LINEAR EQUATIONS WITH THE EULER TOTIENT FUNCTION AMS

... Problems of a similar nature were considered previously, most notably in the series of papers [4], [5], [6] and [7], where the sets of positive integers n such that φ(n) = φ(n + 1), or σ(n) = σ(n + 1), or ω(n) = ω(n + 1), or Ω(n) = Ω(n + 1) or τ (n) = τ (n + 1), where σ, ω, Ω and τ are the sum of di ...

... Problems of a similar nature were considered previously, most notably in the series of papers [4], [5], [6] and [7], where the sets of positive integers n such that φ(n) = φ(n + 1), or σ(n) = σ(n + 1), or ω(n) = ω(n + 1), or Ω(n) = Ω(n + 1) or τ (n) = τ (n + 1), where σ, ω, Ω and τ are the sum of di ...

THE DISTRIBUTION OF PRIME NUMBERS Andrew Granville and K

... equivalent to a theorem about an analytic function... A proof of such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems clear. We have cer ...

... equivalent to a theorem about an analytic function... A proof of such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems clear. We have cer ...

1. Modular arithmetic

... Can you make a guess about the general pattern? 3.1. Euler's function. How does Fermat's Little Theorem generalize to numbers m that are not prime? The key thing in the proof is that f1; 2; ; p 1g are coprime with p so that they have multiplicative inverses in Zp. It is then natural to de ne ...

... Can you make a guess about the general pattern? 3.1. Euler's function. How does Fermat's Little Theorem generalize to numbers m that are not prime? The key thing in the proof is that f1; 2; ; p 1g are coprime with p so that they have multiplicative inverses in Zp. It is then natural to de ne ...

William Stallings, Cryptography and Network Security 3/e

... Answer Brahmagupta’s question: (7th century AD) An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, ...

... Answer Brahmagupta’s question: (7th century AD) An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, ...

The Fibonacci Numbers

... Let’s use this formula: Fn2 = Fn · Fn+1 − Fn · Fn−1 To find out what the sum of the first few SQUARES of Fibonacci numbers is: F12 + F22 + F32 + · · · + Fn2 =???? ...

... Let’s use this formula: Fn2 = Fn · Fn+1 − Fn · Fn−1 To find out what the sum of the first few SQUARES of Fibonacci numbers is: F12 + F22 + F32 + · · · + Fn2 =???? ...

x,y

... Prime Numbers Definition A number a if prime if the only factors it has are 1 and a Examples 6 is not a prime: it has factors 2 and 3 5 is a prime Checking for primality of number N Naive method: test all numbers 2 ,…, N-1 for factors Suffices to test only up to √N 25 tests to make! Too slow ...

... Prime Numbers Definition A number a if prime if the only factors it has are 1 and a Examples 6 is not a prime: it has factors 2 and 3 5 is a prime Checking for primality of number N Naive method: test all numbers 2 ,…, N-1 for factors Suffices to test only up to √N 25 tests to make! Too slow ...

Section 9.3 – The Complex Plane and De Moivre`s Theorem

... To prove this, we will need to use a method called mathematical induction. In mathematical induction, you first show that the statement is true for some initial value of n, usually n = 1. You then assume the statement is true for value k and then show it to be true for k + 1. Part I: n = 1 z(1) = r( ...

... To prove this, we will need to use a method called mathematical induction. In mathematical induction, you first show that the statement is true for some initial value of n, usually n = 1. You then assume the statement is true for value k and then show it to be true for k + 1. Part I: n = 1 z(1) = r( ...

ON THE DISTRIBUTION OF EXTREME VALUES

... function ζ(s) in the t-aspect, Dirichlet L-functions in the q-aspect, and L-functions attached to primitive holomorphic cusp forms of weight 2 in the level aspect. For each family we show that the L-values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We al ...

... function ζ(s) in the t-aspect, Dirichlet L-functions in the q-aspect, and L-functions attached to primitive holomorphic cusp forms of weight 2 in the level aspect. For each family we show that the L-values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We al ...

# 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.