"Iron Table" by Steinhaus

The famous Polish mathematician Steinhaus constructed a table of random numbers using the golden proportion. For this purpose he multiplied 10 000 integral numbers from 1 to 10 000 by the number of w = t - 1 = 0.61803398... where t is the golden proportion. In this connection he got the sequence of numbers multiplied by w, i.e.

1w, 2w, 3w, ... , 4181w, ... , 6765w, ... , 10 000w.

Steinhaus called this number sequence the "golden numbers". Each "golden number" contains integral and fractional parts. For example the number 1000w = 618.03398 has the integral part 618 and the fractional part 0.03398, the number 4181w = 2584.0001. Moreover, there does not exist the "golden number" with fractional part equal to 0 and do not exist two "golden numbers" with the equal fractional parts. Thus, the unique fractional part corresponds to each "golden number".

If we dispose of the "golden numbers" in accordance with their fractional parts we can see that the number of 4181w has the least fractional part and the number of 6765w has the largest one. If we arrange 10 000 natural numbers in accordance with the fractional parts of the corresponding "golden numbers" we can get the following table of natural numbers:

 4181 8362 1597 5778 9959 3194 7365 0610 4791 8972 ........ ........ ........ ........ ........ 8739 1974 6155 3571 7752 0987 5168 9349 2584 6765

Steinhaus names this table the "Iron Table" taking into consideration a number of its unique properties. The "Iron Table" demonstrates deep connections with the Fibonacci numbers. The first property consists of the fact that the difference between the neighboring numbers of the "Iron Table" is equal always to one of three numbers: 4181, 6765 and 2584. In fact, we have:

8362 - 4181 = 4181, 8362 - 1597 = 6765, 5778 - 1597 = 4181, ...

9349 -5168 = 4181, 9349 - 2584 = 6765, 6765 - 2584 = 4181.

It is easy to find the numbers of 2584, 4181 and 6765 if we consider the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 897, 1597, 2584, 4181, 6765, ...

Hence, the numbers of 2584, 4181, 6765 are the neighboring Fibonacci numbers

F18 = 2584, F19 = 4181, F20 = 6765.

We can see from the "Iron Table" that it begins from the Fibonacci number F18 = 2584 and is completed by the Fibonacci numbers F19 = 4181 and F20 = 6765.

It is clear that the "Iron Table" may be constructed for any arbitrary number N of natural numbers. Grejzdelsky analyzed the "Iron Tables" for N = Fn where Fn is the Fibonacci number. He discovered an interesting regularity, which arises at the passage from the "Iron Table" with N = Fn-1 to the next "Iron Table" with N = Fn. In this connection the next "Iron Table" with N = Fn is "moved apart" in comparison to the preceding "Iron Table" with N = Fn-1 creating strictly certain positions in the new "Iron Table" for the new numbers Fn-1 +1, Fn-1 + 2, ... , Fn - 1, Fn .

In Grejzdelsky's opinion, such method of the "Iron Table" construction "reminds the functioning of all the radiation spectra of Nature".