstrate the Z-property for the above-considered example. In fact, if we replace in (17) all the golden ratio powers by the corresponding Fibonacci numbers we get the following sum of Fibonacci numbers:

F4 + F2 + F-2 + F-4.

But because F-4 = -F4 and F-2 = -F2 then we have:

F4 + F2 + F-2 + F-4 = F4 + F2 - F2 - F4 = 0.

Thus we have discovered the new property of natural numbers! Their representations in the t-code (1) possess the "natural" check property, the Z-property. And we can use this property in computers. Let's imagine that we use in computers only natural numbers for representation of numerical information in computer. If we represent all natural numbers in the t-code then we will design automatically self-checking computer based on the Z-property. And we consider later "golden" arithmetic for such "golden" computers. Follow us!