Short intervals with a given number of primes
... What GPY showed is that, for arbitrarily small λ and m “ 2, (1.2) holds for infinitely many n. Only very recently has the breakthrough of Maynard [6] on bounded gaps between primes shown that, for every choice of λ and m, (1.2) holds for infinitely many n. This statement does not preclude the possib ...
... What GPY showed is that, for arbitrarily small λ and m “ 2, (1.2) holds for infinitely many n. Only very recently has the breakthrough of Maynard [6] on bounded gaps between primes shown that, for every choice of λ and m, (1.2) holds for infinitely many n. This statement does not preclude the possib ...
Rational values of the arccosine function
... For even n, a different method is suggested in that book, distinguishing between the cases√n = 2j and n not a power of 2. Thus, it is obtained that (1/π) arccos(1/ n) is rational if and only if n ∈ {1, 2, 4}. The relation (2) can be read in term of Chebyshev polynomials of the first kind. These poly ...
... For even n, a different method is suggested in that book, distinguishing between the cases√n = 2j and n not a power of 2. Thus, it is obtained that (1/π) arccos(1/ n) is rational if and only if n ∈ {1, 2, 4}. The relation (2) can be read in term of Chebyshev polynomials of the first kind. These poly ...
The Circle Method
... Previously we considered the question of determining the smallest number of perfect k th powers needed to represent all natural numbers as a sum of k th powers. One can consider the analogous question for other sets of numbers. Namely, given a set A, is there a number sA such that every natural numb ...
... Previously we considered the question of determining the smallest number of perfect k th powers needed to represent all natural numbers as a sum of k th powers. One can consider the analogous question for other sets of numbers. Namely, given a set A, is there a number sA such that every natural numb ...
LANDAU`S PROBLEMS ON PRIMES 1. Introduction In his invited
... much more about the origin of Goldbach’s Conjecture, however there are some interesting (and partly not well known) facts to mention concerning its origin. In a letter to Euler, written June 7, 1742, Goldbach formulated his conjecture in two different forms. The first one asserted that if a number N ...
... much more about the origin of Goldbach’s Conjecture, however there are some interesting (and partly not well known) facts to mention concerning its origin. In a letter to Euler, written June 7, 1742, Goldbach formulated his conjecture in two different forms. The first one asserted that if a number N ...
Note 3
... sum of the first n odd numbers is not just a perfect square, but is equal precisely to n2 ! Motivated by this discovery, let’s try something counterintuitive: Let us try to show the following stronger claim. Theorem 3.4. For all n ≥ 1, the sum of the first n odd numbers is n2 . Proof. We proceed by ...
... sum of the first n odd numbers is not just a perfect square, but is equal precisely to n2 ! Motivated by this discovery, let’s try something counterintuitive: Let us try to show the following stronger claim. Theorem 3.4. For all n ≥ 1, the sum of the first n odd numbers is n2 . Proof. We proceed by ...
Booklet of lecture notes, exercises and solutions.
... even all powers of 2. Make sure when you use . . . that it really is clear to the reader exactly what needs to be filled in. What comes next after 1, 2, 4, 8, 16, . . .? ...
... even all powers of 2. Make sure when you use . . . that it really is clear to the reader exactly what needs to be filled in. What comes next after 1, 2, 4, 8, 16, . . .? ...
A Course on Number Theory - School of Mathematical Sciences
... Number theory is about properties of the natural numbers, integers, or rational numbers, such as the following: • Given a natural number n, is it prime or composite? • If it is composite, how can we factorise it? • How many solutions do equations like x2 + y2 = n or xn + yn = zn have for fixed n, wh ...
... Number theory is about properties of the natural numbers, integers, or rational numbers, such as the following: • Given a natural number n, is it prime or composite? • If it is composite, how can we factorise it? • How many solutions do equations like x2 + y2 = n or xn + yn = zn have for fixed n, wh ...
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"".