Solutions
... Since PTS = 90o, QTP = TSR, and hence ∆QTP ~ ∆RST. If we let QT = x, then TR = 10 – x, and so x/8 = 3/(10 – x) leading to x2 – 10x + 24 = 0, so (x – 6)(x – 4) = 0 and hence x = 4 or 6. The area of ∆PTS = ½ (8 + 3) × 10 – 4x – ½ × 3 (10 – x) = 55 – 4x – ½ × 3 (10 – x) If x = 4, [PTS] = 55 – 16 – 9 = ...
... Since PTS = 90o, QTP = TSR, and hence ∆QTP ~ ∆RST. If we let QT = x, then TR = 10 – x, and so x/8 = 3/(10 – x) leading to x2 – 10x + 24 = 0, so (x – 6)(x – 4) = 0 and hence x = 4 or 6. The area of ∆PTS = ½ (8 + 3) × 10 – 4x – ½ × 3 (10 – x) = 55 – 4x – ½ × 3 (10 – x) If x = 4, [PTS] = 55 – 16 – 9 = ...
ON LARGE RATIONAL SOLUTIONS OF CUBIC THUE EQUATIONS
... (un , vn ) which increase without bound as n increases without bound. Such cubic equations are birationally equivalent to elliptic curves of the form y 2 = x3 − D. The rational points on an elliptic curve form an abelian group, so a “large” rational point (u, v) maps to a rational point (x, y) of “a ...
... (un , vn ) which increase without bound as n increases without bound. Such cubic equations are birationally equivalent to elliptic curves of the form y 2 = x3 − D. The rational points on an elliptic curve form an abelian group, so a “large” rational point (u, v) maps to a rational point (x, y) of “a ...
Pythagorean triangles with legs less than n
... triangles with hypotenuse less than n are shown in [5,10 –12]. Finally, in [2], we can 6nd estimates for the number of Pythagorean triangles with a common hypotenuse, leg-sum, etc., less than n. However, so far as we know, the problem of estimating the number of Pythagorean triangles with both legs ...
... triangles with hypotenuse less than n are shown in [5,10 –12]. Finally, in [2], we can 6nd estimates for the number of Pythagorean triangles with a common hypotenuse, leg-sum, etc., less than n. However, so far as we know, the problem of estimating the number of Pythagorean triangles with both legs ...
1, N(3)
... largest prime P satisfying K S N - K < P _<_ N. It follows that P divides t and hence that n ? P for all n e S 2 . Hence all of the numbers in S2 lie between P and N . The number of numbers in S2 is thus S Z I <_ N - P <_ P5/8 < N518 <_ (log t) 3/a = O (log t/log log t), where, in obtaining the seco ...
... largest prime P satisfying K S N - K < P _<_ N. It follows that P divides t and hence that n ? P for all n e S 2 . Hence all of the numbers in S2 lie between P and N . The number of numbers in S2 is thus S Z I <_ N - P <_ P5/8 < N518 <_ (log t) 3/a = O (log t/log log t), where, in obtaining the seco ...
Public-Key Crypto Basics Paul Garrett
... Since the 1970’s, better methods have been found (but not polynomial-time): quadratic sieve: the most elementary of modern factorization methods, and still very good by comparison to other methods. Descended from Dixon’s algorithm. elliptic curve sieve: to factor n, this replaces the group Z/n× with ...
... Since the 1970’s, better methods have been found (but not polynomial-time): quadratic sieve: the most elementary of modern factorization methods, and still very good by comparison to other methods. Descended from Dixon’s algorithm. elliptic curve sieve: to factor n, this replaces the group Z/n× with ...
- ESAIM: Proceedings
... By Fatou’s lemma, E(W ) ≤ lim inf n→∞ E(Wn ) = 1. It is, however, possible that W = 0. So it is important to know when W is non-degenerate. We make the trivial remark that W = 0 if the system dies out. In particular, by Theorem 1.1, we have W = 0 a.s. if m ≤ 1. What happens if m > 1? We start with t ...
... By Fatou’s lemma, E(W ) ≤ lim inf n→∞ E(Wn ) = 1. It is, however, possible that W = 0. So it is important to know when W is non-degenerate. We make the trivial remark that W = 0 if the system dies out. In particular, by Theorem 1.1, we have W = 0 a.s. if m ≤ 1. What happens if m > 1? We start with t ...
R The Topology of Chapter 5 5.1
... Definition 5.2.2. A set S ⊂ R is disconnected if there are two open intervals U and V such that U ∩ V = ∅, U ∩ S ̸= ∅, V ∩ S ̸= ∅ and S ⊂ U ∪ V . Otherwise, it is connected. The sets U ∩ S and V ∩ S are said to be a separation of S. In other words, S is disconnected if it can be written as the union ...
... Definition 5.2.2. A set S ⊂ R is disconnected if there are two open intervals U and V such that U ∩ V = ∅, U ∩ S ̸= ∅, V ∩ S ̸= ∅ and S ⊂ U ∪ V . Otherwise, it is connected. The sets U ∩ S and V ∩ S are said to be a separation of S. In other words, S is disconnected if it can be written as the union ...
Primes and Greatest Common Divisors
... Discrete Mathematics & Mathematical Reasoning Primes and Greatest Common Divisors ...
... Discrete Mathematics & Mathematical Reasoning Primes and Greatest Common Divisors ...
![arXiv:1412.5920v1 [math.CO] 18 Dec 2014](http://s1.studyres.com/store/data/007906890_1-968d1291ae5654c6eb06790a1cfb5c04-300x300.png)
powerpoint
... Note on the format for doing calculations. Between each pair of successive formulas, we write “=’’ followed by an indented hint; the hint says what we used in showing that the first formula equals the second. Always put such hints in, because it will help you later in reading your own proof and beca ...
... Note on the format for doing calculations. Between each pair of successive formulas, we write “=’’ followed by an indented hint; the hint says what we used in showing that the first formula equals the second. Always put such hints in, because it will help you later in reading your own proof and beca ...
Chapter 3 - Eric Tuzin Math 4371 Portfolio
... Two integers are said to be relatively prime if the biggest integer that divides both of them is 1. Exercise. List all pairs of integers from 100 to 105 that are relatively prime. * You might find the programs at the end of this chapter to be interesting. 100 and 101, 100 and 103, 101 and 102, 101 a ...
... Two integers are said to be relatively prime if the biggest integer that divides both of them is 1. Exercise. List all pairs of integers from 100 to 105 that are relatively prime. * You might find the programs at the end of this chapter to be interesting. 100 and 101, 100 and 103, 101 and 102, 101 a ...
Note 2/V Noncollinear Points Determine at Least
... first position is arbitrary; it is more natural to continue the rotation both ...
... first position is arbitrary; it is more natural to continue the rotation both ...
Some transcendence results from a harmless irrationality theorem
... problem in pure mathematics to prove the transcendence of naturally occurring numbers.2 As a result, in 1873 Hermite proved that er is transcendental for all rational r 6= 0 (in particular, e is transcendental) [5]. In 1882, Lindemann proved the following extension of Hermite’s result [9]. Lemma 1 ( ...
... problem in pure mathematics to prove the transcendence of naturally occurring numbers.2 As a result, in 1873 Hermite proved that er is transcendental for all rational r 6= 0 (in particular, e is transcendental) [5]. In 1882, Lindemann proved the following extension of Hermite’s result [9]. Lemma 1 ( ...
Sequences of enumerative geometry: congruences and asymptotics
... (instantons, etc) and must satisfy the condition of modularity covariance in order to obtain the same amplitude when two worldsheets have the same intrinsic geometry. Modular forms, as is well known, have Fourier coefficients satisfying many interesting congruences (think of Ramanujan’s congruences ...
... (instantons, etc) and must satisfy the condition of modularity covariance in order to obtain the same amplitude when two worldsheets have the same intrinsic geometry. Modular forms, as is well known, have Fourier coefficients satisfying many interesting congruences (think of Ramanujan’s congruences ...
Presentation on Weierstrass M-Test
... Any sequence that satisfies the Cauchy Criterion is known as a Cauchy sequence. The above theorem also shows that every convergent sequence is Cauchy, and every Cauchy sequence is convergent. COROLLARY 1: If is a Cauchy sequence that converges to Z, and N is chosen such that every such that , then f ...
... Any sequence that satisfies the Cauchy Criterion is known as a Cauchy sequence. The above theorem also shows that every convergent sequence is Cauchy, and every Cauchy sequence is convergent. COROLLARY 1: If is a Cauchy sequence that converges to Z, and N is chosen such that every such that , then f ...
![arXiv:math/0412079v2 [math.NT] 2 Mar 2006](http://s1.studyres.com/store/data/013294887_1-d94fad656ee5fb5bde358f5c8c1d35cf-300x300.png)
arXiv:math/0412079v2 [math.NT] 2 Mar 2006
... to the rules of logic: the discourse of logos. When such a proof is not available, other kinds of arguments can make mathematicians expect that statements will be true, even though these arguments fall well short of proofs. These are called heuristic arguments—the word comes from the Greek root Eurh ...
... to the rules of logic: the discourse of logos. When such a proof is not available, other kinds of arguments can make mathematicians expect that statements will be true, even though these arguments fall well short of proofs. These are called heuristic arguments—the word comes from the Greek root Eurh ...
Periodicity and Correlation Properties of d
... radar systems, signal synchronization, simulation, and cryptography. The pseudorandom sequences in a good family should (a) be easy to generate (possibly with hardware or software), (b) have good distribution properties which make them appear (statistically) to be ‘‘random’’, (c) have low crosscorre ...
... radar systems, signal synchronization, simulation, and cryptography. The pseudorandom sequences in a good family should (a) be easy to generate (possibly with hardware or software), (b) have good distribution properties which make them appear (statistically) to be ‘‘random’’, (c) have low crosscorre ...