More properties in Goldbach`s Conjecture
... 1. Introduction Goldbach’s conjecture [1] states that “every even natural number > 4 can be written as a sum of two primes”, namely: 2n= p + q where n>2, and p , q are prime numbers, ...
... 1. Introduction Goldbach’s conjecture [1] states that “every even natural number > 4 can be written as a sum of two primes”, namely: 2n= p + q where n>2, and p , q are prime numbers, ...
S Chowla and SS Pillai
... ² The non-vanishing at s = 1 of L(s; Â) for non-trivial Dirichlet characters is the key fact used in the proof of Dirichlet's famous theorem on existence of in¯nitely many primes in any arithmetic progression an + b with (a; b) = 1. ² Kenkichi Iwasawa [12] showed in 1975 that the above result has co ...
... ² The non-vanishing at s = 1 of L(s; Â) for non-trivial Dirichlet characters is the key fact used in the proof of Dirichlet's famous theorem on existence of in¯nitely many primes in any arithmetic progression an + b with (a; b) = 1. ² Kenkichi Iwasawa [12] showed in 1975 that the above result has co ...
Sums of squares, sums of cubes, and modern number theory
... Theorem 1.1 (Fermat (1640)). A prime p is a sum of two squares if and only if p ≡ 1 mod 4 or p = 2. More generally, a positive integer n is a sum of two squares if and only if any prime factor of n which is 3 mod 4 occurs to an even power in the prime factorization of n. One way to go about proving ...
... Theorem 1.1 (Fermat (1640)). A prime p is a sum of two squares if and only if p ≡ 1 mod 4 or p = 2. More generally, a positive integer n is a sum of two squares if and only if any prime factor of n which is 3 mod 4 occurs to an even power in the prime factorization of n. One way to go about proving ...
A Musician`s Guide to Prime Numbers
... of congruences. Before we do so, let us first try to compile the musical conjectures into one mathematical conjecture. Conjecture 1.4. There are infinitely many prime numbers of the form 12k +r, for some k ∈ N and r ∈ R = {1, 5, 7, 11}. For any positive integer n ≤ 12 with n∈ / R, 12k + n is never p ...
... of congruences. Before we do so, let us first try to compile the musical conjectures into one mathematical conjecture. Conjecture 1.4. There are infinitely many prime numbers of the form 12k +r, for some k ∈ N and r ∈ R = {1, 5, 7, 11}. For any positive integer n ≤ 12 with n∈ / R, 12k + n is never p ...
Seed and Sieve of Odd Composite Numbers with
... with factorization of an odd composite number in terms of its seed. It shows that factorization of a big odd number can be converted into that of small numbers incorporated with the big number’s seed. The article also makes an investigation on new characteristics of odd numbers as well as their fact ...
... with factorization of an odd composite number in terms of its seed. It shows that factorization of a big odd number can be converted into that of small numbers incorporated with the big number’s seed. The article also makes an investigation on new characteristics of odd numbers as well as their fact ...
Why Is the 3X + 1 Problem Hard? - Department of Mathematics, CCNY
... Z2 is the inverse limit of the sequence of rings Z/2k and the connecting homomorphisms ρk . This means, first, that each a ∈ Z2 is uniquely described by the coherent sequence of congruence classes {[a]k : k = 1, 2, ...} where a sequence {αk ∈ Z/2k : k = 1, 2, ...} is called coherent when ρk αk = αk− ...
... Z2 is the inverse limit of the sequence of rings Z/2k and the connecting homomorphisms ρk . This means, first, that each a ∈ Z2 is uniquely described by the coherent sequence of congruence classes {[a]k : k = 1, 2, ...} where a sequence {αk ∈ Z/2k : k = 1, 2, ...} is called coherent when ρk αk = αk− ...
Bases for Sets of Integers
... We take as A such a set which at the same time is typical according to Theorem 2 . Thus MA < n/2 log n . For A' we take the aforementioned n 3/4 exceptions together with the nearest multiples of K - [n'/ 2 ] to the other members . Thus A' is very near to A and yet, as previously indicated, m A ' < 1 ...
... We take as A such a set which at the same time is typical according to Theorem 2 . Thus MA < n/2 log n . For A' we take the aforementioned n 3/4 exceptions together with the nearest multiples of K - [n'/ 2 ] to the other members . Thus A' is very near to A and yet, as previously indicated, m A ' < 1 ...
HILBERT`S TENTH PROBLEM: What can we do with Diophantine
... It is clear that any solution of the latter equation in integers yields a solution of the former equation in natural numbers. Conversely, every solution of (6) in natural numbers x , . . . , xm can be obtained in this way from some solution of equation (7) in integers w , . . . , zm because by Lagra ...
... It is clear that any solution of the latter equation in integers yields a solution of the former equation in natural numbers. Conversely, every solution of (6) in natural numbers x , . . . , xm can be obtained in this way from some solution of equation (7) in integers w , . . . , zm because by Lagra ...
On the expansions of a real number to several integer bases Yann
... of a given irrational number in two multiplicatively independent bases. We address the following question. Problem 1. Are there irrational real numbers having a ‘simple’ expansion in two multiplicatively independent bases? We have to explain what we mean by ‘simple’, since there are several ways to ...
... of a given irrational number in two multiplicatively independent bases. We address the following question. Problem 1. Are there irrational real numbers having a ‘simple’ expansion in two multiplicatively independent bases? We have to explain what we mean by ‘simple’, since there are several ways to ...
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"".