Lecture 2 - Thursday June 30th
... Sketch Proof. That (1) is true follows from the fact that (1) holds for n = 0, since {k ∈ N|k ≥ 1} is an inductive set, so it equals N which shows k ≥ 1 for all k ∈ N. Therefore since every integer x > 0 is a natural number and we just showed x ≥ 1, (0, 1) contains no integer. Now we consider (n, n ...
... Sketch Proof. That (1) is true follows from the fact that (1) holds for n = 0, since {k ∈ N|k ≥ 1} is an inductive set, so it equals N which shows k ≥ 1 for all k ∈ N. Therefore since every integer x > 0 is a natural number and we just showed x ≥ 1, (0, 1) contains no integer. Now we consider (n, n ...
PMAT 527/627 Practice Midterm
... 9. Square roots modulo p. Executing the algorithms to compute the square root of an integer a modulo p. 10. Randomized algorithms. The difference between Atlantic City, Monte Carlo and Las Vegas algorithms. Definitions of expected running time and worst case running time. 11. Fermat’s little theorem ...
... 9. Square roots modulo p. Executing the algorithms to compute the square root of an integer a modulo p. 10. Randomized algorithms. The difference between Atlantic City, Monte Carlo and Las Vegas algorithms. Definitions of expected running time and worst case running time. 11. Fermat’s little theorem ...
Name: Math 111 - Midterm 1 Review Problems
... for a Tetrahedron? What about for an Octahedron, or a Dodecahedron? 6. Describe in words how to tell if a number is divisible by 300. 7. It is a fact that 17 × 24 = 408. Use this information to help find 413 mod 24 without a calculator. 8. If a 1/2 scale model of a tank holds 10 gallons, how many ga ...
... for a Tetrahedron? What about for an Octahedron, or a Dodecahedron? 6. Describe in words how to tell if a number is divisible by 300. 7. It is a fact that 17 × 24 = 408. Use this information to help find 413 mod 24 without a calculator. 8. If a 1/2 scale model of a tank holds 10 gallons, how many ga ...
lecture1.5
... Disproving something: counterexamples If we are asked to show that a proposition is False, then we just need to provide one counter-example for which the proposition is False In other words, to show that x P(x) is False, we can just show x P(x) = x P(x) to be True Example: “Every positive inte ...
... Disproving something: counterexamples If we are asked to show that a proposition is False, then we just need to provide one counter-example for which the proposition is False In other words, to show that x P(x) is False, we can just show x P(x) = x P(x) to be True Example: “Every positive inte ...
Modular forms and Diophantine questions
... when Richard Taylor and Andrew Wiles released a modified version of the proof that circumvented the gap. The new proof was divided into two articles, one by Wiles alone and one a collaboration by Taylor and Wiles [32], [29]. The two articles were published together in 1995. The proof presented in th ...
... when Richard Taylor and Andrew Wiles released a modified version of the proof that circumvented the gap. The new proof was divided into two articles, one by Wiles alone and one a collaboration by Taylor and Wiles [32], [29]. The two articles were published together in 1995. The proof presented in th ...
Non-congruent numbers, odd graphs and the Birch–Swinnerton
... for Q( D) is defined as the simple directed graph with vertices {D1 , . . . , Dt } ...
... for Q( D) is defined as the simple directed graph with vertices {D1 , . . . , Dt } ...
Mathematical Proof - College of the Siskiyous | Home
... Mathematical Proof If you find it easy to follow the proof, the “helpful” parts of the writing may be unnecessary for you, even distracting. But when the proof is challenging for you to understand, it helps to know: 1) The goal of the proof, 2) The logical “flow” of the proof, and 3) The reasons th ...
... Mathematical Proof If you find it easy to follow the proof, the “helpful” parts of the writing may be unnecessary for you, even distracting. But when the proof is challenging for you to understand, it helps to know: 1) The goal of the proof, 2) The logical “flow” of the proof, and 3) The reasons th ...
On the number of parts of integer partitions lying in given residue
... Tbr,N (n) − TbN −r,N (n), which turns out to be given by the quotient of an explicit weight 1 Eisenstein series for the principal congruence subgroup Γ(N ) by the Dedekind eta function. The generating function of the individual terms Tbr,N (n) however has absolutely no modularity properties so that ...
... Tbr,N (n) − TbN −r,N (n), which turns out to be given by the quotient of an explicit weight 1 Eisenstein series for the principal congruence subgroup Γ(N ) by the Dedekind eta function. The generating function of the individual terms Tbr,N (n) however has absolutely no modularity properties so that ...
SUMS OF DISTINCT UNIT FRACTIONS
... since bm„+t= dji+t, and the ¿'s are greater than the corresponding ö"s, which in turn are greater than the a's. By (8) the y's do not change the situation. Their number is at most C22xtz. But by (2) there are at least 2k'/k] > 2x~1/x\ ...
... since bm„+t= dji+t, and the ¿'s are greater than the corresponding ö"s, which in turn are greater than the a's. By (8) the y's do not change the situation. Their number is at most C22xtz. But by (2) there are at least 2k'/k] > 2x~1/x\ ...
Assignment 9 (for submission in the week beginning 5
... (b) Does every polynomial of degree greater 2 have a fixed point? Give reasons for your answer. 7 (Project) Let r be an irrational real number greater than 1. The Beatty sequence Br is the sequence of natural √ numbers whose nth term is bnrc. For example, if r = 2 = 1.41421 . . ., then the terms of ...
... (b) Does every polynomial of degree greater 2 have a fixed point? Give reasons for your answer. 7 (Project) Let r be an irrational real number greater than 1. The Beatty sequence Br is the sequence of natural √ numbers whose nth term is bnrc. For example, if r = 2 = 1.41421 . . ., then the terms of ...
THE BINOMIAL THEOREM FOR HYPERCOMPLEX NUMBERS
... for any vector x ∈ Cn . For this reason vectors are also called hypercomplex numbers. The theory of complex variables is based on considerations of z m . Leutwiler has in [5], [6], [7] and [8] generalized the Cauchy–Riemann equations to Cn with xm as one of the main solutions. But since Cn is not co ...
... for any vector x ∈ Cn . For this reason vectors are also called hypercomplex numbers. The theory of complex variables is based on considerations of z m . Leutwiler has in [5], [6], [7] and [8] generalized the Cauchy–Riemann equations to Cn with xm as one of the main solutions. But since Cn is not co ...
Cryptography issues – elliptic curves
... Why do we need it ? Who uses cryptographic services ? How is it done classically ? Elliptic curves How is it done using elliptic curves ? Summary ...
... Why do we need it ? Who uses cryptographic services ? How is it done classically ? Elliptic curves How is it done using elliptic curves ? Summary ...
3. CATALAN NUMBERS Corollary 1. cn = 1
... Proposition 2. The number of triangulations of a convex (n + 2)-gon by diagonals is cn . To count triangulations correctly one should have the vertices of the polygon numbered. For example, for a square with the vertices 1, 2, 3, 4 there are c2 = 2 triangulations: a diagonal 1; 3 or a diagonal 2; 4. ...
... Proposition 2. The number of triangulations of a convex (n + 2)-gon by diagonals is cn . To count triangulations correctly one should have the vertices of the polygon numbered. For example, for a square with the vertices 1, 2, 3, 4 there are c2 = 2 triangulations: a diagonal 1; 3 or a diagonal 2; 4. ...
Full text
... The subtraction function x - y or the sgn(^r) function can now be used to obtain a characteristic function for the primes. A characteristic function for a set is a two-valued function taking value 1 on the set and value 0 on the complement of the set. The proper subtraction function x - y is defined ...
... The subtraction function x - y or the sgn(^r) function can now be used to obtain a characteristic function for the primes. A characteristic function for a set is a two-valued function taking value 1 on the set and value 0 on the complement of the set. The proper subtraction function x - y is defined ...
ADDING AND COUNTING Definition 0.1. A partition of a natural
... BY JOEY KRAISLER AND CHARLES WALKER Abstract. This paper is a summary of a talk on partition numbers given by Ken Ono during the MASS Colloquium on November 7, 2013. We would like to thank Professor Ono for providing the lecture slides of which this summary is based. ...
... BY JOEY KRAISLER AND CHARLES WALKER Abstract. This paper is a summary of a talk on partition numbers given by Ken Ono during the MASS Colloquium on November 7, 2013. We would like to thank Professor Ono for providing the lecture slides of which this summary is based. ...
Fermat`s two square theorem for rationals
... he attributes the theorem (without reference) to Fermat, who didn’t have Hasse-Minkowski available. So my question is, how did Fermat prove this theorem? And part 2 of the question is, what is the simplest direct proof? I googled for this result and found a manuscript with a proof that doesn’t use H ...
... he attributes the theorem (without reference) to Fermat, who didn’t have Hasse-Minkowski available. So my question is, how did Fermat prove this theorem? And part 2 of the question is, what is the simplest direct proof? I googled for this result and found a manuscript with a proof that doesn’t use H ...
series with non-zero central critical value
... the work of Kolyvagin [13], as supplemented by the work of Murty and Murty [17] or that of Bump, Friedberg and Hoffstein [3] (see also [10] for a shorter proof), that if E is a modular elliptic curve and, if L(E, 1) 6= 0, then the rank of E is 0. Thus, if f has the property that a positive proportio ...
... the work of Kolyvagin [13], as supplemented by the work of Murty and Murty [17] or that of Bump, Friedberg and Hoffstein [3] (see also [10] for a shorter proof), that if E is a modular elliptic curve and, if L(E, 1) 6= 0, then the rank of E is 0. Thus, if f has the property that a positive proportio ...
The pigeonhole principle
... the N, S, W, E. How many trails must be traversed before some room is visited more than once? ...
... the N, S, W, E. How many trails must be traversed before some room is visited more than once? ...