Example sheet 1
... mod q, and also congruent to ±1 mod 8. Use this to factor 211 − 1 = 2047. 7. We say that a natural number n is perfect if the sum of all the positive divisors of n is equal to 2n. Prove that a positive even integer n is perfect if and only if it can be written in the form n = 2q−1 (2q − 1), where 2q ...
... mod q, and also congruent to ±1 mod 8. Use this to factor 211 − 1 = 2047. 7. We say that a natural number n is perfect if the sum of all the positive divisors of n is equal to 2n. Prove that a positive even integer n is perfect if and only if it can be written in the form n = 2q−1 (2q − 1), where 2q ...
Review guide for Exam 2
... be done using the Euclidean algorithm. (5) Compute Z× n for a few n, as in Exercise 4. (6) Recall problems 5-7 from the book. Hints: the extra exercise on the course website from that section is a hint for problem 5, problem 6 requires factoring the polynomial x2 − 1 and using the facts mentioned ab ...
... be done using the Euclidean algorithm. (5) Compute Z× n for a few n, as in Exercise 4. (6) Recall problems 5-7 from the book. Hints: the extra exercise on the course website from that section is a hint for problem 5, problem 6 requires factoring the polynomial x2 − 1 and using the facts mentioned ab ...
Prime Numbers - Winchester College
... have been true to start off with. Hence, SOMETHING is true. ...
... have been true to start off with. Hence, SOMETHING is true. ...
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"".