Week 1 - Mathematics and Computer Studies
... When such a cryptosystem is used, the public key is known to everybody. It can be used to encrypt messages. However, only the holder of the private key is able to decrypt such a ciphertext. Nowadays, internet data security is based on such cryptographic techniques. Public key cryptography is not onl ...
... When such a cryptosystem is used, the public key is known to everybody. It can be used to encrypt messages. However, only the holder of the private key is able to decrypt such a ciphertext. Nowadays, internet data security is based on such cryptographic techniques. Public key cryptography is not onl ...
Arithmetic Operations in the Polynomial Modular Number System
... Diffie-Hellman key exchange [6], need fast arithmetic modulo integers of size 1024 to roughly 15000 bits. For the same level of security, elliptic curves defined over prime fields, require operations modulo prime numbers whose size range approximately from 160 to 500 bits [8]. Classic implementation ...
... Diffie-Hellman key exchange [6], need fast arithmetic modulo integers of size 1024 to roughly 15000 bits. For the same level of security, elliptic curves defined over prime fields, require operations modulo prime numbers whose size range approximately from 160 to 500 bits [8]. Classic implementation ...
Full text
... E-mail address: [email protected] Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267 E-mail address: [email protected] Department of Mathematical Sciences, Clemson University, Clemson, SC 29634 E-mail address: [email protected] Department of Mathematics and Statistics ...
... E-mail address: [email protected] Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267 E-mail address: [email protected] Department of Mathematical Sciences, Clemson University, Clemson, SC 29634 E-mail address: [email protected] Department of Mathematics and Statistics ...
Full text
... In one of his famous results, Fermat showed that there exists no Pythagorean triangle with integer sides whose area is an integer square. His elegant method of proof is one of the first known examples in the history of the theory of numbers where the method of infinite descent is employed. Mohanty [ ...
... In one of his famous results, Fermat showed that there exists no Pythagorean triangle with integer sides whose area is an integer square. His elegant method of proof is one of the first known examples in the history of the theory of numbers where the method of infinite descent is employed. Mohanty [ ...
Waring`s problem, taxicab numbers, and other sums of powers
... the interplay of addition and multiplication. One important class of such problems is those in which we ask which numbers can be expressed as sums of some numbers which are defined multiplicatively. Such classes of numbers include nth powers (in this paper, for any fixed n; in a more general treatment ...
... the interplay of addition and multiplication. One important class of such problems is those in which we ask which numbers can be expressed as sums of some numbers which are defined multiplicatively. Such classes of numbers include nth powers (in this paper, for any fixed n; in a more general treatment ...
b - FSU Computer Science
... indicates that the integers a and b fall into the same congruence class modulo n = means that integer a is the reminder of the division of integer b by integer n. Example: 14 2 mod 3 and 2 = 14 mod 3 ...
... indicates that the integers a and b fall into the same congruence class modulo n = means that integer a is the reminder of the division of integer b by integer n. Example: 14 2 mod 3 and 2 = 14 mod 3 ...
Full text
... Furthermore, e^ + e^+i > 0 for each i, which means that there is never a gap greater than one among the Fibonacci numbers constituting any representation, which is evident from the "drip-feed" tree-coloring procedure. Deleting the 1 from each leaf node, in each representation, one obtains integer re ...
... Furthermore, e^ + e^+i > 0 for each i, which means that there is never a gap greater than one among the Fibonacci numbers constituting any representation, which is evident from the "drip-feed" tree-coloring procedure. Deleting the 1 from each leaf node, in each representation, one obtains integer re ...
Author`s preface
... Let us remark that the number m thus constructed is sometimes a prime (2.3.5.7+1=211), and sometimes a composite number (2.3.5.7.11.13+1 = 30 031 = 59 509). When we look into Elements, we will see that Euclid states this theorem in a slightly different form. As Greek mathematicians did not use the n ...
... Let us remark that the number m thus constructed is sometimes a prime (2.3.5.7+1=211), and sometimes a composite number (2.3.5.7.11.13+1 = 30 031 = 59 509). When we look into Elements, we will see that Euclid states this theorem in a slightly different form. As Greek mathematicians did not use the n ...
Brownian Motion and Kolmogorov Complexity
... An infinite binary sequence x is autocomplex if there is a function f : N → N with limn f (n) = ∞, f computable from x, and K (x n) ≥ f (n). A sequence x is Martin-Löf random if x 6∈ ∩n Un for any uniformly Σ01 sequence of open sets Un with µUn ≤ 2−n . A sequence x is Kurtz random if x 6∈ C for a ...
... An infinite binary sequence x is autocomplex if there is a function f : N → N with limn f (n) = ∞, f computable from x, and K (x n) ≥ f (n). A sequence x is Martin-Löf random if x 6∈ ∩n Un for any uniformly Σ01 sequence of open sets Un with µUn ≤ 2−n . A sequence x is Kurtz random if x 6∈ C for a ...
Did I get it right? COS 326 Andrew W. Appel Princeton University
... slides copyright 2013-2015 David Walker and Andrew W. Appel ...
... slides copyright 2013-2015 David Walker and Andrew W. Appel ...
Number Theory Begins - Princeton University Press
... what makes that kind of number special. For square numbers we have lost in modern times that “feel” of the geometric quality that makes them special. Fermat undoubtedly had a much fuller appreciation of both the algebraic and geometric nature of square numbers than we do today. One of the great math ...
... what makes that kind of number special. For square numbers we have lost in modern times that “feel” of the geometric quality that makes them special. Fermat undoubtedly had a much fuller appreciation of both the algebraic and geometric nature of square numbers than we do today. One of the great math ...
Full tex
... If the biggest part is ≥ 2k + 1 take two from the part of it that was not fixed, two from the second biggest part, and so on, until there is a part from which only one (or nothing) can be taken. If there is one, we take it. From the “taken” twos and possible one we make a new part for the new partit ...
... If the biggest part is ≥ 2k + 1 take two from the part of it that was not fixed, two from the second biggest part, and so on, until there is a part from which only one (or nothing) can be taken. If there is one, we take it. From the “taken” twos and possible one we make a new part for the new partit ...
Euclid`s algorithm and multiplicative inverse
... this mean that x is a multiplicative inverse of 13 modulo 18 if and only if x ≡ 7 (mod 18) ? NO. What it means is this: IF x is a multiplicative inverse of 13 modulo 18 THEN x ≡ 7 (mod 18). This argument does NOT in itself prove that x = 7 or x = 25 or x = −11 etc. are solutions. What we proved is t ...
... this mean that x is a multiplicative inverse of 13 modulo 18 if and only if x ≡ 7 (mod 18) ? NO. What it means is this: IF x is a multiplicative inverse of 13 modulo 18 THEN x ≡ 7 (mod 18). This argument does NOT in itself prove that x = 7 or x = 25 or x = −11 etc. are solutions. What we proved is t ...
The Number of Topologies on a Finite Set
... Proof. This proof is shorter than that in Stephen [4]. First, there is a bijective correspondence between the k−ordered partitions (partitions having k blocks) of the set X and the chains of subsets of X having k (non empty and different from X) members: the chain φ 6= A1 $ A2 $ A3 . . . $ Ak $ X, ...
... Proof. This proof is shorter than that in Stephen [4]. First, there is a bijective correspondence between the k−ordered partitions (partitions having k blocks) of the set X and the chains of subsets of X having k (non empty and different from X) members: the chain φ 6= A1 $ A2 $ A3 . . . $ Ak $ X, ...
Lecture 6 (powerpoint): finding a gigantic prime number
... Assume n is Carmichael, for all a, aB = 1 mod n. Property: Carmichael number is the product of distinct prime. Thus, let n = p1p2..pk. Let g’ is a generator of Zp1*. Let a = (g’, 1), i.e., a = g’ (mod p1), a = 1 (mod p2..pr), by CRT By assumption, aB = 1 (mod n). It implies g’B = 1 (mod p1) (why?). ...
... Assume n is Carmichael, for all a, aB = 1 mod n. Property: Carmichael number is the product of distinct prime. Thus, let n = p1p2..pk. Let g’ is a generator of Zp1*. Let a = (g’, 1), i.e., a = g’ (mod p1), a = 1 (mod p2..pr), by CRT By assumption, aB = 1 (mod n). It implies g’B = 1 (mod p1) (why?). ...
Full text
... to Fam (see [ 5 ] , p. 30). So/? is not a divisor of Fam_i because/? is a divisor of Fam and Fr = 0 (mod/?). From this follows a = r by reason of definition of a = a(p). Thus/? is divisible by a = a(p). Should it happen that/? is divisible by a = a(p), then, due to the Vorobev's previous theorem, Fn ...
... to Fam (see [ 5 ] , p. 30). So/? is not a divisor of Fam_i because/? is a divisor of Fam and Fr = 0 (mod/?). From this follows a = r by reason of definition of a = a(p). Thus/? is divisible by a = a(p). Should it happen that/? is divisible by a = a(p), then, due to the Vorobev's previous theorem, Fn ...
THE ARITHMETIC LARGE SIEVE WITH AN APPLICATION TO THE
... . In the 1960’s Burgess gave the estimate np p 4 ...
... . In the 1960’s Burgess gave the estimate np p 4 ...
PERIODIC DECIMAL FRACTIONS A Thesis Presented to the Faculty
... an alternate representation for a terminating decimal fraction by subtracting 1 from the last digit of the decimal represen tation and affixing a repeating sequence of nines at the end. Such an instance would be .3 written as .29. ...
... an alternate representation for a terminating decimal fraction by subtracting 1 from the last digit of the decimal represen tation and affixing a repeating sequence of nines at the end. Such an instance would be .3 written as .29. ...
Structure and randomness in the prime numbers
... Despite not having a good exact formula for the sequence of primes, we do have a fairly good inexact formula: Prime number theorem (Hadamard, de la Vallée Poussin, 1896) pn is approximately equal to n ln n. (More precisely: nplnn n converges to 1 as n → ∞.) ln n is the logarithm of n to the natura ...
... Despite not having a good exact formula for the sequence of primes, we do have a fairly good inexact formula: Prime number theorem (Hadamard, de la Vallée Poussin, 1896) pn is approximately equal to n ln n. (More precisely: nplnn n converges to 1 as n → ∞.) ln n is the logarithm of n to the natura ...
Irrational numbers
... A real number that can be expressed as ba with b 6= 0 is termd as a rational number. A real number that is not rational is said to be an irrational. ...
... A real number that can be expressed as ba with b 6= 0 is termd as a rational number. A real number that is not rational is said to be an irrational. ...