Z-property of natural numbers Let's consider the representation of natural number
The representation of number Note that the discrete variables 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:
where the discrete variables Let's remember that Fibonacci and Lucas numbers are the infinite number series determined for positive and negative values of the indices of
By investigating Table 1 we can find the following mathematical properties for Fibonacci and Lucas numbers:
and also
Then if we substitute the expression (2) instead t
where
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 Let's consider now the expression (5). This expression is highly extraordinarily. In fact, it follows from Table 1 that the sum
Taking into consideration the identity (8) the expression (5) can be represented also in the following form:
where Taking into consideration the expressions (6) and (7) we can represent the expression (9) as the following:
But now we can use the expression (3) and then the expression (10) can be represented in the form
The expression (11) is called the As the binary numerals of the expressions (1) and (11) coincide, it follows from here that the Let'us represent now the
where the term
Taking into consideration the identity (4) the expression (13) can be represented in the following form:
The expression (9) is called the As the binary numerals of the expressions (1), (14) coincide it follows from here that the The abridged notation of the sums (1), (11), and (14) has the following form:
The expressions (1), (11), (14) give three different methods of the binary representation of the same natural number 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):
For the t-code (1) code combination (16) has the following algebraic interpretation:
By using Binet's formula (2) we can represent the sum (17) as the following:
If we take into consideration the following correlations connecting the Fibonacci and Lucas numbers (see Table 1)
we can get from the expression (18) the following result: Let's consider now the interpretation of the code combination (16) as the 10 = 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 t This unusual natural number property arising at its representation in the t-code (1) is called the Let's demonstrate the
But because
Thus we have discovered the new property of natural numbers! Their representations in the t-code (1) possess the "natural" check property, the |