√ 2 IS IRRATIONAL Recall the well ordering principle: Every non
... Remark. There is a more standard proof that 2 is irrational that uses the fact that every positive integer can be expressed uniquely as the product of prime numbers. The latter statement is called the fundamental theorem of arithmetic. We will prove the FTA later in this course, but in the meantime, ...
... Remark. There is a more standard proof that 2 is irrational that uses the fact that every positive integer can be expressed uniquely as the product of prime numbers. The latter statement is called the fundamental theorem of arithmetic. We will prove the FTA later in this course, but in the meantime, ...
(1) Find all prime numbers smaller than 100. (2) Give a proof by
... (1) Find all prime numbers smaller than 100. (2) Give a proof by induction (instead of a proof by contradiction given in class) that any natural number > 1 has a unique (up to order) factorization as a product of primes. (3) Give a proof by induction that if a ≡ b( mod m) then an ≡ bn ( mod m) for a ...
... (1) Find all prime numbers smaller than 100. (2) Give a proof by induction (instead of a proof by contradiction given in class) that any natural number > 1 has a unique (up to order) factorization as a product of primes. (3) Give a proof by induction that if a ≡ b( mod m) then an ≡ bn ( mod m) for a ...
1. Prove that 3n + 2 and 5 n + 3 are relatively prime for every positive
... b) d bxce = d xe 11. Factor 51, 948 into a product of primes. 12. Prove that n5 − n is divisble by 30 for every integer n. 13. Prove that n, n + 2, n + 4 are all primes if and only if n = 3. n 14. The integer Fn = 22 +1 is called the nth Fermat number. Show that for n = 1, 2, 3, 4 Fn is prime. 15. S ...
... b) d bxce = d xe 11. Factor 51, 948 into a product of primes. 12. Prove that n5 − n is divisble by 30 for every integer n. 13. Prove that n, n + 2, n + 4 are all primes if and only if n = 3. n 14. The integer Fn = 22 +1 is called the nth Fermat number. Show that for n = 1, 2, 3, 4 Fn is prime. 15. S ...
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"".