ARITHMETIC TRANSLATIONS OF AXIOM SYSTEMS
... that 1.4-1.5 (but not necessarily 1.1-1.3) hold. Notwithstanding ...
... that 1.4-1.5 (but not necessarily 1.1-1.3) hold. Notwithstanding ...
Statistics of incomplete quotients of continued fractions of quadratic
... Otherwise, considering equality (11) modulo 4 or modulo p, we get the equation x2 ≡ −1; it is well known that the latter equation is neither solvable modulo 4 nor modulo p. Now Theorem 3.4 is reduced to the assertion that all primes in the form 4k + 1 belong to K. This fact was proved by Legendre (s ...
... Otherwise, considering equality (11) modulo 4 or modulo p, we get the equation x2 ≡ −1; it is well known that the latter equation is neither solvable modulo 4 nor modulo p. Now Theorem 3.4 is reduced to the assertion that all primes in the form 4k + 1 belong to K. This fact was proved by Legendre (s ...
MATH 100 V1A
... 3. A number a is called a fixed point of a function f is f (a) = a. Prove that if f is differentiable and f 0 (x) 6= 1 for all x, then f can have at most one fixed point. (Note that I’ve explicitly added the assumption that f is differentiable to make the problem less ambiguous). Solution: It turns ...
... 3. A number a is called a fixed point of a function f is f (a) = a. Prove that if f is differentiable and f 0 (x) 6= 1 for all x, then f can have at most one fixed point. (Note that I’ve explicitly added the assumption that f is differentiable to make the problem less ambiguous). Solution: It turns ...
Slide 1
... It is important to be aware that there are a number of words that mean essentially the same thing as the word “theorem,” but which are used in slightly different ways. • Theorem – In general the word “theorem” is reserved for a statement that is considered important or significant (the Pythagorean T ...
... It is important to be aware that there are a number of words that mean essentially the same thing as the word “theorem,” but which are used in slightly different ways. • Theorem – In general the word “theorem” is reserved for a statement that is considered important or significant (the Pythagorean T ...
Working Notes for Week 5
... Lemma 2.35 (Limit is an accumulation point) If (xn ) converges to x ∈ R, then x is an accumulation point. Of course, if (xn ) converges to x, then (by definition), given ε > 0 there exists N ∈ N such that all xn with n > N lie ε-close to x (and thus in particular infinitely many). But let us formall ...
... Lemma 2.35 (Limit is an accumulation point) If (xn ) converges to x ∈ R, then x is an accumulation point. Of course, if (xn ) converges to x, then (by definition), given ε > 0 there exists N ∈ N such that all xn with n > N lie ε-close to x (and thus in particular infinitely many). But let us formall ...
IN THE WAKE OF CARDANO`S FORMULAS 1. Completing the cube
... all quadratics, cubics, and quartics have their solutions there, it was natural to expect that all solutions to all polynomials can be found in the complex plane. This proved to be true. The great Euler sketched out some ideas for what is now known as the Fundamental Theorem of Algebra, which can be ...
... all quadratics, cubics, and quartics have their solutions there, it was natural to expect that all solutions to all polynomials can be found in the complex plane. This proved to be true. The great Euler sketched out some ideas for what is now known as the Fundamental Theorem of Algebra, which can be ...
Applications of Number Theory to Fermat`s Last Theorem
... more complicated than the proof of n = 4. This proof was first published by Leonard Euler (1707-1783), but it was incomplete in an important respect. We will discuss Euler’s proof and correct his mistake. Euler’s proof also uses the method of infinite descent. He shows that if positive whole numbers ...
... more complicated than the proof of n = 4. This proof was first published by Leonard Euler (1707-1783), but it was incomplete in an important respect. We will discuss Euler’s proof and correct his mistake. Euler’s proof also uses the method of infinite descent. He shows that if positive whole numbers ...
On the Reciprocal of the Binary Generating Function for the Sum of
... If A is a set of natural numbers containing 0, then there is a unique nonempty “reciprocal” set B of natural numbers (containing 0) such that every positive integer can be written in the form a + b, where a ∈ A and b ∈ B, in an even number of ways. Furthermore, the generating functions for A and B o ...
... If A is a set of natural numbers containing 0, then there is a unique nonempty “reciprocal” set B of natural numbers (containing 0) such that every positive integer can be written in the form a + b, where a ∈ A and b ∈ B, in an even number of ways. Furthermore, the generating functions for A and B o ...
i+1
... Pick x between 0 and 1, so non-zero digits follow decimal point First fractional digit of f(1) is 1 Pick first fractional digit of x to be different, say 2 Second fractional digit of f(2) is 4 Pick second fractional digit of x to be different, say 6 And so on …. ...
... Pick x between 0 and 1, so non-zero digits follow decimal point First fractional digit of f(1) is 1 Pick first fractional digit of x to be different, say 2 Second fractional digit of f(2) is 4 Pick second fractional digit of x to be different, say 6 And so on …. ...
Lecture 3 - Duke Computer Science
... Four guys want to cross a bridge that can only hold two people at one time. It is pitch dark and they only have one flashlight, so people must cross either alone or in pairs (bringing the flashlight). Their walking speeds allow them to cross in 1, 2, 5, and 10 minutes, respectively. Is it possible ...
... Four guys want to cross a bridge that can only hold two people at one time. It is pitch dark and they only have one flashlight, so people must cross either alone or in pairs (bringing the flashlight). Their walking speeds allow them to cross in 1, 2, 5, and 10 minutes, respectively. Is it possible ...
PDF
... Because a Pratt certificate requires the factorization of n − 1, it is generally only used for small numbers, with “small” being roughly defined as being less than about a billion. We’ll use a much smaller number for our example, one for which it would actually be faster to just perform trial divisi ...
... Because a Pratt certificate requires the factorization of n − 1, it is generally only used for small numbers, with “small” being roughly defined as being less than about a billion. We’ll use a much smaller number for our example, one for which it would actually be faster to just perform trial divisi ...
Continued fractions Yann BUGEAUD Let x0,x1,... be real numbers
... partial quotients are bounded. However, much more is known: the sequence of their partial quotients is ultimately periodic. Theorem 1.14. The real irrational number ξ = [a0 ; a1 , a2 , . . .] has a periodic continued fraction expansion (that is, there exist integers k ≥ 0 and n ≥ 1 such that am+n = ...
... partial quotients are bounded. However, much more is known: the sequence of their partial quotients is ultimately periodic. Theorem 1.14. The real irrational number ξ = [a0 ; a1 , a2 , . . .] has a periodic continued fraction expansion (that is, there exist integers k ≥ 0 and n ≥ 1 such that am+n = ...
Document
... finite fields, the real numbers, and the complex numbers. In 1985, after mathematicians had been working on Fermat’s Last Theorem for about 350 years, Gerhard Frey suggested that if we assumed Fermat’s Last Theorem was false, the existence of an elliptic curve y 2 x( x a n )( x bn ) where a, b ...
... finite fields, the real numbers, and the complex numbers. In 1985, after mathematicians had been working on Fermat’s Last Theorem for about 350 years, Gerhard Frey suggested that if we assumed Fermat’s Last Theorem was false, the existence of an elliptic curve y 2 x( x a n )( x bn ) where a, b ...
An asymptotic for the representation of integersas sums of triangular
... Figure 1. The fundamental domains for 0 (left) and for 00 (4) (right). The full modular group has a single cusp at i∞. From the fundamental domain for 00 (4) shown in Figure 1, we see that 00 (4) has three cusps, namely 0, 21 and i∞. 1C. Modular forms. Modular forms are holomorphic functions on H wh ...
... Figure 1. The fundamental domains for 0 (left) and for 00 (4) (right). The full modular group has a single cusp at i∞. From the fundamental domain for 00 (4) shown in Figure 1, we see that 00 (4) has three cusps, namely 0, 21 and i∞. 1C. Modular forms. Modular forms are holomorphic functions on H wh ...
Bloom`s Taxonomy applied to understanding the Pythagorean
... 2. Would the Pythagorean Theorem hold true in three dimensions? If so, what would it say and how would it work? If not, why? 3. Use the Pythagorean Theorem to explain why the graph of x 2 y 2 r 2 is a circle. 4. Create a story problem the solution of which involves use of the Pythagorean Theorem ...
... 2. Would the Pythagorean Theorem hold true in three dimensions? If so, what would it say and how would it work? If not, why? 3. Use the Pythagorean Theorem to explain why the graph of x 2 y 2 r 2 is a circle. 4. Create a story problem the solution of which involves use of the Pythagorean Theorem ...
CSE 20 * Discrete Mathematics
... Yes, but m has to be the same as n. Yes, and m,n can be different but for each kind of coefficient that appears in both, it has to agree. That is, a0 = b0, a1 = b1, etc. Yes, if m=n and all the coefficients agree. More than one of the above / none of the above. ...
... Yes, but m has to be the same as n. Yes, and m,n can be different but for each kind of coefficient that appears in both, it has to agree. That is, a0 = b0, a1 = b1, etc. Yes, if m=n and all the coefficients agree. More than one of the above / none of the above. ...