RU2619527C1 - Method of flowing generation of the sequence of figure numbers used in training the solution of the fermat equation - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: method comprises the steps of selecting the generating function for the first operand and calculating the square or cube of the first operand on the successive multiplier with the first ALU, storing the intermediate result in the multiplier register. Simultaneously, with the help of the second ALU, the first operand is transferred to the additional code up to 2 and summed with the intermediate result from the multiplication register. Repeating the actions for the second operand. At the same time, by means of the third ALU, the quotient on a sequential divisor is calculated. Writing a shape number from the first operand to the storage device. Performing actions for the second operand. Repeating the entire sequence of actions for the remaining operands.

EFFECT: efficient generation of a sequence of curly numbers of a given type.

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Description

The invention relates to the field of computer technology, and in particular to means for generating digital data in the form of numerical sequences. The main areas of application of this technical solution are teaching number theory using the Great Fermat equation as an example, mathematical research in the field of this theory, as well as cryptography and signal processing.

A number of author's mathematical methods prof. Shikhayeva K.N. and prof. Anokhina V.A. based on the properties of curly numbers and require the calculation of their sequences.

For learning to solve the Great Fermat equation, and in some other cases, it is assumed to use sequences of triangular and pyramidal numbers.

Figured triangular and pyramidal numbers, as well as their sequences, can be obtained on the basis of well-known expressions (N. Vilenkin Combinatorics. M.: Nauka Publishing House, 1969, p. 124):

- k-е треугольное число;Where

- k-th triangular number;

k is a natural number.

- k-е пирамидальное число;Where

- k-th pyramidal number;

k is a natural number.

The generating functions (1), (2) give a sequence of triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, ..., ∞ and a sequence of pyramidal numbers 1, 4, 10, 20, 35, 56, 84, 120, ..., ∞. The sequences of curly numbers obtained using expressions (1), (2) are devoid of zero, which makes these expressions unsuitable, in particular, for working with indefinite equations of number theory.

One of the main properties of curly numbers, an infinitely growing arithmetic square, is their streaming property, when any curly number is always equal to the sum of two already known numbers, that is, the number standing above it, plus the number standing in front of it.

The authors found that it is advisable to calculate these sequences for natural numbers 1, 2, 3, 4, 5, ..., ∞ on the basis of the generating functions (3) and (4), respectively, for triangular and pyramidal numbers.

where x is a natural number.

For x = 1, expressions (3) and (4) take respectively the form:

For x> 1, expressions (3) and (4) generate curly triangular and pyramidal numbers:

All of the above gives endless sequences:

последовательность фигурных треугольных чисел =0, 1, 3, 6, 10, 15, 21, 28, 36, 45, …, ∞,

a sequence of curly triangular numbers = 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, ..., ∞,

последовательность фигурных пирамидальных чисел =0, 1, 4, 10, 20, 35, 56, 84, 120, …, ∞,

a sequence of curly pyramidal numbers = 0, 1, 4, 10, 20, 35, 56, 84, 120, ..., ∞,

including zero and subsets of natural numbers (hereinafter, these sequences are simplistically named respectively as a sequence of triangular numbers and a sequence of pyramidal numbers), which are the unique solvers of the indefinite equations of number theory from the equations of doubling the square and the equation of the cube to the Fermat equation.

In particular, it is possible to study the Fermat’s Great Equation for educational purposes on the basis of the Pythagorean identity, taken by the authors in curly numbers (5).

where x, y, z are natural numbers;

L _{2 (x)} , L _{2 (y)} , L _{2 (z)} are the corresponding triangular numbers.

Identity (5) is obtained from the elementary identity (6) for x and similar identities for y and z.

where x is a natural number;

L _{2 (x)} is the corresponding x triangular number.

For example, for x equal to 10, using expression (3), we get L _{2 (x)} = 45, and identity (6) takes the form 10 ² = 10 + 2-45, that is, 100 = 100, which means the fidelity of this identity.

For example, taking x = 3, y = 4 and z = 5 and taking in identity (5) x ² , y ² , z ² instead of x, y, z, we get:

.

.

Substitution of numbers on the left side of this identity gives:

x ⁴ + y ⁴ -z ⁴ = 3 ⁴ +4 ⁴ -5 ⁴ = 81 + 256-625 = -288.

The right side of the same identity has the following form:

The correctness of the identity is confirmed by the equality of its left and right sides.

Identity (5) allows one to obtain Fermat equations of any degree and provides their solution thanks to the study of the right-hand side of the indicated identity, which is the Fermat equation given in curly numbers, which expands the educational possibilities.

In particular, students perform the following sequences of transformations.

From expression (5):

2 ² +3 ² -4 ² = 4 + 9-16 = -3

2 + 3-4 = 1

2 (L _{2 (2)} + L _{2 (3)} -L _{2 (4)} ) = 1 + 3-6 = -2

-4 + 1 = -3

-3 = -3

Identity obtained.

Also from the expression (5) in the linear form of recording calculations:

3 ² +4 ² -5 ² = 0 = 3 + 4-5 = 2 + 2 (L _{2 (3)} + L _{2 (4)} -L _{2 (5)} = 3 + 6-10 = -1) = - 2 + 2 = 0

Identities obtained.

Further, the student is invited to take in identity (5) instead of the numbers x, y, z, the numbers x ² , y ² , z ² , x ³ , y ³ , z ³ , ... x ⁿ , y ⁿ , z ⁿ and construct the corresponding equations.

For numbers x ⁿ , y ⁿ , z ^{n, the} student receives:

If we assume that this equation has an integer solution, then we will get:

That is, 1 = 0, which cannot be, and the equation x ⁿ + y ⁿ = z ⁿ cannot have integer solutions.

As an example:

The identity is compiled, and dividing the numbers of its right-hand side (equation in curly numbers) by the number 25, you will get:

That is, 1 = 0, which cannot be, on the basis of which the student concludes that the equation x ⁴ + y ⁴ = z ⁴ , where x = 3, y = 4, z = 5, cannot have integer solutions.

The sequence of pyramidal numbers L _{3 (x)} also finds its application.

The generation of sequences of curly numbers, especially including large-digit numbers in mathematical research, in cryptography and in signal processing, makes it necessary to use highly efficient computer calculations.

From patent document RU 2549129 C1 of 04/20/2015, a device for testing simplicity of numbers is known that implements the calculation of the expression N _n = (n ² -n) / 2, where n is the number to be tested. The known device is characterized by increased performance when testing numbers for simplicity, however, the device has limited functionality, which makes it impossible to use it to generate a sequence of pyramidal numbers and memorize any numerical sequences. In addition, the design of the known device is not optimized for the computational mode in the stream, the need for which is dictated by practice.

The task is to effectively generate a sequence of curly numbers of a given type and memorize this sequence for future use.

The technical result provided by the present invention is to expand the functionality of the device for generating a numerical sequence based on both expression (3) and expression (4), with storing the results obtained; also the technical result is to increase the performance of the specified computing device, subject to the conditions of simplicity of its design and its scalability.

The technical result is achieved due to the fact that the method of streaming generating a sequence of curly numbers by the generating function (3) and / or (4) based on a given set of natural numbers x, forming at least the first and second operands, is characterized by the fact that the first operand is converted to binary form, select the generating function and calculate the square or cube of the first operand on a serial multiplier with the first arithmetic logic unit (ALU), storing the intermediate result in the multiplication register. At the same time, using the second ALU, the first operand is transferred to an additional code of up to 2, after which it is summed with the intermediate result from the multiplication register, storing the intermediate result in the summation register. Then, the above actions are performed for the second operand, at the same time, using the third ALU, the quotient of dividing the intermediate result in the summing register by the number 2 or 6 in binary form on a serial divider is calculated and the figured number from the first operand is written to the storage device. After that, the above actions are performed for the second operand and the entire sequence of actions for the remaining operands is repeated.

The invention is illustrated by the following illustrations.

FIG. 1: block diagram of a computing device.

FIG. 2: the main steps of the figure number generation algorithm.

The implementation of the present invention is shown in the example of the design of the computing device.

For streaming the generation of a sequence of curly numbers by the generating function (3) or (4), a computing device is used (Fig. 1).

The computing device contains an operand register 1, a generating function selection register 2, a multiplication register 3, a summing register 4, a division register 5, a first ALU 6 multiplication, a second ALU 7 summation, a third ALU 8 division, a control device 9 and a storage device 10.

Registers 3 and 5 are shift. Register 3 is configured to perform a shift to the right, and register 5 is configured to carry out a shift to the right and shift to the left. The storage device 10 is configured to store a sequence of numbers in binary form.

The first ALU 6 and register 3, together with the control device 9, form a serial number multiplier in binary form to calculate the square or cube of the operand.

The second ALU 7, register 4 and control device 9, together form a functional unit for translating the operand into an additional code of up to 2 and summing while maintaining an intermediate result.

The third ALU 8, the register 5 and the control device 9 form a sequential divider of numbers in binary form.

Register 1 of the operand, a serial multiplier, a functional unit for translating the operand into an additional code of up to 2 and summing, a serial divider, as well as a storage device 10 are informationally connected to each other sequentially through their inputs and outputs.

In this case, the information output of register 1 is electrically connected to the first input of ALU 7, the second inputs of ALU 6 and register 3. The output of register 2 is connected to the first input of device 9. The output of ALU 6 is connected to the first input of register 3. The first output of register 3 is connected to the first input ALU 6 and the second input of ALU 7. The second output of register 3 is connected to the second input of device 9. The output of ALU 7 is connected to the second input of ALU 8 through register 4. The output of ALU 8 is connected to the first input of register 5. The second input of register 5 is connected to device 9 The first output of register 5 is connected with the first input of ALU 8. In The second output of register 5 is connected to the third input of device 9 and device 10. ALU 6, 7, and 8, as well as registers 3, 5 are connected to device 9 through their control inputs and outputs.

In the process of functioning of the described computing device, a method for streaming the generation of a sequence of curly numbers by the generating function (3) and / or (4) is implemented.

First of all, a set of natural numbers is set, consisting of at least two numbers, on the basis of which a sequence of curly numbers is to be generated. It is permissible to use a set of numbers that is set dynamically during the operation of a computing device. Numbers of this set are used as operands.

From the indicated set, numbers are selected, and the generating functions are also set and this data is supplied to the input of the computing device. The rules for choosing numbers and their associated functions are determined by the task. At the output of the computing device receive a sequence of curly numbers recorded in the memory of the storage device. Using this sequence of numbers, carry out mathematical research in the field of number theory or teach this theory, etc.

Choosing a positive integer translates it into binary form, and then writes it to register 1. At the same time, the generating function (3) or (4) is also chosen. If function (3) is selected, then the number 0 is written in register 2, and if function (4) is selected, the number 1 is written in register 2.

After writing to the registers 1, 2, the device 9 starts its operation, which sets and controls the course of the computing process.

The operand is received at the information input of the serial multiplier (ALU 6 and register 3). The device 9 controls the operations of the shift to the right and the entries in the register 3, reading its current contents. Moreover, if the number 0 is written in register 2, the operand is squared, and if the number 1 is written in register 2, the operand is cube, resulting in the value "x ² " or "x ³ " for function (3) or (4) respectively, is calculated and stored in register 3 as an intermediate result.

Simultaneously with the indicated actions, the ALU 7 translates the operand into an additional code of up to 2 (complement of two) by inverting and shifting, obtaining the value “-x” for generating functions. After that, the resulting binary number is summed with the intermediate result from register 3. As a result of the operation of the functional unit, which includes ALU 7 and register 4, the value “x ² -x” or “x ³ -x” for function (3) or respectively, for function (4). This value is stored as an intermediate result in register 4, from which it then enters the input of the serial divider.

The contents of register 4 are fed to one of the information inputs of the serial divider of numbers in binary form (ALU 8 and register 5). If the number 0 was written to register 2, then the number 2 is sent to the other information input of the serial divider in binary form, and if to the register 2 written the number 1, then serves the number 6 in binary form. The device 9 controls the operations of the shift to the right and left, as well as the entries in the register 5, reading its current contents. As a result, the value of the generating function of the number x is obtained and written to the storage device 10, forming in it a sequence of curly numbers in binary form. Also, the device 10 stores data characterizing the corresponding operand and the generating function.

The next operand is placed in register 1 immediately after receiving the intermediate result in register 4 and the above actions are repeated, which ensures the streaming mode of operation of the computing device.

Thus, the computing device allows you to generate a numerical sequence both on the basis of expression (3) and expression (4), with the storage of the results, as a result of which the functionality of this device is expanded.

Thanks to the streaming mode of operation involving three ALUs operating simultaneously, two shift and one simple registers, the performance of the computing device is improved. At the same time, its design does not contain superfluous and / or complex elements and has good scalability for adaptation to work with high-digit numbers, which generally characterizes the high efficiency of generating a sequence of curly numbers.

Claims (1)

A method for streaming the generation of a sequence of curly numbers by the generating function L _{2 (x)} = (x ^2- x) / 2 and / or L _{3 (x)} = (x ^3- x) / 6 based on a given set of natural numbers x, forming at least at least the first and second operands, characterized in that they convert the first operand into binary form, select the generating function and calculate the square or cube of the first operand on a serial multiplier with the first ALU, storing the intermediate result in the multiplication register, at the same time using the second ALU translate the first operand to There is an additional code up to 2, after which they are summed up with the intermediate result from the multiplication register, storing the intermediate result in the summation register, then the above actions are performed for the second operand, at the same time, using the third ALU, the quotient of dividing the intermediate result in the summation register by 2 is calculated or 6 in binary form on a serial divider and write the curly number from the first operand to the storage device, after which the above actions are performed for the second operand and repeat the entire sequence of actions for the remaining operands.

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