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**1 - 4**of**4**### University of Białystok Sorting Operators for Finite Sequences Yatsuka Nakamura

"... Summary. Two kinds of sorting operators, descendent one and ascendent one are introduced for finite sequences of reals. They are also called rearrangement of finite sequences of reals. Maximum and minimum values of finite sequences of reals are also defined. We also discuss relations between these c ..."

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Summary. Two kinds of sorting operators, descendent one and ascendent one are introduced for finite sequences of reals. They are also called rearrangement of finite sequences of reals. Maximum and minimum values of finite sequences of reals are also defined. We also discuss relations between these concepts. MML Identifier: RFINSEQ2.

### University of Białystok Sorting Operators for Finite Sequences Yatsuka Nakamura

"... Summary. Two kinds of sorting operators, descendent one and ascendent one are introduced for finite sequences of reals. They are also called rearrangement of finite sequences of reals. Maximum and minimum values of finite sequences of reals are also defined. We also discuss relations between these c ..."

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Summary. Two kinds of sorting operators, descendent one and ascendent one are introduced for finite sequences of reals. They are also called rearrangement of finite sequences of reals. Maximum and minimum values of finite sequences of reals are also defined. We also discuss relations between these concepts. MML Identifier: RFINSEQ2.

### Witt’s Proof of the Wedderburn Theorem 1

"... notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every natural number a and for every real number q such that 1 < q and q a = 1 holds a = 0. (2) For all natural numbers a, k, r and for every real number x such that 1 < x and 0 < k holds ..."

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notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every natural number a and for every real number q such that 1 < q and q a = 1 holds a = 0. (2) For all natural numbers a, k, r and for every real number x such that 1 < x and 0 < k holds x a·k+r = x a · x a·(k− ′ 1)+r. (3) For all natural numbers q, a, b such that 0 < a and 1 < q and q a − ′ 1 | q b − ′ 1 holds a | b. (4) For all natural numbers n, q such that 0 < q holds q n = q n. (5) Let f be a finite sequence of elements of N and i be a natural number. If for every natural number j such that j ∈ dom f holds i | f j, then i | ∑ f. (6) Let X be a finite set, Y be a partition of X, and f be a finite sequence of elements of Y. Suppose f is one-to-one and rng f = Y. Let c be a finite sequence of elements of N. Suppose lenc = len f and for every natural number i such that i ∈ domc holds c(i) = f(i). Then cardX = ∑c.

### Properties of Primes and Multiplicative Group of a Field

"... Summary. In the [16] has been proven that the multiplicative group Z/pZ ∗ is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it ..."

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Summary. In the [16] has been proven that the multiplicative group Z/pZ ∗ is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group. Meanwhile, in cryptographic system like RSA, in which security basis depends upon the difficulty of factorization of given numbers into prime factors, it is important to employ integers that are difficult to be factorized into prime factors. If both p and 2p + 1 are prime numbers, we call p as Sophie Germain prime, and 2p + 1 as safe prime. It is known that the product of two safe primes is a composite number that is difficult for some factoring algorithms to factorize into prime factors. In addition, safe primes are also important in cryptography system because of their use in discrete logarithm based techniques like Diffie-Hellman key exchange. If p is a safe prime, the multiplicative group of numbers modulo p has a subgroup of large prime order. However, no definitions have not been established yet with the safe prime and Sophie Germain prime. So it is important to give definitions of the Sophie Germain prime and safe prime. In this article, we prove finite subgroup of the multiplicative group of a field is a cyclic group, and, further, define the safe prime and Sophie Germain prime, and prove several facts about them. In addition, we define Mersenne number (Mn), and some facts about Mersenne numbers and prime numbers are proven.