Amicable Numbers
... An extension of the preceding table would throw further light upon such a conjecture. In support one may cite the series of amicable pairs defined by the ancient formula of Thabit ben Korrah. (See Ore [5, pp. 98-99].) If they represent an infinite set of amicable numbers they must satisfy (8 ). Obse ...
... An extension of the preceding table would throw further light upon such a conjecture. In support one may cite the series of amicable pairs defined by the ancient formula of Thabit ben Korrah. (See Ore [5, pp. 98-99].) If they represent an infinite set of amicable numbers they must satisfy (8 ). Obse ...
Fibonacci Numbers
... Wherever we stop, we will always get a rectangle, since the next square to add is determined by the longest edge on the current rectangle. Also, those longest edges are just the sum of the latest two sides-of-squares to be added. Since we start with squares of sides 1 and 1, this tells us why the sq ...
... Wherever we stop, we will always get a rectangle, since the next square to add is determined by the longest edge on the current rectangle. Also, those longest edges are just the sum of the latest two sides-of-squares to be added. Since we start with squares of sides 1 and 1, this tells us why the sq ...
4CCM115A and 5CCM115B Numbers and Functions
... Which functions in the above examples are injective? Definition 2.3. Let f : A → B be a function. Then f (A), the image (or range) of A under the function f , is the set {y ∈ B | y = f (x) for some x ∈ A} ⊂ B . Example. What are the images of the functions in the above examples? It is sometimes the ...
... Which functions in the above examples are injective? Definition 2.3. Let f : A → B be a function. Then f (A), the image (or range) of A under the function f , is the set {y ∈ B | y = f (x) for some x ∈ A} ⊂ B . Example. What are the images of the functions in the above examples? It is sometimes the ...
Group action
... The first answer, similar to 4(a), is: if N = 3k p1i1 p2i2 ... pnin q1j1 q2j2 ... qmjm , where pi are of type 3s+1, and qj are of form 3s+2, then it depends on the parity of jl. If at least one of them is odd, then it is impossible; if all of them are even, then the number of representatio ...
... The first answer, similar to 4(a), is: if N = 3k p1i1 p2i2 ... pnin q1j1 q2j2 ... qmjm , where pi are of type 3s+1, and qj are of form 3s+2, then it depends on the parity of jl. If at least one of them is odd, then it is impossible; if all of them are even, then the number of representatio ...
The imaginary unit
... The number i is by no means alone! By taking multiples of this imaginary unit, we can create infinitely many more pure imaginary numbers. For example, 3i, i√5, and −12i are all examples of pure imaginary numbers, or numbers of the form bi, where b is a nonzero real number. Taking the squares of these ...
... The number i is by no means alone! By taking multiples of this imaginary unit, we can create infinitely many more pure imaginary numbers. For example, 3i, i√5, and −12i are all examples of pure imaginary numbers, or numbers of the form bi, where b is a nonzero real number. Taking the squares of these ...
appendix B
... Another important difference between floating point numbers and real numbers is their density. o Between any 2 real numbers, x and y, is another real number, no matter how close x is to y. The real numbers form a continuum. o Floating point numbers, in contrast, do not form a continuum. Exactly ...
... Another important difference between floating point numbers and real numbers is their density. o Between any 2 real numbers, x and y, is another real number, no matter how close x is to y. The real numbers form a continuum. o Floating point numbers, in contrast, do not form a continuum. Exactly ...
Chapter 0 – Section 03 - Dr. Abdullah Almutairi
... as (x + 2)(2x – 5) = 2x2 – x – 10. That is, starting with the expression 2x2 – x – 10, we would like to factor it to get the expression (x + 2)(2x – 5). An expression of the form ax2 + bx + c, where a, b, and c are real numbers, is called a quadratic expression in x. Thus, given a quadratic expressi ...
... as (x + 2)(2x – 5) = 2x2 – x – 10. That is, starting with the expression 2x2 – x – 10, we would like to factor it to get the expression (x + 2)(2x – 5). An expression of the form ax2 + bx + c, where a, b, and c are real numbers, is called a quadratic expression in x. Thus, given a quadratic expressi ...
A formally verified proof of the prime number theorem
... Let R(x) = ψ(x) − x denote the “error term.” By Chebyshev’s equivalences the prime number theorem amounts to the assertion limx→∞ R(x)/x = 0. With some delicate calculation, the symmetry formula yields: |R(x)| ln x ≤ 2 ...
... Let R(x) = ψ(x) − x denote the “error term.” By Chebyshev’s equivalences the prime number theorem amounts to the assertion limx→∞ R(x)/x = 0. With some delicate calculation, the symmetry formula yields: |R(x)| ln x ≤ 2 ...
Supplementary Ex_S1_..
... (a) If there are 8 layers of logs piled up as shown in the figure, how many logs are there in total? (b) If two more layers of logs are added now, how many logs are added? (c) Among these 10 layers of logs, the upper 6 layers will be exported to Japan and the remaining will be exported to Singapore. ...
... (a) If there are 8 layers of logs piled up as shown in the figure, how many logs are there in total? (b) If two more layers of logs are added now, how many logs are added? (c) Among these 10 layers of logs, the upper 6 layers will be exported to Japan and the remaining will be exported to Singapore. ...
Erratum
... ERRATUM F O R "COMPLEX FIBONACCI AND LUCAS NUMBERS, CONTINUED FRACTIONS, AND THE SQUARE R O O T O F THE GOLDEN R A T I O " The Fibonacci Quarterly 31.1 (1993):7-20 It has been pointed out to me by a correspondent who wished to remain anonymous that the number 185878941, which was printed in the "loo ...
... ERRATUM F O R "COMPLEX FIBONACCI AND LUCAS NUMBERS, CONTINUED FRACTIONS, AND THE SQUARE R O O T O F THE GOLDEN R A T I O " The Fibonacci Quarterly 31.1 (1993):7-20 It has been pointed out to me by a correspondent who wished to remain anonymous that the number 185878941, which was printed in the "loo ...
