![arXiv:math/0510054v2 [math.HO] 17 Aug 2006](http://s1.studyres.com/store/data/014467696_1-4b3b027c0fc1865fd923af6e7da50abe-300x300.png)
arXiv:math/0510054v2 [math.HO] 17 Aug 2006
... s(1 − ππ )(1 − 4ππ )(1 − 9ππ ) etc. before the pentagonal number theorem in this letter, and this is also discussed in the previous two letters in the Euler-Niklaus I Bernoulli correspondence. Euler also notes in this letter that the coefficients of the terms in the series 1 + 1n + 2n2 + 3n3 + 5n4 + ...
... s(1 − ππ )(1 − 4ππ )(1 − 9ππ ) etc. before the pentagonal number theorem in this letter, and this is also discussed in the previous two letters in the Euler-Niklaus I Bernoulli correspondence. Euler also notes in this letter that the coefficients of the terms in the series 1 + 1n + 2n2 + 3n3 + 5n4 + ...
Modular Arithmetic
... Problem 2. How do we generate a random number? There is nothing random about random() or rand() function. The standard technique is to create a pseudorandom sequence. The most commonly used procedure for generating pseudorandom numbers is the linear congruential method (Lehmer, 1949). We generate a ...
... Problem 2. How do we generate a random number? There is nothing random about random() or rand() function. The standard technique is to create a pseudorandom sequence. The most commonly used procedure for generating pseudorandom numbers is the linear congruential method (Lehmer, 1949). We generate a ...
The Goldston-Pintz-Yıldırım sieve and some applications
... distributed among the arithmetic progressions modulo q, for q up to (almost) xθ . The Bombieri-Vinogradov theorem states that θ = 1/2 is an admissible level of distribution for the primes, and the Elliot-Halberstam conjecture goes beyond this to assert that the primes have level of distribution θ = ...
... distributed among the arithmetic progressions modulo q, for q up to (almost) xθ . The Bombieri-Vinogradov theorem states that θ = 1/2 is an admissible level of distribution for the primes, and the Elliot-Halberstam conjecture goes beyond this to assert that the primes have level of distribution θ = ...
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 ...
3 Congruence arithmetic
... remainders) are 0, 1, . . . , n − 1. For example for n = 7 we would like to say that 5 mod 7 + 3 mod 7 = 1. However this is technically incorrect as 5 mod 7 + 3 mod 7 = 5 + 3 = 8 6= 1. In other words, usual integer addition doesn’t make sense on the integers mod n. So one approach is to write equati ...
... remainders) are 0, 1, . . . , n − 1. For example for n = 7 we would like to say that 5 mod 7 + 3 mod 7 = 1. However this is technically incorrect as 5 mod 7 + 3 mod 7 = 5 + 3 = 8 6= 1. In other words, usual integer addition doesn’t make sense on the integers mod n. So one approach is to write equati ...
local copy - Harvard Mathematics Department
... the chance being a solution to both becomes the product of the probabilities for each. To get the probability, start with the full set of integers on some large interval, then look at the probability changes, when primes are successively added to a growing list of primes to be considered. Now, every ...
... the chance being a solution to both becomes the product of the probabilities for each. To get the probability, start with the full set of integers on some large interval, then look at the probability changes, when primes are successively added to a growing list of primes to be considered. Now, every ...
A Few New Facts about the EKG Sequence
... aN the sequence can cross the border of 2p upwards only π(p) times — for every prime smaller than p at most once. Now we will show that for large p, the sequence must have crossed this border downwards Θ(p) times, which is asymptotically bigger than π(p). We will try to be tight on the constants, to ...
... aN the sequence can cross the border of 2p upwards only π(p) times — for every prime smaller than p at most once. Now we will show that for large p, the sequence must have crossed this border downwards Θ(p) times, which is asymptotically bigger than π(p). We will try to be tight on the constants, to ...
Explicit Estimates in the Theory of Prime Numbers
... It is no small thing, the gratitude I have for my supervisor Dr. Tim Trudgian. In fact, a significant effort was required to keep this thesis from becoming a treatise on his brilliance as a mentor. Before I had even started my PhD, Tim had quite the hold on my email inbox, providing me with judiciou ...
... It is no small thing, the gratitude I have for my supervisor Dr. Tim Trudgian. In fact, a significant effort was required to keep this thesis from becoming a treatise on his brilliance as a mentor. Before I had even started my PhD, Tim had quite the hold on my email inbox, providing me with judiciou ...
and let A,B be finitely generated graded S-modules. If T is a
... the ideals I generated by quadrics such that m2 ⊂ I + L for every ideal L generated by n − q − 1 independent linear forms. In Section 7 we study powers of linearly presented ideals. The following conjecture sparked this entire paper: CONJECTURE 1.1 (Eisenbud and Ulrich). If I ⊂ S is a linearly prese ...
... the ideals I generated by quadrics such that m2 ⊂ I + L for every ideal L generated by n − q − 1 independent linear forms. In Section 7 we study powers of linearly presented ideals. The following conjecture sparked this entire paper: CONJECTURE 1.1 (Eisenbud and Ulrich). If I ⊂ S is a linearly prese ...
How to Recognize Whether a Natural Number is a Prime
... no uniform way of predicting, for all primes p, which is the smallest primitive root modulo p. However, several results were known about the size of gp . In 1944, Pillai proved that there exist infinitely many primes p, such that gp > C log log p (where C is a positive constant). In particular, lim s ...
... no uniform way of predicting, for all primes p, which is the smallest primitive root modulo p. However, several results were known about the size of gp . In 1944, Pillai proved that there exist infinitely many primes p, such that gp > C log log p (where C is a positive constant). In particular, lim s ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".