An introduction to the Smarandache Square

... Case 1. According to the theorem 7 Ssc(n)=n and Ssc(n+1)=n+1 that implies that Ssc(n)<>Ssc(n+1) Case 2. Without loss of generality let's suppose that: n = pa ⋅ q b n + 1 = p a ⋅ qb + 1 = sc ⋅ t d where p,q,s and t are distinct primes. According to the theorem 4: Ssc( n) = Ssc ( p a ⋅ q b ) = p odd ( ...

... Case 1. According to the theorem 7 Ssc(n)=n and Ssc(n+1)=n+1 that implies that Ssc(n)<>Ssc(n+1) Case 2. Without loss of generality let's suppose that: n = pa ⋅ q b n + 1 = p a ⋅ qb + 1 = sc ⋅ t d where p,q,s and t are distinct primes. According to the theorem 4: Ssc( n) = Ssc ( p a ⋅ q b ) = p odd ( ...

Pi and the Fibonacci Numbers

... Since 60° is a sixth of a full turn (360°) then 60° = 2 Pi/ 6 = Pi/3 radians = (Pi/3)r and so 30° = Pi/6 radians = (Pi/6)r. ...

... Since 60° is a sixth of a full turn (360°) then 60° = 2 Pi/ 6 = Pi/3 radians = (Pi/3)r and so 30° = Pi/6 radians = (Pi/6)r. ...

Problem Set 1 Solutions

... Induction step: Assume P (n) is true, so by definition Q(k) is true for all k ∈ N with 1 ≤ k ≤ n. Thus our assumption guarantees Q(n + 1) is true. We see that both P (n) and Q(n + 1) are both true, and since the statement ‘P (n) and Q(n + 1) are both true’ is equivalent to P (n + 1), we conclude P ( ...

... Induction step: Assume P (n) is true, so by definition Q(k) is true for all k ∈ N with 1 ≤ k ≤ n. Thus our assumption guarantees Q(n + 1) is true. We see that both P (n) and Q(n + 1) are both true, and since the statement ‘P (n) and Q(n + 1) are both true’ is equivalent to P (n + 1), we conclude P ( ...

Discrete Mathematics Lecture Notes Incomplete Preliminary Version

... authors’ contempt for their readers’ intelligence; the result is a multitude of unnecessary case distinctions, destroying a fundamental element of mathematical aesthetics. (To these authors, for instance, x | x does not hold for all x; there is an exception: x = 0. And then, to them, x − y does not ...

... authors’ contempt for their readers’ intelligence; the result is a multitude of unnecessary case distinctions, destroying a fundamental element of mathematical aesthetics. (To these authors, for instance, x | x does not hold for all x; there is an exception: x = 0. And then, to them, x − y does not ...

13(4)

... The California Mathematics Council All subscription correspondence should be addressed to Professor Leonard Klosinski, Mathematics Department, University of Santa Clara, Santa Clara, California 95053. All checks ($12.00 per year) should be made out t o the Fibonacci Association or the Fibonacci Quar ...

... The California Mathematics Council All subscription correspondence should be addressed to Professor Leonard Klosinski, Mathematics Department, University of Santa Clara, Santa Clara, California 95053. All checks ($12.00 per year) should be made out t o the Fibonacci Association or the Fibonacci Quar ...

Introduction to analytic number theory

... tables have been found, dating back to 2000 BC. When ancient civilizations reached a level which provided leisure time to ponder about things, some people began to speculate about the nature and properties of numbers. This curiosity developed into a sort of numbermysticism or numerology, and even to ...

... tables have been found, dating back to 2000 BC. When ancient civilizations reached a level which provided leisure time to ponder about things, some people began to speculate about the nature and properties of numbers. This curiosity developed into a sort of numbermysticism or numerology, and even to ...

34(5)

... Lagrange interpolation must be considered as describing the properties of a linear operator sending a function of one variable to a symmetric function. It can be written as a summation on a set or as a product of divided differences; it is this latter version that we shall use here. In fact, in Sect ...

... Lagrange interpolation must be considered as describing the properties of a linear operator sending a function of one variable to a symmetric function. It can be written as a summation on a set or as a product of divided differences; it is this latter version that we shall use here. In fact, in Sect ...

local copy - Harvard Mathematics Department

... [24]. Also distance 2 is sometimes included as a twin. Their appearance on Im(z) = 1 was studied by [48] who gave the corresponding Hardy-Littlewood constant. While one does not know whether infinitely many Gaussian prime twins exist, one can estimate that there are asymptotically Ct r/ log2 (r) of ...

... [24]. Also distance 2 is sometimes included as a twin. Their appearance on Im(z) = 1 was studied by [48] who gave the corresponding Hardy-Littlewood constant. While one does not know whether infinitely many Gaussian prime twins exist, one can estimate that there are asymptotically Ct r/ log2 (r) of ...