Lecture 2: Irrational numbers
... We want to appreciate one of the great moments of mathematics: the insight that there are numbers which are irrational. It was the Pythagoreans, who realized this first and - according to legend - tried even to ”cover the discovery up” and kill Hippasus, one of the earlier discoverers. We have seen ...
... We want to appreciate one of the great moments of mathematics: the insight that there are numbers which are irrational. It was the Pythagoreans, who realized this first and - according to legend - tried even to ”cover the discovery up” and kill Hippasus, one of the earlier discoverers. We have seen ...
Numbers: Fun and Challenge
... • The equation x2 + y 2 = z 2 has lots of integer solutions. The primitive ones with x odd and y even are given by the formula x = s2 − t2 , y = 2st, z = s2 + t2 . • (Fermat) The equation x4 − y 4 = z 2 has no non-trivial integer solution. We will come back to this shortly. • (Fermat’s Last Theorem) ...
... • The equation x2 + y 2 = z 2 has lots of integer solutions. The primitive ones with x odd and y even are given by the formula x = s2 − t2 , y = 2st, z = s2 + t2 . • (Fermat) The equation x4 − y 4 = z 2 has no non-trivial integer solution. We will come back to this shortly. • (Fermat’s Last Theorem) ...
All numbers are integers.
... n = pe11 · · · perr all the exponents e1 , . . . , er are even. 3. A rational number is one that can be written as a quotient of two integers. If r = a/b is a rational number b ̸= 0, we say it is written in lowest terms or in reduced form iff b > 0 and gcd(a, b) = 1. It is easy to see that every rati ...
... n = pe11 · · · perr all the exponents e1 , . . . , er are even. 3. A rational number is one that can be written as a quotient of two integers. If r = a/b is a rational number b ̸= 0, we say it is written in lowest terms or in reduced form iff b > 0 and gcd(a, b) = 1. It is easy to see that every rati ...
Here
... blanks. Definition 1. A sequence L1 , L2 , L3 , . . . , Ln , . . . of real numbers is said to converge to L ∈ R if, for every real > 0, there exists an N ∈ N such that |Ln − L| < for all n > N . The number L is called the limit of the sequence. Please draw parentheses on the definition to indica ...
... blanks. Definition 1. A sequence L1 , L2 , L3 , . . . , Ln , . . . of real numbers is said to converge to L ∈ R if, for every real > 0, there exists an N ∈ N such that |Ln − L| < for all n > N . The number L is called the limit of the sequence. Please draw parentheses on the definition to indica ...
![Primes in the Interval [2n, 3n]](http://s1.studyres.com/store/data/017368795_1-688e9118f5c89a0cb6fb7aedda33fc32-300x300.png)
Sample pages 2 PDF
... These will all be in the same orbit and everything in x’s orbit is of this form. If we keep going we just get the same strings over again since T p x = x. There are at most p elements in these orbits then. We show that when p is prime there are exactly p elements. If there were less than p then for ...
... These will all be in the same orbit and everything in x’s orbit is of this form. If we keep going we just get the same strings over again since T p x = x. There are at most p elements in these orbits then. We show that when p is prime there are exactly p elements. If there were less than p then for ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".