A SIMPLE RULE TO DISTINGUISH PRIME FROM COMPOSITE
... For more than two millennia, the topic of prime numbers and the related issues is one of the most important topics in mathematics and has caught the attention of many great mathematicians. Finding a solution for calculating Pm+1 from Pm, in which Pm is the mth prime number, is still unresolved. The ...
... For more than two millennia, the topic of prime numbers and the related issues is one of the most important topics in mathematics and has caught the attention of many great mathematicians. Finding a solution for calculating Pm+1 from Pm, in which Pm is the mth prime number, is still unresolved. The ...
Hilbert Type Deductive System for Sentential Logic, Completeness
... Proof: The following is a proof of α→α α→[(α→α)→α], {α→[(α→α)→ α]}→{[α→(α→α)]→(α→α)], (α→(α→α))→(α→α), α→(α→α), α→α The first wff is an instance of Axiom (i), the second––of Axiom (ii), the third is inferred from the first two via modus ponens, the fourth is an instance of Axiom (i) and the fifth i ...
... Proof: The following is a proof of α→α α→[(α→α)→α], {α→[(α→α)→ α]}→{[α→(α→α)]→(α→α)], (α→(α→α))→(α→α), α→(α→α), α→α The first wff is an instance of Axiom (i), the second––of Axiom (ii), the third is inferred from the first two via modus ponens, the fourth is an instance of Axiom (i) and the fifth i ...
REVERSE MATHEMATICS Contents 1. Introduction 1 2. Second
... In reverse mathematics, subsystems of second-order arithmetic (Z2 ) are most often used. Z2 is a formal system consisting of language L2 and some axioms. From these axioms, we can deduce formulas, called theorems of Z2 . A subsystem of second-order arithmetic is a formal system consisting of languag ...
... In reverse mathematics, subsystems of second-order arithmetic (Z2 ) are most often used. Z2 is a formal system consisting of language L2 and some axioms. From these axioms, we can deduce formulas, called theorems of Z2 . A subsystem of second-order arithmetic is a formal system consisting of languag ...
LOGARITHMS OF MATRICES Theorem 1. If M=E(A), N = EiB
... lary to [2, Theorem 36, p. 30]). Theorem 10. If {Ai}T-o *s a commutative sequence of linearly independent logarithms of I and {ai}|"_0 is a sequence of real numbers such that 2^a-o a,v4,-£Log I, then ao is a rational number. Proof. Suppose {^4i}^0 and {o,}f_0 satisfy the hypothesis and that ao is an ...
... lary to [2, Theorem 36, p. 30]). Theorem 10. If {Ai}T-o *s a commutative sequence of linearly independent logarithms of I and {ai}|"_0 is a sequence of real numbers such that 2^a-o a,v4,-£Log I, then ao is a rational number. Proof. Suppose {^4i}^0 and {o,}f_0 satisfy the hypothesis and that ao is an ...
SUMS OF DISTINCT UNIT FRACTIONS PAUL ERDŐS AND
... Herbert S . Wilf raises several questions about R-bases, including : Does an R-basis necessarily have a positive density? If S consists of all positive integers and f (n) is the least number required to represent n, what, in some average sense, is the growth of f(n)? These two questions are answered ...
... Herbert S . Wilf raises several questions about R-bases, including : Does an R-basis necessarily have a positive density? If S consists of all positive integers and f (n) is the least number required to represent n, what, in some average sense, is the growth of f(n)? These two questions are answered ...
A Primer on Proving
... In a classroom setting, a problem that starts off with “prove this theorem” is an exercise in which students are assured that what they are working on can be proven by the word “theorem” in the problem statement. Neither teachers nor mathematicians call a statement “theorem” unless it’s been proved. ...
... In a classroom setting, a problem that starts off with “prove this theorem” is an exercise in which students are assured that what they are working on can be proven by the word “theorem” in the problem statement. Neither teachers nor mathematicians call a statement “theorem” unless it’s been proved. ...
Lecture 11: the Euler φ-function In the light of the previous lecture
... In the light of the previous lecture, we shall now look in more detail at the function defined there. Let n ≥ 1 be a natural number. Recall that we defined φ(n) to be the number of natural numbers 1 ≤ m ≤ n and coprime to n. This function is called Euler’s φ-function. For example, let’s calculate φ( ...
... In the light of the previous lecture, we shall now look in more detail at the function defined there. Let n ≥ 1 be a natural number. Recall that we defined φ(n) to be the number of natural numbers 1 ≤ m ≤ n and coprime to n. This function is called Euler’s φ-function. For example, let’s calculate φ( ...
Full text
... Definition 2. Let D ∗ (n) denote the total number of necklaces of n beads in two colors, say black and white. It is known that D ∗ (n) = 1 + D(n) [4]. Definition 3. Let D ∗∗ (n) denote the total number of necklaces of n beads in two colors, with the proviso that at least one bead must be black. Clea ...
... Definition 2. Let D ∗ (n) denote the total number of necklaces of n beads in two colors, say black and white. It is known that D ∗ (n) = 1 + D(n) [4]. Definition 3. Let D ∗∗ (n) denote the total number of necklaces of n beads in two colors, with the proviso that at least one bead must be black. Clea ...
p-3 q. = .pq = p,
... This theorem being not explicitly mentioned in Lewis's Symbolic Logic, I shall prove it here. Throughout this paper we shall follow Lewis's practice of ignoring the distinction, which is characteristic of Huntington's ...
... This theorem being not explicitly mentioned in Lewis's Symbolic Logic, I shall prove it here. Throughout this paper we shall follow Lewis's practice of ignoring the distinction, which is characteristic of Huntington's ...
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.