Lecture 13 - Basic Number Theory.
... root X, and we want to find X (or −X). We can assume P is odd (if P is the only even prime, namely two, then we can easily solve this problem mod P ). P (mod 4) can be either 1 or 3. We start with the case that P (mod 4) = 3. That is, P = 4T + 3. In this case we claim that AT +1 is a square root of ...
... root X, and we want to find X (or −X). We can assume P is odd (if P is the only even prime, namely two, then we can easily solve this problem mod P ). P (mod 4) can be either 1 or 3. We start with the case that P (mod 4) = 3. That is, P = 4T + 3. In this case we claim that AT +1 is a square root of ...
Hilbert`s Tenth Problem
... In 1900, David Hilbert gave an address at the International Congress of Mathematicians in which he gave a list of 23 problems that would influence the world of mathematics. In this project, we will focus on his tenth problem. In his tenth problem, Hilbert talks about Diophantine equations, and if th ...
... In 1900, David Hilbert gave an address at the International Congress of Mathematicians in which he gave a list of 23 problems that would influence the world of mathematics. In this project, we will focus on his tenth problem. In his tenth problem, Hilbert talks about Diophantine equations, and if th ...
Six more gems that every FP1 teacher should know
... angle of approxirnately45"-but clearly only one of theseanswerscan be dght. Somesimptenumericalwork showsthat the correctsolution is in facr Codetia's, and it is then fairly easy to seethat each of the other methodsdependson the erroneousassumptlonthat the limit or lendency of a quotient is equal to ...
... angle of approxirnately45"-but clearly only one of theseanswerscan be dght. Somesimptenumericalwork showsthat the correctsolution is in facr Codetia's, and it is then fairly easy to seethat each of the other methodsdependson the erroneousassumptlonthat the limit or lendency of a quotient is equal to ...
Math 5330 Spring 2013 Elementary factoring algorithms The RSA
... The RSA cryptosystem is founded on the idea that, in general, factoring is hard. Where as with Fermat’s Little Theorem and some related ideas, one can usually tell very quickly if a composite number is, in fact, composite, actually producing a factorization of a composite number is a very different ...
... The RSA cryptosystem is founded on the idea that, in general, factoring is hard. Where as with Fermat’s Little Theorem and some related ideas, one can usually tell very quickly if a composite number is, in fact, composite, actually producing a factorization of a composite number is a very different ...
Solving a linear equation in a set of integers I
... use induction. The statement is obviously true for n = 1. We establish it for n assuming its validity for n − 1. Suppose that there is aP solution, and let S denote the set of thosePsubscripts for which xi = un . If i∈S ai 6= 0, this contradicts to (2.4). If i∈S ai = 0, then by replacing each occurr ...
... use induction. The statement is obviously true for n = 1. We establish it for n assuming its validity for n − 1. Suppose that there is aP solution, and let S denote the set of thosePsubscripts for which xi = un . If i∈S ai 6= 0, this contradicts to (2.4). If i∈S ai = 0, then by replacing each occurr ...
Full tex
... In recent time there has been much progress made on the problem of determining sufficiency conditions for a positive rational termed series to converge to either an irrational or transcendental number (see [1], [4], [6] and the references cited therein). Surprisingly, in comparison, very little atte ...
... In recent time there has been much progress made on the problem of determining sufficiency conditions for a positive rational termed series to converge to either an irrational or transcendental number (see [1], [4], [6] and the references cited therein). Surprisingly, in comparison, very little atte ...
Bachet`s Equation - Math-Boise State
... Zp , k , |S|, and |S| ∪ IDg Where Zp is the range to which we confine our search k is the specific constant we are looking at |S| is the solution size And |S| ∪ IDg is the solution size including the identity element. ...
... Zp , k , |S|, and |S| ∪ IDg Where Zp is the range to which we confine our search k is the specific constant we are looking at |S| is the solution size And |S| ∪ IDg is the solution size including the identity element. ...
Chapter 4
... We very often encounter binary operations in mathematics, and nearly all of these are associative: addition, multiplication, composition etc. In this chapter we introduce a sufficiently abstract notion to deal with all such operations. 4.1. Definition of a Semigroup 4.1.1 Definition A semigroup is a ...
... We very often encounter binary operations in mathematics, and nearly all of these are associative: addition, multiplication, composition etc. In this chapter we introduce a sufficiently abstract notion to deal with all such operations. 4.1. Definition of a Semigroup 4.1.1 Definition A semigroup is a ...
MATHEMATICAL INDUCTION
... of the axioms was so designed as to incorporate induction as a method of proof. In other words, the intuitive way to deal with induction below is actually a legitimate technique. In what follows, the theory is presented in short sections, each with its own problems. These are rather easy especially ...
... of the axioms was so designed as to incorporate induction as a method of proof. In other words, the intuitive way to deal with induction below is actually a legitimate technique. In what follows, the theory is presented in short sections, each with its own problems. These are rather easy especially ...
135. Some results on 4-cycle packings, Ars Combin. 93, 2009, 15-23.
... to find an integer a in a sequence of integers A such that for each b ∈ A and b = a, the greatest common divisor of a and b is 1. For convenience, in what follows, we say a and b are coprime and a is a relprime of A. A very fundamental argument can show that for any sequence of at most 16 consecuti ...
... to find an integer a in a sequence of integers A such that for each b ∈ A and b = a, the greatest common divisor of a and b is 1. For convenience, in what follows, we say a and b are coprime and a is a relprime of A. A very fundamental argument can show that for any sequence of at most 16 consecuti ...
PARTITION STATISTICS EQUIDISTRIBUTED WITH THE NUMBER OF HOOK DIFFERENCE ONE CELLS
... Conjecture 1.6. The statistics h1,1 and a2 are equidistributed on the set of partitions of n with 2-core {k, k 1, · · · , 1} for all non-negative integers n and k. This refines Theorem 1.1. We propose a generalization of this as Conjecture 6.3, and prove an analogous special case in Theorem 6.5. 1.3 ...
... Conjecture 1.6. The statistics h1,1 and a2 are equidistributed on the set of partitions of n with 2-core {k, k 1, · · · , 1} for all non-negative integers n and k. This refines Theorem 1.1. We propose a generalization of this as Conjecture 6.3, and prove an analogous special case in Theorem 6.5. 1.3 ...
For printing - Mathematical Sciences Publishers
... of distinct prime factors of n. Let I be an integer larger than one and let 6 be a positive real number. Let 2 = Pi 5 P2,..- be the sequence of prime numbers in increasing order and let m be that positive integer for which p x -pm < N < p\ p m +i. In [3], Erdόs, Pomerance, Sarkozy and Stewart proved ...
... of distinct prime factors of n. Let I be an integer larger than one and let 6 be a positive real number. Let 2 = Pi 5 P2,..- be the sequence of prime numbers in increasing order and let m be that positive integer for which p x -pm < N < p\ p m +i. In [3], Erdόs, Pomerance, Sarkozy and Stewart proved ...
Modified Stern-Brocot Sequences
... continued fractions could be used for the same purpose [5], which sparked an interest in the connection between the two. Indeed, it was later discovered that the mediant could also be expressed as an operation on the continued fraction expansion of two fractions, whose continued fractions were alrea ...
... continued fractions could be used for the same purpose [5], which sparked an interest in the connection between the two. Indeed, it was later discovered that the mediant could also be expressed as an operation on the continued fraction expansion of two fractions, whose continued fractions were alrea ...
PRIMES OF THE FORM x2 + ny 2 AND THE GEOMETRY OF
... 1This definition can be shown to be equivalent to the definitions of Euler and Gauss. ...
... 1This definition can be shown to be equivalent to the definitions of Euler and Gauss. ...
THE CHINESE REMAINDER THEOREM CLOCK FIGURE 1. The
... the periodic nature of remainders. Moreover, some small arithmetical problems may be visualized on the Clock dial (see Section 4). Last but not least the Clock can be entertaining (also for non-mathematicians): anybody who simply wants to play around with numbers may have found something for him. Fu ...
... the periodic nature of remainders. Moreover, some small arithmetical problems may be visualized on the Clock dial (see Section 4). Last but not least the Clock can be entertaining (also for non-mathematicians): anybody who simply wants to play around with numbers may have found something for him. Fu ...