Euler's Function

I. Concept of interconnectivity

1. Definition

Mutual Quality(Relatively prime sale) is also calledMutual Element. If the maximum public factor of N integers is 1, the N integers are mutually qualitative.

For example, the maximum public factor of 8, 10 is 2, not 1, so it is not an integer mutual quality.

The maximum public factor of 7, 10, 13 is 1, so this is an integer mutual quality.

5 and 5 are not mutually qualitative, because the public factors of 5 and 5 are 1 and 5.

1. It is a multiple relationship with any number, but it is mutually dependent with any number. Because the factor of 1 is only 1, and the principle of the prime number is: as long as the public factor of two numbers is only 1, the two numbers are the prime numbers. 1 has only one factor (So 1 is neither a prime number nor a combination of numbers), and the other public factors of 1 and other numbers cannot be found,So 1 and any number are mutually qualitative (except 0 ).

Description: for example, if C and M are mutually qualitative, write (C, M) = 1.

The elementary school mathematics textbook defines the number of interchange items as follows: "There is only one common number, which is called the number of interchange items ."

The "two numbers" here refer to natural numbers.

"There is only one common appointment ."

2. Discriminant Method

(1) Two Different prime numbers must be mutual prime numbers.

For example, 2, 7, 13, and 19.

(2) One prime number, and the other is not a multiple of them. These two numbers are the prime numbers.

For example, 3, 10, 5, and 26.

(3) 1. It is neither a prime nor a combination. It is a mass number together with any natural number. For example, 1 and 9908.

(4) two adjacent natural numbers are mass numbers. For example, 15 and 16.

(5) The two adjacent odd numbers are mass numbers. For example, 49 and 51.

(6) The larger number is the prime number. For example, 97 and 88.

(7) The two numbers are all equal numbers (two numbers have big difference). Compared with all the prime factors in decimal places, these two numbers are not the approximate numbers of large numbers.

For example, 357 and 715,357 = 3 × 7 × 17, and 3, 7, and 17 are not 715 digrams.

(8) The two numbers are all equal (the difference between two numbers is small). All prime factors of the difference between the two numbers are not equal to the approximate number of decimal places. For example, 85 and 78. 85-78 = 7 is not the approximate number of 78, and these two numbers are the mutual number.

(9) The two numbers are all Union numbers. All prime factors with a larger number divided by the remainder of a decimal number (not "0" and greater than "1") are not equal to the decimal number, these two numbers are mutual prime numbers. For example, 462 and 221

462 bytes 221 = 2 ...... 20,

20 = 2 × 2 × 5.

Neither 2 nor 5 is an appointment of 221. These two numbers are mutual numbers.

(10) subtraction. For example, 255 and 182.

255-182 = 73, observed 73182.

182-(73 × 2) = 36, apparently 3673.

73-(36 × 2) = 1,

(255,182) = 1.

Therefore, these two numbers are mutual numbers.

There are two differences between the three or more natural numbers: one is that these natural numbers are mutually qualitative. Such as 2, 3, and 5. The other is not of mutual quality. For example, 6, 8, and 9.

Ii. Euler's Function 1. Basic Content In number theory, for a positive integer N, the Euler's function is the number of numbers that are less than or equal to the number of numbers that intersect with N. It is also known as Euler's totient, function, Phi function, and Euler's business number. For example, Phi (8) = 4, because 1, 3, 5, 7 and 8 are mutually qualitative. The fact of the ring theory derived from the Euler's function and the Laplace Theorem constitute the proof of the Euler's theorem. The general formula of the function: φ( x) = x (1-1/P1) (1-1/P2) (1-1/P3) (1-1/P4) ..... (1-1/PN), where p1, p2 ...... All prime factors whose PN is X, and whose X is not an integer of 0. (1) = 1 (the number of unique and 1 mutual quality is 1 itself ). (Note: Each prime factor is only one. For example, 12 = 2*2*3, then PHI (12) = 12 * (1-1/2) * (1-1/3) = 4) if n is the k power of prime number P, Phi (n) = P ^ K-P ^ (k-1) = (PM) P ^ (k-1 ), except for the multiples of P, all other numbers are in the same quality as N. Assume that N is a positive integer and the number of positive integers not greater than N and with N is expressed as the Euler's function value of N. Here the function is: n→n, n → PHI (n) is called the Euler's function. Euler's function is a product function-if M, N are mutually qualitative, Phi (Mn) = PHI (m) PHI (n ). Special Properties: WHEN n is an odd number, Phi (2n) = PHI (n) proves to be similar to the above. 2. proof: Let A, B, and C be a set of numbers with M, N, and Mn. According to the Chinese Remainder Theorem, A * B and C can establish a one-to-one correspondence relationship. Therefore, we can use the basic arithmetic theorem for the values of PHI (N). If n = Beijing P ^ (α (subscript p) p | n then PHI (n) = Beijing (p-1) P ^ (α (subscript P)-1) = n squared (1-1/p) p | n e. g. PHI (72) = PHI (2 ^ 3 × 3 ^ 2) = (2-1) 2 ^ (3-1) × (3-1) 3 ^ (2-1) = 24 Relationship Between Euler's theorem and ferma's small theorem to any positive integers a, m, m> = 2 with a ^ PHI (m) 1_1 (mod m) that is, the Euler's theorem. When M is a prime number P, this formula is: a ^ (p-1) 1_1 (mod m), that is, the Fermat's theorem.