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On a Density for Sets of Integers 1 Introduction 2 A
On a Density for Sets of Integers 1 Introduction 2 A

... (3) and upper versions, simply by replacing the above n n→∞ limit by lim inf and lim sup, respectively. The aim of this work is to investigate the properThe asymptotic density is said to exist if and only if both the lower and upper asymptotic densities do ties of the Dirichlet density as defined in ...
On prime factors of integers which are sums or shifted products by
On prime factors of integers which are sums or shifted products by

... The proofs of both (8) and (9) depend on estimates for linear forms in the logarithms of algebraic numbers, see [32]. Győry and Sárközy [17] proved that the conjecture holds in the special case that at least one of the numbers a, b, c, a/b, b/c, a/c has bounded prime factors. This work, later ref ...
A New 3n − 1 Conjecture Akin to Collatz Conjecture
A New 3n − 1 Conjecture Akin to Collatz Conjecture

Solutions to Assignment 7 22.3 Let S be the set of all infinite
Solutions to Assignment 7 22.3 Let S be the set of all infinite

"Topology Without Tears" by Sidney A. Morris
"Topology Without Tears" by Sidney A. Morris

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Wilson quotients for composite moduli

On Divisors of Lucas and Lehmer Numbers
On Divisors of Lucas and Lehmer Numbers

... for n sufficiently large in terms of the number of distinct prime factors of ab. We remark that the factor 104 which occurs on the right hand side of (7) has no arithmetical significance. Instead it is determined by the current quality of the estimates for linear forms in p-adic logarithms of algebr ...
Diskrete Mathematik für Informatik (SS 2017)
Diskrete Mathematik für Informatik (SS 2017)

... According to legend, the power of exponential growth was already known by the Brahmin Sissa ibn Dahir (ca. 300-400 AD): As a reward for the invention of the game of chess (or its Indian predecessor Chaturanga) he asked his king to place one grain of rice in the first square of a chessboard, two in t ...
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Introduction to Discrete Mathematics

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ON THE ERROR TERM OF THE LOGARITHM OF THE LCM OF A

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The Math Encyclopedia of Smarandache Type Notions / Vol. 1

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1 Q1: A wire that weighs 24 kilograms is cut into two pieces so that

... A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D. EACH statement ALONE is sufficient. E. Statemen ...
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Document

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with 1 - American Mathematical Society
with 1 - American Mathematical Society

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Document

Extremely Abundant Numbers and the Riemann Hypothesis
Extremely Abundant Numbers and the Riemann Hypothesis

Introduction to Number Theory 2
Introduction to Number Theory 2

University of Chicago “A Textbook for Advanced Calculus”
University of Chicago “A Textbook for Advanced Calculus”

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Theorem



In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.
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