For a pdf file
... Clearly, n > p. Since p is the largest prime number, n cannot be a prime number. In other words, n is composite. Let q be any prime number. Because of the way n is constructed, when n is divided by q the remainder is 1. That is, n is not a multiple of q. This contradicts the Fundamental Theorem of A ...
... Clearly, n > p. Since p is the largest prime number, n cannot be a prime number. In other words, n is composite. Let q be any prime number. Because of the way n is constructed, when n is divided by q the remainder is 1. That is, n is not a multiple of q. This contradicts the Fundamental Theorem of A ...
Proof Example: The Irrationality of √ 2 During the lecture a student
... By definition of x (that is: the positive number y such that y 2 = x), we obtain n2 /m2 = 2 and from this n2 = 2 m2 . Thus 2 divides n2 , from here 2 divides n (because 2 is prime), therefore there exists a k such that n = 2 k. We replace this into the previous equality and obtain 4 k 2 = 2 m2 , thu ...
... By definition of x (that is: the positive number y such that y 2 = x), we obtain n2 /m2 = 2 and from this n2 = 2 m2 . Thus 2 divides n2 , from here 2 divides n (because 2 is prime), therefore there exists a k such that n = 2 k. We replace this into the previous equality and obtain 4 k 2 = 2 m2 , thu ...
For a pdf file
... The above equation implies that b2 is even and hence b is even. Since we know a is even this means that a and b have 2 as a common factor which contradicts the assumption that a and b have no common factors. ...
... The above equation implies that b2 is even and hence b is even. Since we know a is even this means that a and b have 2 as a common factor which contradicts the assumption that a and b have no common factors. ...
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"".