x,y
... Lemma If aN-1 1 mod N for some a relatively prime to N, then it must hold for at least half the choices of a < N Proof Fix some value of a such that aN-1 1 mod N. Suppose b < N Satisfies the test, i.e., bN-1 1 mod N. Then, (a·b)N-1 aN-1·bN-1 aN-1 1 mod N Let S be the set of all b < N tha ...
... Lemma If aN-1 1 mod N for some a relatively prime to N, then it must hold for at least half the choices of a < N Proof Fix some value of a such that aN-1 1 mod N. Suppose b < N Satisfies the test, i.e., bN-1 1 mod N. Then, (a·b)N-1 aN-1·bN-1 aN-1 1 mod N Let S be the set of all b < N tha ...
MATH 3240Q Practice Problems for First Test Mathematicians have
... Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.- Leonhard Euler. Please note: 1. Calculators are not allowed in the exam. 2. You may assume the fo ...
... Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.- Leonhard Euler. Please note: 1. Calculators are not allowed in the exam. 2. You may assume the fo ...
A Transition to Advanced Mathematics
... Excerpts from the Preface to the First Edition “I understand mathematics but I just can’t do proofs.” Our experience has led us to believe that the remark above, though contradictory, expresses the frustration many students feel as they pass from beginning calculus to a more rigorous level of mathem ...
... Excerpts from the Preface to the First Edition “I understand mathematics but I just can’t do proofs.” Our experience has led us to believe that the remark above, though contradictory, expresses the frustration many students feel as they pass from beginning calculus to a more rigorous level of mathem ...
Fast Exponentiation with Precomputation: Algorithms and Lower
... Tables 1 and 2 summarize the effects of the various methods presented above on the storage and complexity of the parameters that might be used for the DSS and Brickell-McCurley schemes, namely 160 and 512 bit exponents respectively. The larger sets of multipliers were found by a computer search. Lar ...
... Tables 1 and 2 summarize the effects of the various methods presented above on the storage and complexity of the parameters that might be used for the DSS and Brickell-McCurley schemes, namely 160 and 512 bit exponents respectively. The larger sets of multipliers were found by a computer search. Lar ...
Lower bound theorems for general polytopes
... A polytope is the convex hull of a finite set. They are encountered as the feasible regions for linear programming problems. The simplex algorithm works by moving from one vertex to another, so it is interesting to bound the lengths of paths in polytopes. This motivated the Hirsch conjecture (1957) ...
... A polytope is the convex hull of a finite set. They are encountered as the feasible regions for linear programming problems. The simplex algorithm works by moving from one vertex to another, so it is interesting to bound the lengths of paths in polytopes. This motivated the Hirsch conjecture (1957) ...
13(4)
... In [ 2 ] , it has been shown that the only numbers which are both triangular, i.e., = ( " ) for some n, and tetrahedral, i.e., = I" ) for some n, are 1, 10, 120, 1540 and 7140. The first two are trivial and the last three were also found by the computer, giving a check on the search procedure. The c ...
... In [ 2 ] , it has been shown that the only numbers which are both triangular, i.e., = ( " ) for some n, and tetrahedral, i.e., = I" ) for some n, are 1, 10, 120, 1540 and 7140. The first two are trivial and the last three were also found by the computer, giving a check on the search procedure. The c ...
arXiv:math/0703236v1 [math.FA] 8 Mar 2007
... norm. Recall that a point P of a compact convex set K is exposed if there is a hyperplane that meets K only in P ; P is extreme if it is not the midpoint of any two other points of K. Theorem 1.2. Let K be the unit ball of the space CΛ and let P ∈ K. (a) The point P is an exposed point of K if and o ...
... norm. Recall that a point P of a compact convex set K is exposed if there is a hyperplane that meets K only in P ; P is extreme if it is not the midpoint of any two other points of K. Theorem 1.2. Let K be the unit ball of the space CΛ and let P ∈ K. (a) The point P is an exposed point of K if and o ...
Integers without large prime factors in short intervals: Conditional
... except that the bound for S(t) will be different. Remark 1. Recently Soundararajan [So10] has improved the result substantially on√RH alone. He proves, on RH, that there are Xα -smooth numbers in intervals of length c(α) X. Remark 2. Our proof shows that the number of Xα -smooth numbers in the inter ...
... except that the bound for S(t) will be different. Remark 1. Recently Soundararajan [So10] has improved the result substantially on√RH alone. He proves, on RH, that there are Xα -smooth numbers in intervals of length c(α) X. Remark 2. Our proof shows that the number of Xα -smooth numbers in the inter ...
full - CS.Duke
... number theorem was proved which gives an asymptotic estimate for the number of primes not exceeding x. Prime Number Theorem: The ratio of the number of primes not exceeding x and x/ln x approaches as x grows without bound. (ln x is the natural logarithm of x) – The theorem tells us that the number o ...
... number theorem was proved which gives an asymptotic estimate for the number of primes not exceeding x. Prime Number Theorem: The ratio of the number of primes not exceeding x and x/ln x approaches as x grows without bound. (ln x is the natural logarithm of x) – The theorem tells us that the number o ...
Notes for Number theory (Fall semester)
... The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by ”number theory”. The word ”arithmetic” (from the Greek, arithmos which means ”number”) is used by the general public to mean ”elementary calculations”; it has also acquired other meanings in mat ...
... The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by ”number theory”. The word ”arithmetic” (from the Greek, arithmos which means ”number”) is used by the general public to mean ”elementary calculations”; it has also acquired other meanings in mat ...
Lecture 13
... within some reasonable long time interval) to reverse a key, which is used for the encryption. • One of the most popular, efficient (and simple!) modern “open key” encryption methods is RSA named for its inventors ...
... within some reasonable long time interval) to reverse a key, which is used for the encryption. • One of the most popular, efficient (and simple!) modern “open key” encryption methods is RSA named for its inventors ...
A Few New Facts about the EKG Sequence
... Theorem 4.2. For any primes p, q > 2 the term qp appears in the sequence after 2p. Proof. We will prove Theorem 4.2 by contradiction. Let us assume that p is the first prime such that for some N we have aN +1 = p and aN = qp, where q > 2. From Lemmas 3.1 and 3.2 we know that these are the first term ...
... Theorem 4.2. For any primes p, q > 2 the term qp appears in the sequence after 2p. Proof. We will prove Theorem 4.2 by contradiction. Let us assume that p is the first prime such that for some N we have aN +1 = p and aN = qp, where q > 2. From Lemmas 3.1 and 3.2 we know that these are the first term ...
Constructibility and the construction of a 17-sided
... those of dividing the circle into n equal arcs and constructing regular polygons[1]. In 1801, Carl Frederich Gauss published his book Disquisitiones Arithmeticae, part of which addressed the problem of dividing the circle[4]. Then in the late 1800s, Felix Klein wrote Famous Problems of Elementary Ge ...
... those of dividing the circle into n equal arcs and constructing regular polygons[1]. In 1801, Carl Frederich Gauss published his book Disquisitiones Arithmeticae, part of which addressed the problem of dividing the circle[4]. Then in the late 1800s, Felix Klein wrote Famous Problems of Elementary Ge ...
HW7 - NYU (Math)
... p = 5 + 8t, we must have 5 + 8t ≡ ±2 mod 5, so −2t ≡ ±2 mod 5, so t = 5s ± 1 and p = 5 + 8(5s ± 1) or p ≡ 13 or − 3 mod 40. Writing p = 3 + 8t, we get 3 + 8t ≡ ±2 mod 5, so −2t ≡ 2 ± 2 mod 5, so t = 5s or t = −2 + 5s. This gives p = 3 + 40s or p = −13 + 40s, so p ≡ 3 or 27 mod 40. Summarizing, the p ...
... p = 5 + 8t, we must have 5 + 8t ≡ ±2 mod 5, so −2t ≡ ±2 mod 5, so t = 5s ± 1 and p = 5 + 8(5s ± 1) or p ≡ 13 or − 3 mod 40. Writing p = 3 + 8t, we get 3 + 8t ≡ ±2 mod 5, so −2t ≡ 2 ± 2 mod 5, so t = 5s or t = −2 + 5s. This gives p = 3 + 40s or p = −13 + 40s, so p ≡ 3 or 27 mod 40. Summarizing, the p ...
Book of Proof - people.vcu.edu
... Until this point in your education, mathematics has probably been presented as a primarily computational discipline. You have learned to solve equations, compute derivatives and integrals, multiply matrices and find determinants; and you have seen how these things can answer practical questions abou ...
... Until this point in your education, mathematics has probably been presented as a primarily computational discipline. You have learned to solve equations, compute derivatives and integrals, multiply matrices and find determinants; and you have seen how these things can answer practical questions abou ...
http://waikato.researchgateway.ac.nz/ Research Commons at the
... the Mersenne number Mp is prime if and only if Mp divides up−1. This can be tested by computing the residue modulo Mp of the un . Today this method is still used. For instance, on September 4, 2006, Curtis Cooper and Steven Boone discovered the 44th known Mersenne prime, 232,582,657 − 1, just nearly ...
... the Mersenne number Mp is prime if and only if Mp divides up−1. This can be tested by computing the residue modulo Mp of the un . Today this method is still used. For instance, on September 4, 2006, Curtis Cooper and Steven Boone discovered the 44th known Mersenne prime, 232,582,657 − 1, just nearly ...