Small Symbol
Four (three plus one) numbers 4 play very important role in this website. Their stories represent irrationality in the website. The Small Symbol was created in their honour. Main roles play here four fractions with four digits denominators and numerators and a symbol √2. The Small Symbol by a graphical way tells the stories about two numerical networks that were created in the end of the last century.
Half Symbol
Understanding the natural numbers makes it easier to understand prime numbers, numerical structures, the special types of topology as well as irrational symbols such as the square root of two.
Our website is characterized by four key numbers: 16, 64, 256 and 4096. It was a good inspiration to asked four natural numbers (two pairs) to form Half Symbol. The numbers in the first pair have 16 digits. The numbers in the second pair have 64 digits. One number in each of pair got two roles - a role of an odd number and a role of an even number. Both pairs created two fractions. These two fractions form two points of a corridor in which √2 runs.
Square root of two is a symbol of irrationality while those fractions are symbols for transformation irrationality into rationality with positive or a negative deviation depending on our choice and with managing (required) accuracy of the deviation
- Half Symbol is based only on the a half of four key numbers (16 and 64)
- There are four types of fractions for expressing value that it very closed to value of √2 (four types of points of corridor in which √2 runs) but only two of them are displayed in Half Symbol.
Four Groups of Natural Numbers
- Natural numbers that are denominators and numerators of the fractions presented in the Small Symbol and Half Symbol belong to four groups of natural numbers.
- Three of these groups play very important role as regards a distribution of prime numbers.
- All these four groups of natural numbers can be characterized by the group of numbers called F numbers,
(F = 4x2 + y2). Exactly by F numbers belong to the sequence 1, 5, 145, 4901, 166465, 5654885, 192099601, 6525731525, 221682772225 .... - Number 5 is single prime number in the group of F numbers (it is very easy to prove it).
May be it will sound very stupid for many people educated "in mathematics science" but the natural numbers belong to the group of F numbers can be described by an attribute that provides the simplest explanation of the last Fermat theorem.
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From the book "Twelve Coils and Stories about Prime Numbers":
In any n-th generation, the absolute value of each of the four deviations is always lower than the absolute value of any deviation in previous generations.