On smooth integers in short intervals under the Riemann Hypothesis
... 1. Introduction. We say a natural number n is y-smooth if every prime factor p of n satisfies p ≤ y. Let Ψ (x, y) denote the number of y-smooth integers up to x. The function Ψ (x, y) is of great interest in number theory and has been studied by many researchers. Let Ψ (x, z, y) = Ψ (x + z, y) − Ψ ( ...
... 1. Introduction. We say a natural number n is y-smooth if every prime factor p of n satisfies p ≤ y. Let Ψ (x, y) denote the number of y-smooth integers up to x. The function Ψ (x, y) is of great interest in number theory and has been studied by many researchers. Let Ψ (x, z, y) = Ψ (x + z, y) − Ψ ( ...
http://waikato.researchgateway.ac.nz/ Research Commons at the
... In 1548-1626, Cataldi [28] proved that all perfect numbers given by Euclid’s form end in 6 or 8. In 1603, Cataldi [78] found and listed all primes ≤ 750, then proved that 217 −1 = 131071 is a prime because 131071 < 562500 = 7502 , and he could check the number with his list of primes (≤ 750) to show ...
... In 1548-1626, Cataldi [28] proved that all perfect numbers given by Euclid’s form end in 6 or 8. In 1603, Cataldi [78] found and listed all primes ≤ 750, then proved that 217 −1 = 131071 is a prime because 131071 < 562500 = 7502 , and he could check the number with his list of primes (≤ 750) to show ...
13(4)
... The class of solutions determined by 4 + 2^JB is the same as the class determined by 4 - 2\/5, i.e.,the class Is a m biguous, in the terminology of [ 5 ] . Hence all solutions are given by u-x + v-rJ5 = (-1 + y/5)(9 + 4y/5)'', ...
... The class of solutions determined by 4 + 2^JB is the same as the class determined by 4 - 2\/5, i.e.,the class Is a m biguous, in the terminology of [ 5 ] . Hence all solutions are given by u-x + v-rJ5 = (-1 + y/5)(9 + 4y/5)'', ...
Values of the Carmichael Function Equal to a Sum of Two Squares
... holds with some absolute constant c1 > 0 for all sufficiently large values of x . Our proof of the upper bound of Theorem 1 (see Section 4) uses ideas from [1], where similar bounds have been obtained for the Euler function ϕ(n) and for the sum of divisors function σ(n). One difference in our case is t ...
... holds with some absolute constant c1 > 0 for all sufficiently large values of x . Our proof of the upper bound of Theorem 1 (see Section 4) uses ideas from [1], where similar bounds have been obtained for the Euler function ϕ(n) and for the sum of divisors function σ(n). One difference in our case is t ...
Introduction to Programming Languages and Compilers
... Now suppose Lemma 1 is true for n. • That is, 9 b’ such that |h(b’)-a2| < e, |h(g(b’))-a3| < e ... |h(g(n-1)(b’))-a3| < e . Hence 9 b>0 such that |h(b)-a1|< e and g(b) = b’. Therefore the result holds for n+1. QED • Why are we doing this? We wish to show that any finite collection of n real numbers ...
... Now suppose Lemma 1 is true for n. • That is, 9 b’ such that |h(b’)-a2| < e, |h(g(b’))-a3| < e ... |h(g(n-1)(b’))-a3| < e . Hence 9 b>0 such that |h(b)-a1|< e and g(b) = b’. Therefore the result holds for n+1. QED • Why are we doing this? We wish to show that any finite collection of n real numbers ...
Generalizations of Carmichael numbers I
... [48] showed that every Lehmer number n must be odd and square-free, and that the number of distinct prime factors of n must be greater than 6. However, no Lehmer numbers are known up to date, and computations by Pinch [61] show that any examples must be greater than 1030 . In 1977 Pomerance [64] sho ...
... [48] showed that every Lehmer number n must be odd and square-free, and that the number of distinct prime factors of n must be greater than 6. However, no Lehmer numbers are known up to date, and computations by Pinch [61] show that any examples must be greater than 1030 . In 1977 Pomerance [64] sho ...
Littlewood-Richardson rule
... for some tableau T and U of shapes λ and µ with T ·U = V0 . By construction, entries of T0 are less than the entries of S, so the second tableau in (3.2) is T0 by the product construction of tableau via row-insertion. In (3.3), it can be seen that the second tableau is U0 by using Theorem 3.1. This ...
... for some tableau T and U of shapes λ and µ with T ·U = V0 . By construction, entries of T0 are less than the entries of S, so the second tableau in (3.2) is T0 by the product construction of tableau via row-insertion. In (3.3), it can be seen that the second tableau is U0 by using Theorem 3.1. This ...
Chapter 5: Rational Numbers
... are neither terminating nor repeating, such as the numbers below, are called irrational because they cannot be written as fractions. You will learn more about irrational numbers in Chapter 9. ...
... are neither terminating nor repeating, such as the numbers below, are called irrational because they cannot be written as fractions. You will learn more about irrational numbers in Chapter 9. ...
