On the relationship of prime and irrational numbers

After some of my research on prime numbers, I found an interesting connection with irrational numbers. This relationship answers the question of why prime numbers are so “chaotic” and why they are so complex. Under the cat an explanation of this connection and a variant of the improved RSA algorithm.


Consider the set $ \ lbrace 2n + 3m \ vert n, m \ in \ mathbb {N} \ rbrace $. Now let's try to organize it. That is, find a way to find the next pair of numbers n and m, knowing the previous one. It is obvious that: 2 + 2 + 2 = 3 + 3 and 2 + 2> 3, 2 <3. Thus, the pairs of numbers are distributed as follows:

(1.0), (0.1), (2.0) , (1,1), (3,0), (2,1), (4,0), (3,1), (5,0) ...

We note that the order and, accordingly, the method of obtaining the following pairs of numbers. There are no problems here and the task is trivial.
Now consider the set$ \ lbrace 2n + \ pi m \ vert n, m \ in \ mathbb {N} \ rbrace $. Unfortunately or fortunately, this set will not be arranged in the same sense as the previous one:

(1.0), (0.1), (2.0), (1.1), (3.0), ( 0,2), (2,1), (4,0), (3,1), (0,3) ...

If you decide that you have found the exact order, then finish these pairs further and see that it is broken. The “chaos” of these pairs of numbers is directly related to the irrationality of the number$ \ pi $, proved by Johann Lambert in 1761. Indeed, in order to line up the pairs, we first try to lay a segment of length 2 into a segment of length$ \ pi $. We try to put the obtained residue into a segment of length 2. It holds together only once. This means that our remnant will “play” its role already on a segment of length$ 2 \ pi $where it is no longer two segments of length 2, but three. Making this operation further, it becomes clear that as soon as we get the impression that we have found order, it will break down after a certain number of steps. Since the last, not yet used, the remainder will sooner or later “play” its role and the order will change. Therefore, the question of finding a “good” algorithm for this problem remains open.

Few definitions

Let be $ (\ mathbb {R}, +) \ simeq (\ mathbb {R} _ {> 0}, \ oplus) $where $ f $ - isomorphism such that:
$ f (x \ oplus y) = f (x) + f (y) $
And, accordingly, for $ g $ - back to $ f $:
$ g (x + y) = g (x) \ oplus g (y) $.
Now we define the set of interest to us:
$ W _ {\ oplus} = \ lbrace a_ {2} f (2) + a_ {3} f (3) + \ dots + a_ {n} f (n) + \ dots \ vert \ forall m \ in \ mathbb {N}, a_ {m} \ in \ mathbb {N} \ rbrace \ setminus \ lbrace 0 \ rbrace $
$ \ Rightarrow W _ {\ oplus} \ subset \ mathbb {R} _ {> 0} $
Let it go $ F (x, y) = f (x) + f (y) $. Then:
$ g (F (x, y)) = x \ oplus y $
AND $ \ mathbb {T} $ - the image of the set $ W _ {\ oplus} $ to display $ g $.
And finally$ \ mathbb {P} _ {\ oplus} = \ lbrace p \ in \ mathbb {T} \ vert \ forall w_ {1}, w_ {2} \ in W _ {\ oplus}, g (w_ {1} + w_ {2}) \ neq p \ rbrace $ - set of prime numbers for operation $ \ oplus $.
Now it is easy to clarify these definitions on our usual example. For the multiplication operation,$ f (x) = log_ {2} (x) $. And many$ W $ - this $ log_ {2} (\ mathbb {N}) $. It is worthwhile to stop here and explain why this is important.

Connection itself

In fact, using isomorphism, we have obtained that the complexity of all problems about primes is equivalent to problems about sums of logarithms that are irrational. That is, as we saw in the example with a set of numbers$ \ pi $and 2, it is the irrationality that introduces chaos. Likewise, the irrationality of logarithms distributes prime numbers on the number line in a practically random fashion. There is a difficulty in ordering the pairs n and m in the set, for example,$ \ lbrace n + mlog_ {2} 3 \ rbrace $. In other words, the simplicity of a number directly depends on, for example, some decimal place in the number$ log_ {2} 3 $. But we have defined primes not only for multiplication, but in general for an arbitrary binary operation. I did this to show that our primes are not unique.


For binary operation x + xy + y:
$ \ mathbb {P} = \ lbrace 2,4,6,10,12,16,18,22,28,30,36,40,42,46 ... \ rbrace $.
The randomness of a given set is characterized by the irrational values ​​of isomorphism on natural numbers. Moreover, isomorphism, apparently, is not expressed through elementary functions. Here we have constructed other primes for the operation, the distribution of which obviously does not depend on the distribution of ordinary primes. This allows us to construct RSA on an arbitrary binary operation such that the isomorphism is irrational. After all, the logarithm function is too “good” for cryptanalysts. And here she behaves in an absolutely unpredictable way. It is also possible, and vice versa, to construct an isomorphism by which a commutative binary operation will be defined.

Based on arbitrary primes, we change the problem of decomposing a composite number into factors by the task of decomposing an almost arbitrary irrational number by the sum of two others from a given set. Something tells me that this task should belong to the class NP.


Mankind has not yet solved many problems about primes, as mathematics throws up an infinite number of similar problems. It will naturally be asked what to do with it? My proposal is to consider all the theorems of Number Theory not for addition and multiplication, but for addition and an arbitrary commutative binary operation closed on positive integers. Then each statement about prime numbers would be only a consequence of certain properties of the operation. For example, the infinity of prime numbers would be a consequence of the monotony of the operation and its quite rapid growth. But this is a topic for a separate article. Thanks for attention.

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