THE SUPER CATALAN NUMBERS S(m, m + s) FOR s ≤... SOME INTEGER FACTORIAL RATIOS
... (3) We draw a non-self-intersecting clockwise closed contour (following the order of B(p, q)) for each block of the partition such that the region enclosed stays in the annulus. (4) Regions enclosed by different contours are mutually disjoint. In other words, different blocks of the partition do not ...
... (3) We draw a non-self-intersecting clockwise closed contour (following the order of B(p, q)) for each block of the partition such that the region enclosed stays in the annulus. (4) Regions enclosed by different contours are mutually disjoint. In other words, different blocks of the partition do not ...
A Friendly Introduction to My Thesis
... Incumbent in this problem are two actual questions: 1.) Does such an n exist regardless of the power k that I’ve chosen? 2.) For a given power k, what is n? Question 1 was answered in the affirmative by David Hilbert5 in 1908. For the second question, however, Hilbert’s method was ineffective; for i ...
... Incumbent in this problem are two actual questions: 1.) Does such an n exist regardless of the power k that I’ve chosen? 2.) For a given power k, what is n? Question 1 was answered in the affirmative by David Hilbert5 in 1908. For the second question, however, Hilbert’s method was ineffective; for i ...
n - Washington University in St. Louis
... The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p That is to find i such that b = ai (mod p) This is written as i = dloga b (mod p) If a is a primitive root then it always exists, otherwise it may not, e.g., x = log3 4 mod 13 has no answer x = log2 3 mod 13 ...
... The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p That is to find i such that b = ai (mod p) This is written as i = dloga b (mod p) If a is a primitive root then it always exists, otherwise it may not, e.g., x = log3 4 mod 13 has no answer x = log2 3 mod 13 ...
Numbers: Rational and Irrational
... Liouville, Appendix C by the method of Cantor. The techniques are markedly different and the reader will be well rewarded if he follows each. The proof in Chapter 7 is laden with unavoidable technical de tails ; and, even more than in the earlier chapters, the reader will have to use pencil and pap ...
... Liouville, Appendix C by the method of Cantor. The techniques are markedly different and the reader will be well rewarded if he follows each. The proof in Chapter 7 is laden with unavoidable technical de tails ; and, even more than in the earlier chapters, the reader will have to use pencil and pap ...
Chapter 10 Number Theory and Cryptography
... The set Zn is also called the set of residues modulo n, because if b = a mod n, b is sometimes called the residue of a modulo n. Modular arithmetic in Zn , where operations on the elements of Zn are performed mod n, exhibits properties similar to those of traditional arithmetic, such as the associat ...
... The set Zn is also called the set of residues modulo n, because if b = a mod n, b is sometimes called the residue of a modulo n. Modular arithmetic in Zn , where operations on the elements of Zn are performed mod n, exhibits properties similar to those of traditional arithmetic, such as the associat ...
Full text
... any two summands of a decomposition (i) cannot be members of the same bin and (ii) must be at least s bins away from each other. We call this the (s, b)-Generacci sequence (see Definition 5.2) and the Fibonacci numbers are the (1, 1)-Generacci sequence. In this paper we consider the case s = 1, b = ...
... any two summands of a decomposition (i) cannot be members of the same bin and (ii) must be at least s bins away from each other. We call this the (s, b)-Generacci sequence (see Definition 5.2) and the Fibonacci numbers are the (1, 1)-Generacci sequence. In this paper we consider the case s = 1, b = ...
32(2)
... 54* - (01)4. In the sequel, in the proofs of Lemma 5 and Theorem 1, certain closed formulas will be given for (/w + 1)* and (m-2)*. The relationships between m* and (m±j)* can be "translated" easily into well-known identities. For example, the assertion that, if m - (10)^ 1 for some k>0, then (m +1) ...
... 54* - (01)4. In the sequel, in the proofs of Lemma 5 and Theorem 1, certain closed formulas will be given for (/w + 1)* and (m-2)*. The relationships between m* and (m±j)* can be "translated" easily into well-known identities. For example, the assertion that, if m - (10)^ 1 for some k>0, then (m +1) ...
39(1)
... fashion. In those eight conferences she was author or coauthor of sixteen papers, at least one at every conference, finishing on a high note at her last conference in Rochester with three papers. This previous and very special lady, whose humility was so natural that we took it for granted, loved ma ...
... fashion. In those eight conferences she was author or coauthor of sixteen papers, at least one at every conference, finishing on a high note at her last conference in Rochester with three papers. This previous and very special lady, whose humility was so natural that we took it for granted, loved ma ...
Could Euler have conjectured the prime number theorem?
... x + 1 is composite and 1 if x + 1 is prime. But the right side is some real number between 0 and 1, which does not “know” anything about prime numbers; rather, it’s decaying smoothly. So (6) is nonsense, but, to quote Gilbert and Sullivan [GS10], “oh, what precious nonsense!” Modern mathematicians u ...
... x + 1 is composite and 1 if x + 1 is prime. But the right side is some real number between 0 and 1, which does not “know” anything about prime numbers; rather, it’s decaying smoothly. So (6) is nonsense, but, to quote Gilbert and Sullivan [GS10], “oh, what precious nonsense!” Modern mathematicians u ...
22(1)
... Theorem 2.1 (Zeckendorf Theorem for double-ended sequences): Let p > 1 be a positive integer, and let un + 2 = un+1 ...
... Theorem 2.1 (Zeckendorf Theorem for double-ended sequences): Let p > 1 be a positive integer, and let un + 2 = un+1 ...
MATH 289 PROBLEM SET 4
... is an integer, p1 < p2 < · · · < pk are distinct primes and a1 , . . . , ak are nonnegative integers. Give a formula for the number of divisors of A. Exercise 6. *** We start with a deck of 52 cards. We put all the cards in one row, face down. In the first round we turn all the cards around. In the ...
... is an integer, p1 < p2 < · · · < pk are distinct primes and a1 , . . . , ak are nonnegative integers. Give a formula for the number of divisors of A. Exercise 6. *** We start with a deck of 52 cards. We put all the cards in one row, face down. In the first round we turn all the cards around. In the ...
Lectures on Sieve Methods - School of Mathematics, TIFR
... I have tried to overcome in most cases by presenting the simplest approach in details and a sketch of the more sophisticated results if their proof would have required too much time. Nevertheless I have decided to include a chapter an the history of the large sieve upto Bombieri’s first paper, becau ...
... I have tried to overcome in most cases by presenting the simplest approach in details and a sketch of the more sophisticated results if their proof would have required too much time. Nevertheless I have decided to include a chapter an the history of the large sieve upto Bombieri’s first paper, becau ...
The Rabin-Miller Primality Test - University of San Diego Home Pages
... This statement is absolute: There are no exceptions. Unfortunately, the inverse statement is not always true. Inverse to Fermat’s Little Theorem (not always true): If a n − 1 ≡ 1 (mod n) for some a with a ≡/ 0 (mod n), then n is prime. ...
... This statement is absolute: There are no exceptions. Unfortunately, the inverse statement is not always true. Inverse to Fermat’s Little Theorem (not always true): If a n − 1 ≡ 1 (mod n) for some a with a ≡/ 0 (mod n), then n is prime. ...
