= 0, = 1 and = + 2 , ≥ 0 (1.1) and
= 0, = 1 and = + 2 , ≥ 0 (1.2)
Although the two sequences of **natural** **numbers** were introduced by E. Jacobsthal 2 in 1919, they have not drawn much attention until their applications to curve was studied by Horadam 3,4,5 in 1988. B. Srinivasa Rao 6 has proved the squares in the two pairs of sequences given in (1.1) and

In this paper, I will argue that this view about reference to **numbers** in **natural** language is fundamentally mistaken. **Natural** language presents a very different view of the ontological status of **natural** **numbers**. On this view, **numbers** are not primarily abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of **numbers**. **Natural** language moreover provides support for another view of the ontological status of **numbers**, on which **natural** **numbers** do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to **numbers**, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language that reference to pure **numbers** is permitted.

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Fibonacci **numbers** are well known for **some** of its interesting properties [1]. Golden ratio is one of the amazing property. Fibonacci **numbers** and Golden ratio have applications in physics, astrophysics, biology, chemistry and technology [2]. This article proves property of determinant of Fibonacci **numbers** , geometric consideration for Golden ratio and construction of Fibonacci subsequence from a Fibonacci sequence. The determinant of first n 2 n >= 2 of a Fibonacci **numbers** is zero. The golden ratio is shown to be sequence of lines converging to a line with slope as golden ratio. Method of constructing a subsequence from a Fibonacci sequence is presented. Examples presented in [2] is not exhaustive list of applications. One may find other applications in different domains of science.

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Types of **square** roots of negative worksheet links to consolidate the problem. Door into the **square** roots negative **numbers** answers are place value chart, and centimeter variations for distilling complex or the world.
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50 -0.012859 500 -0.012859 1050 -0.000920261
Still, we cannot find a lower limit for D in this case.If D were greater than 0 this would prove the conjecture. The condition is not necessary, as it might be too strong for the statement we want to prove. If D > 0 this asserts that not only are there infinitely many prime **numbers** with continued fraction of a given length but that there are infinitely many which satisfy Theorem 4.1. We can, thus, focus on the lengths of the periods of continued frac- tions for prime **numbers**, no matter the partial quotients of the continued fraction. Numerical testings exhibit very interesting results that I could ver- ify for very large **numbers**, up to 30,000,000. We can, thus, state a conjecture:

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between different "ranks" of coal. "Typical" coal (rank not specified) usually means bituminous coal, the most common fuel for power plants (27 GJ/t).
• **Natural** gas: HHV = 1027 Btu/ft3 = 38.3 MJ/m 3 ; LHV = 930 Btu/ft3 = 34.6 MJ/m 3

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Mathematics is the language of patterns and relationships, and is used to describe anything that can be quantified. The main goal of Number theory is to discover interesting and unexpected relationships. It is devoted primarily to the study of **natural** **numbers** and integers. In Number theory, Pythagorean triangles have been a matter of interest to various mathematicians. For an extensive variety of fascinating problems one may refer [1-3]. Apart from the polygonal **numbers** we have **some** more fascinating patterns of **numbers** namely Jarasandha **numbers**, Nasty **numbers** and Dhuruva **numbers** are presented in [4-7].

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development and education (2), there still room for new research to clarify the mutual relationship between the **numbers** and number patterns.
In **natural** **numbers**, various subsets have been recognized by ancient
mathematicians. **Some** are odd **numbers**, prime **numbers**, oblong **numbers**, triangular **numbers** and squares. These **numbers** shall be identified by number patterns. Recognizing number patterns is also an important problem-solving skill. Working with number patterns leads directly to the concept of functions in mathematics: a formal description of the

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105. If B is the set of integers b such that a ≤ b for all a ∈ A, then we are given that B is nonempty. Since **some** positive number lies in A, we know that B is contained in the positive integers. Therefore B has a minimal element b ∗ by the Well-Ordering Property. If b ∗ ∈ A then b ∗ is a maximal element of A because a ∈ A implies a ≤ b ∗ . To finish the proof, it suffices to eliminate the possibility that b ∗ 6∈ A. In that case a < b ∗ for all a and hence a ≤ b ∗ − 1 for all a ∈ A. This contradicts the defining condition that b ∗ is the least integer which is greater than or equal to each element of A. Hence b ∗ is a maximal element of A.

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interest in **some** area of mathematics, computer science, physics and philosophy ([1], [4], [8], [9], [15], [17] - [21]). There are many situations in the above subjects where it is more convenient to consider a collection like multiset. e.g., the repeated eigen values of a matrix, prime factors of a positive integer, repeated observations in a statistical sample, data structure etc. Although the root of the studies in multiset is in combinatorics from ancient times ([25], [26], [27]), the modern research in this field about the structural development in multiset context is a relatively new concept. **Some** research works on the relations and functions in multiset context ([13], [14],

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OF **NATURAL** **NUMBERS** AND THE RATIO OF GAMMA FUNCTIONS
FENG QI AND BAI-NI GUO
Abstract. In this article, using Stirling’s formula, the series-expansion of digamma functions and other techniques, two inequalities involving the geo- metric mean of **natural** **numbers** and the ratio of gamma functions are obtained.

1. Introduction and main results
It is stated in [18] that the Catalan **numbers** C n for n ≥ 0 form a sequence of **natural** **numbers** that occur in tree enumeration problems such as “In how many ways can a regular n-gon be divided into n − 2 triangles if different orientations are counted separately?” whose solution is the Catalan number C n−2 . The Catalan **numbers** C n can be generated by

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This paper provides a comprehensive description of the situation in the [1, N] setting.
We describe the spectrum of attainable values, establish constructive existence results and obtain characterizations of sets attaining extremal (and **some** near-extremal) values.
It transpires that in order to answer the above questions for subsets of a given interval [1, N] (N ∈ N ), it is helpful to consider the problem from another angle. For s ∈ N , we can ask: what is the range of cardinalities |{(x, y) ∈ S 2 : x + y ∈ S}| that can be attained as S runs through all size-s subsets of N ? What is the smallest N for which all of these values are attained by subsets of [1, N ]? We shall see that the “tipping point” for the problem occurs at N = 2s − 1, and that sum-free sets play a crucial role.

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, 2 8
# , j k
The purpose of this paper is to establish the facts men- tioned in items (B) (see Section 1) and (C) (see Section 2). Bounds for **some** of the larger cases of these **numbers** will be given in a subsequent paper. The methods we use are not complicated, but **some** imagination is required to find an approach that reduces the number of cases to a manageable level.

Giannis Prelorentzos, Professor University of Ioannina
Τ ΗΕ Τ HEORY OF N ATURAL L AW IN T HOMAS A QUINAS Thomas Aquinas’ theory of **natural** law was a great innovation at a time when the world of jurists was still under the inﬂ uence of Augus- tinian legalism, which remained attached to the ecumenical character of canonical law and the authority of the Gospels. Thomas Aquinas’ great contribution was his focusing on the secular nature of law; in other words, the “de-sanctiﬁ cation” of law, the acknowledgement of its role in a secular context, that is, within a society; not in solitude and contemplation. The nature of things lies in the foundation of **natural** law, which is taken for granted in the context nature. On the other hand, positive law has been formulated by the divine or human law, being the result of an agreement.

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1. Adding a single vertex in every step. In its simplest form, this strategy supposes we have added k vertices to the initial graph in G 11 , so at the k + 1-st step we have a graph or order 12 + k if we take into account the vertex v adjacent to all the vertices of the original graph. For the k + 1-st vertex, we need to try all possible connections to the previous k vertices that have been added and then pick a feasible neighborhood that is compatible with the rest of the neighborhoods chosen so far. A nice property of this strategy is that one can assume that, for **some** ordering of the list of feasible neighborhoods, their assignment to the vertices of the extended graph are in non-decreasing order. Intuitively, this means that, at a successful termination of this extension process, the vertices of **some** graph Y ∈ G 7 will have been “discovered” in **some** order defined by the compatible neighborhoods attached to them. A drawback of blindly selecting the adjacency of the k+1-st vertex with respect to the k vertices previously added is that one may end up with a graph of order k + 1 that is not a subgraph of any graph in G 7 .

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Balasubramani Prema Rangasamy
Former Student of Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India
Abstract
My previous work dealt finding **numbers** which relatively prime to factorial value of certain number, high exponents and also find the way for finding mod values on certain number’s exponents. Firstly, I retreat my previous works about Euler’s phi function and **some** works on Fermat’s little theorem.

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A value of 0.60 would be a good number for general use.. general use.[r]

The authors would like to thank the referee for his/her helpful comments. The first author was partially supported by the National **Natural** Science Foundation of China (Grant No. 11371184) and the Nat- ural Science Foundation of Henan Province (Grant No. 162300410086, 2016B259, 172102410069).
The second author was partially supported by the National **Natural** Science Foundation of China (Grant No. 11801451) and the **Natural** Science Basic Research Plan in Shaanxi Province of China (No. 2017JQ1001).

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