A formally verified proof of the prime number theorem
... Overview of the formalization Some statistics regarding length, and time, are given in the associated paper. A lot of time and effort was spent: • Building basic libraries of easy facts. ...
... Overview of the formalization Some statistics regarding length, and time, are given in the associated paper. A lot of time and effort was spent: • Building basic libraries of easy facts. ...
MATH 103A Homework 1 Solutions Due January 11, 2013
... b are relatively prime, show that ab divides c. Show, by example, that if a and b are not relatively prime, then ab need not divide c. Solution: Since a, b are relatively prime gcda, b 1. By Theorem 0.2, this means that there are s, t Z such that 1 as bt. By assumption that ac and bc there a ...
... b are relatively prime, show that ab divides c. Show, by example, that if a and b are not relatively prime, then ab need not divide c. Solution: Since a, b are relatively prime gcda, b 1. By Theorem 0.2, this means that there are s, t Z such that 1 as bt. By assumption that ac and bc there a ...
Induction
... L-shapes subject to conditions 1′ and 2′ . A 1 × 1 grid consists of only one square. This square is s, and we can leave it untiled. Since there are no other squares in the grid to tile, this gives us a valid tiling of a 1 × 1 grid. Now we prove the inductive step using a direct proof. Let n ∈ N and ...
... L-shapes subject to conditions 1′ and 2′ . A 1 × 1 grid consists of only one square. This square is s, and we can leave it untiled. Since there are no other squares in the grid to tile, this gives us a valid tiling of a 1 × 1 grid. Now we prove the inductive step using a direct proof. Let n ∈ N and ...
Prime factorization of integral Cayley octaves
... For most c~ ~ Q the lattice Ca is not a left ideal of C since by a theorem of Mahler, van der Blij and Springer [1] the only left ideals of C are the Cm with m E 7L However, for m (E 7L the principal lattice Cm is a twosided ideal and reduction mod m is a surjective homomorphism of alternative rings ...
... For most c~ ~ Q the lattice Ca is not a left ideal of C since by a theorem of Mahler, van der Blij and Springer [1] the only left ideals of C are the Cm with m E 7L However, for m (E 7L the principal lattice Cm is a twosided ideal and reduction mod m is a surjective homomorphism of alternative rings ...
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"".